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Seminars (FRB)

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Event When Speaker Title Presentation Material
FRBW01 6th January 2014
10:00 to 11:00
Derivation of FBs for tumor growth - 1
Reaction-Diffusion systems - The Stefan problem - Latent heat

When used for biology and medicine, PDEs have to be used with care. Even though some are very classical, as front propagation for invading species, they are always questioned by comparison to observations or experiments. This course aims at showing some examples of free boundary problems motivated by biology and medicine, to concentrate on weak solutions, and to discuss their limitations and the need for further developments.

Suggested reading

[1] M. Belhadj, J.-F. Gerbeau and B. Perthame. A multiscale transport model of colloids with degenerate anisotropic diffusion. Asymptotic Analysis 34(1) (2003) 41--54.

[2] G. Barles and P.E. Souganidis. Front propagation for reaction-diffusion equations arising in combustion theory. Asymptot. Anal., 14:277­292, 1997.

[3] A. Lorz, B. Perthame, P. Markowich, Bernoulli variational problem and beyond. To appear in ARMA (2014). http://hal.upmc.fr/hal-00881760

[4] B. Perthame, F. Quiros, J.-L. Vazquez, The Hele-Shaw asymptotics for mechanical models of tumor growth. To appear in ARMA (2014) http://hal.upmc.fr/hal-00831932

[5] Emeric Bouin, Vincent Calvez, Nicolas Meunier, Sepideh Mirrahimi, Benoît Perthame, Gael Raoul, and Raphaël Voituriez. Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration. C. R. Math. Acad. Sci. Paris, 350(15-16):761­766, 2012.

FRBW01 6th January 2014
11:30 to 12:30
Numerical Methods for FBPs - 1
This tutorial is a tour from classical techniques to recent developments of numerical methods for free boundary problems. The emphasis is on ideas and methods rather than problems.

Lecture 1: Variational Inequalities I

The classical obstacle problem. A priori error analysis in energy and maximum norm. The thin obstacle problem. The fractional obstacle problem.

Lecture 2: Variational Inequalities II

A priori rate of convergence for free boundaries. A posteriori error analysis in the maximum norm. A posteriori barrier sets.

Lecture 3: Discrete Gradient Flows

Evolution PDE: energy solutions, convexity, coercivity. Error analysis of time discretization. Error analysis of space discretization. Applications: parabolic variational inequalities, degenerate parabolic PDE, TV gradient flow.

Lecture 4: Geometric Problems

Shape differential calculus: examples. Geometric gradient flows: mean curvature, surface diffusion, Willmore flow. Parametric approach. Phase field approach. Level set approach.

FRBW01 6th January 2014
13:30 to 14:30
Derivation of FBs for tumor growth - 2
The example of invasion fronts - The approach based on Hamilton-Jacobi equations - Accelerating fronts - The tail problem

When used for biology and medicine, PDEs have to be used with care. Even though some are very classical, as front propagation for invading species, they are always questioned by comparison to observations or experiments. This course aims at showing some examples of free boundary problems motivated by biology and medicine, to concentrate on weak solutions, and to discuss their limitations and the need for further developments.

Suggested reading

[1] M. Belhadj, J.-F. Gerbeau and B. Perthame. A multiscale transport model of colloids with degenerate anisotropic diffusion. Asymptotic Analysis 34(1) (2003) 41--54.

[2] G. Barles and P.E. Souganidis. Front propagation for reaction-diffusion equations arising in combustion theory. Asymptot. Anal., 14:277­292, 1997.

[3] A. Lorz, B. Perthame, P. Markowich, Bernoulli variational problem and beyond. To appear in ARMA (2014). http://hal.upmc.fr/hal-00881760

[4] B. Perthame, F. Quiros, J.-L. Vazquez, The Hele-Shaw asymptotics for mechanical models of tumor growth. To appear in ARMA (2014) http://hal.upmc.fr/hal-00831932

[5] Emeric Bouin, Vincent Calvez, Nicolas Meunier, Sepideh Mirrahimi, Benoît Perthame, Gael Raoul, and Raphaël Voituriez. Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration. C. R. Math. Acad. Sci. Paris, 350(15-16):761­766, 2012.

FRBW01 6th January 2014
15:00 to 16:00
Numerical Methods for FBPs - 2
This tutorial is a tour from classical techniques to recent developments of numerical methods for free boundary problems. The emphasis is on ideas and methods rather than problems.

Lecture 1: Variational Inequalities I

The classical obstacle problem. A priori error analysis in energy and maximum norm. The thin obstacle problem. The fractional obstacle problem.

Lecture 2: Variational Inequalities II

A priori rate of convergence for free boundaries. A posteriori error analysis in the maximum norm. A posteriori barrier sets.

Lecture 3: Discrete Gradient Flows

Evolution PDE: energy solutions, convexity, coercivity. Error analysis of time discretization. Error analysis of space discretization. Applications: parabolic variational inequalities, degenerate parabolic PDE, TV gradient flow.

Lecture 4: Geometric Problems

Shape differential calculus: examples. Geometric gradient flows: mean curvature, surface diffusion, Willmore flow. Parametric approach. Phase field approach. Level set approach.

FRBW01 6th January 2014
16:00 to 17:00
Perspectives on Free Boundaries
FRBW01 7th January 2014
10:00 to 11:00
Derivation of FBs for tumor growth - 3
Mechanical models of tumor growth - boundary stability and instability - variants

When used for biology and medicine, PDEs have to be used with care. Even though some are very classical, as front propagation for invading species, they are always questioned by comparison to observations or experiments. This course aims at showing some examples of free boundary problems motivated by biology and medicine, to concentrate on weak solutions, and to discuss their limitations and the need for further developments.

Suggested reading

[1] M. Belhadj, J.-F. Gerbeau and B. Perthame. A multiscale transport model of colloids with degenerate anisotropic diffusion. Asymptotic Analysis 34(1) (2003) 41--54.

[2] G. Barles and P.E. Souganidis. Front propagation for reaction-diffusion equations arising in combustion theory. Asymptot. Anal., 14:277­292, 1997.

[3] A. Lorz, B. Perthame, P. Markowich, Bernoulli variational problem and beyond. To appear in ARMA (2014). http://hal.upmc.fr/hal-00881760

[4] B. Perthame, F. Quiros, J.-L. Vazquez, The Hele-Shaw asymptotics for mechanical models of tumor growth. To appear in ARMA (2014) http://hal.upmc.fr/hal-00831932

[5] Emeric Bouin, Vincent Calvez, Nicolas Meunier, Sepideh Mirrahimi, Benoît Perthame, Gael Raoul, and Raphaël Voituriez. Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration. C. R. Math. Acad. Sci. Paris, 350(15-16):761­766, 2012.

FRBW01 7th January 2014
11:30 to 12:30
Numerical Methods for FBPs - 3
This tutorial is a tour from classical techniques to recent developments of numerical methods for free boundary problems. The emphasis is on ideas and methods rather than problems.

Lecture 1: Variational Inequalities I

The classical obstacle problem. A priori error analysis in energy and maximum norm. The thin obstacle problem. The fractional obstacle problem.

Lecture 2: Variational Inequalities II

A priori rate of convergence for free boundaries. A posteriori error analysis in the maximum norm. A posteriori barrier sets.

Lecture 3: Discrete Gradient Flows

Evolution PDE: energy solutions, convexity, coercivity. Error analysis of time discretization. Error analysis of space discretization. Applications: parabolic variational inequalities, degenerate parabolic PDE, TV gradient flow.

Lecture 4: Geometric Problems

Shape differential calculus: examples. Geometric gradient flows: mean curvature, surface diffusion, Willmore flow. Parametric approach. Phase field approach. Level set approach.

FRBW01 7th January 2014
13:30 to 14:30
E Varvaruca Geometric approaches to water waves and free surface flows - 1
These lectures aim to present a new geometric approach to the asymptotic behaviour near singularities in some classical free-boundary problems in fluid dynamics. We start by introducing the problems and providing an outline of the methods that have been used to prove existence of solutions. We then present a modern proof, using monotonicity formulas and frequency formulas, of the famous Stokes conjecture from 1880, which asserts that at any stagnation point on the free surface of a two-dimensional steady irrotational gravity water wave, the wave profile necessarily has lateral tangents enclosing a symmetric angle of 120 degrees. (This result was first proved in the 1980s under restrictive assumptions and by somewhat ad-hoc methods.) We then explain how the methods extend to the case of two-dimensional steady gravity water waves with vorticity. Finally, we show how the same methods can be adapted to describe the asymptotic behaviour near singularities in the problem of steady three-dimensional axisymmetric free surface flows with gravity.

References:

[1] Buffoni, B.; Toland, J. F. Analytic theory of global bifurcation. An introduction. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, N.J., 2003.

[2] Constantin, A. Nonlinear water waves with applications to wave-current interactions and tsunamis. CBMS-NSF Regional Conference Series in Applied Mathematics, 81. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, P.A., 2011.

[3] Constantin, A.; Strauss, W. Exact steady periodic water waves with vorticity. Comm. Pure Appl. Math. 57 (2004), no. 4, 481--527.

[4] Varvaruca, E. On the existence of extreme waves and the Stokes conjecture with vorticity. J. Differential Equations 246 (2009), no. 10, 4043--4076.

[5] Varvaruca, E.; Weiss, G. S. A geometric approach to generalized Stokes conjectures. Acta Math. 206 (2011), no. 2, 363--403.

[6] Varvaruca, E.; Weiss, G. S. The Stokes conjecture for waves with vorticity. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), no. 6, 861--885.

[7] Varvaruca, E.; Weiss, G. S. Singularities of steady axisymmetric free surface flows with gravity, to appear in Comm. Pure Appl. Math., http://arxiv.org/abs/1210.3682.

FRBW01 7th January 2014
15:00 to 16:00
Numerical Methods for FBPs - 4
This tutorial is a tour from classical techniques to recent developments of numerical methods for free boundary problems. The emphasis is on ideas and methods rather than problems.

Lecture 1: Variational Inequalities I

The classical obstacle problem. A priori error analysis in energy and maximum norm. The thin obstacle problem. The fractional obstacle problem.

Lecture 2: Variational Inequalities II

A priori rate of convergence for free boundaries. A posteriori error analysis in the maximum norm. A posteriori barrier sets.

Lecture 3: Discrete Gradient Flows

Evolution PDE: energy solutions, convexity, coercivity. Error analysis of time discretization. Error analysis of space discretization. Applications: parabolic variational inequalities, degenerate parabolic PDE, TV gradient flow.

Lecture 4: Geometric Problems

Shape differential calculus: examples. Geometric gradient flows: mean curvature, surface diffusion, Willmore flow. Parametric approach. Phase field approach. Level set approach.

FRBW01 8th January 2014
10:00 to 11:00
Derivation of FBs for tumor growth - 4
The limit of stiff pressure law - Weak solutions of the Hele-Shaw free boundary problem

When used for biology and medicine, PDEs have to be used with care. Even though some are very classical, as front propagation for invading species, they are always questioned by comparison to observations or experiments. This course aims at showing some examples of free boundary problems motivated by biology and medicine, to concentrate on weak solutions, and to discuss their limitations and the need for further developments.

Suggested reading

[1] M. Belhadj, J.-F. Gerbeau and B. Perthame. A multiscale transport model of colloids with degenerate anisotropic diffusion. Asymptotic Analysis 34(1) (2003) 41--54.

[2] G. Barles and P.E. Souganidis. Front propagation for reaction-diffusion equations arising in combustion theory. Asymptot. Anal., 14:277­292, 1997.

[3] A. Lorz, B. Perthame, P. Markowich, Bernoulli variational problem and beyond. To appear in ARMA (2014). http://hal.upmc.fr/hal-00881760

[4] B. Perthame, F. Quiros, J.-L. Vazquez, The Hele-Shaw asymptotics for mechanical models of tumor growth. To appear in ARMA (2014) http://hal.upmc.fr/hal-00831932

[5] Emeric Bouin, Vincent Calvez, Nicolas Meunier, Sepideh Mirrahimi, Benoît Perthame, Gael Raoul, and Raphaël Voituriez. Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration. C. R. Math. Acad. Sci. Paris, 350(15-16):761­766, 2012.

FRBW01 8th January 2014
11:30 to 12:30
E Varvaruca Geometric approaches to water waves and free surface flows - 2
These lectures aim to present a new geometric approach to the asymptotic behaviour near singularities in some classical free-boundary problems in fluid dynamics. We start by introducing the problems and providing an outline of the methods that have been used to prove existence of solutions. We then present a modern proof, using monotonicity formulas and frequency formulas, of the famous Stokes conjecture from 1880, which asserts that at any stagnation point on the free surface of a two-dimensional steady irrotational gravity water wave, the wave profile necessarily has lateral tangents enclosing a symmetric angle of 120 degrees. (This result was first proved in the 1980s under restrictive assumptions and by somewhat ad-hoc methods.) We then explain how the methods extend to the case of two-dimensional steady gravity water waves with vorticity. Finally, we show how the same methods can be adapted to describe the asymptotic behaviour near singularities in the problem of steady three-dimensional axisymmetric free surface flows with gravity.

References:

[1] Buffoni, B.; Toland, J. F. Analytic theory of global bifurcation. An introduction. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, N.J., 2003.

[2] Constantin, A. Nonlinear water waves with applications to wave-current interactions and tsunamis. CBMS-NSF Regional Conference Series in Applied Mathematics, 81. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, P.A., 2011.

[3] Constantin, A.; Strauss, W. Exact steady periodic water waves with vorticity. Comm. Pure Appl. Math. 57 (2004), no. 4, 481--527.

[4] Varvaruca, E. On the existence of extreme waves and the Stokes conjecture with vorticity. J. Differential Equations 246 (2009), no. 10, 4043--4076.

[5] Varvaruca, E.; Weiss, G. S. A geometric approach to generalized Stokes conjectures. Acta Math. 206 (2011), no. 2, 363--403.

[6] Varvaruca, E.; Weiss, G. S. The Stokes conjecture for waves with vorticity. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), no. 6, 861--885.

[7] Varvaruca, E.; Weiss, G. S. Singularities of steady axisymmetric free surface flows with gravity, to appear in Comm. Pure Appl. Math., http://arxiv.org/abs/1210.3682.

FRBW01 8th January 2014
13:30 to 14:30
Poster Session
FRBW01 8th January 2014
15:00 to 16:00
Regularity of Free Boundaries in Obstacle Type Problems - 1
The aim of these lectures is to give an introduction to the regularity theory of free boundaries related to the obstacle problem. Besides the classical obstacle problem, we will consider the problem on harmonic continuation of Newtonian potentials, the thin obstacle problem, and their parabolic counterparts (as much as the time permits).

Lecture 1. In this lecture, we will introduce the problems we will be working on and discuss initial regularity results for the solutions.

Lecture 2. In this lecture, we will discuss the optimal regularity of solutions and give proofs by using monotonicity formulas.

Lecture 3. In this lecture, we will consider the blowups of the solutions at free boundary points. We will then classify the blowups and thereby classify the free boundary points.

Lecture 4. In this lecture, we will show how to prove the regularity of the "regular set" and obtain a structural theorem on the singular set.

Suggested reading:

[1] Petrosyan, Arshak ; Shahgholian, Henrik; Uraltseva, Nina . Regularity of free boundaries in obstacle-type problems. Graduate Studies in Mathematics, 136. American Mathematical Society, Providence, RI, 2012. x+221 pp. ISBN: 978-0-8218-8794-3

[2] Caffarelli, L. A. The obstacle problem revisited. J. Fourier Anal. Appl. 4 (1998), no. 4-5, 383--402.

[3] Weiss, Georg S. A homogeneity improvement approach to the obstacle problem. Invent. Math. 138 (1999), no. 1, 23--50.

[4] Caffarelli, Luis A. ; Karp, Lavi ; Shahgholian, Henrik . Regularity of a free boundary with application to the Pompeiu problem. Ann. of Math. (2) 151 (2000), no. 1, 269--292.

[5] Caffarelli, Luis ; Petrosyan, Arshak ; Shahgholian, Henrik . Regularity of a free boundary in parabolic potential theory. J. Amer. Math. Soc. 17 (2004), no. 4, 827--869.

[6] Garofalo, Nicola ; Petrosyan, Arshak . Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem. Invent. Math. 177 (2009), no. 2, 415­461.

[7] Danielli, Donatella ; Garofalo, Nicola ; Petrosyan, Arshak ; To, Tung . Optimal regularity and the free boundary in the parabolic Signorini problem. arXiv:1306.5213

FRBW01 9th January 2014
10:00 to 11:00
Regularity of Free Boundaries in Obstacle Type Problems - 2
The aim of these lectures is to give an introduction to the regularity theory of free boundaries related to the obstacle problem. Besides the classical obstacle problem, we will consider the problem on harmonic continuation of Newtonian potentials, the thin obstacle problem, and their parabolic counterparts (as much as the time permits).

Lecture 1. In this lecture, we will introduce the problems we will be working on and discuss initial regularity results for the solutions.

Lecture 2. In this lecture, we will discuss the optimal regularity of solutions and give proofs by using monotonicity formulas.

Lecture 3. In this lecture, we will consider the blowups of the solutions at free boundary points. We will then classify the blowups and thereby classify the free boundary points.

Lecture 4. In this lecture, we will show how to prove the regularity of the "regular set" and obtain a structural theorem on the singular set.

Suggested reading:

[1] Petrosyan, Arshak ; Shahgholian, Henrik; Uraltseva, Nina . Regularity of free boundaries in obstacle-type problems. Graduate Studies in Mathematics, 136. American Mathematical Society, Providence, RI, 2012. x+221 pp. ISBN: 978-0-8218-8794-3

[2] Caffarelli, L. A. The obstacle problem revisited. J. Fourier Anal. Appl. 4 (1998), no. 4-5, 383--402.

[3] Weiss, Georg S. A homogeneity improvement approach to the obstacle problem. Invent. Math. 138 (1999), no. 1, 23--50.

[4] Caffarelli, Luis A. ; Karp, Lavi ; Shahgholian, Henrik . Regularity of a free boundary with application to the Pompeiu problem. Ann. of Math. (2) 151 (2000), no. 1, 269--292.

[5] Caffarelli, Luis ; Petrosyan, Arshak ; Shahgholian, Henrik . Regularity of a free boundary in parabolic potential theory. J. Amer. Math. Soc. 17 (2004), no. 4, 827--869.

[6] Garofalo, Nicola ; Petrosyan, Arshak . Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem. Invent. Math. 177 (2009), no. 2, 415­461.

[7] Danielli, Donatella ; Garofalo, Nicola ; Petrosyan, Arshak ; To, Tung . Optimal regularity and the free boundary in the parabolic Signorini problem. arXiv:1306.5213

FRBW01 9th January 2014
11:30 to 12:30
E Varvaruca Geometric approaches to water waves and free surface flows - 3
These lectures aim to present a new geometric approach to the asymptotic behaviour near singularities in some classical free-boundary problems in fluid dynamics. We start by introducing the problems and providing an outline of the methods that have been used to prove existence of solutions. We then present a modern proof, using monotonicity formulas and frequency formulas, of the famous Stokes conjecture from 1880, which asserts that at any stagnation point on the free surface of a two-dimensional steady irrotational gravity water wave, the wave profile necessarily has lateral tangents enclosing a symmetric angle of 120 degrees. (This result was first proved in the 1980s under restrictive assumptions and by somewhat ad-hoc methods.) We then explain how the methods extend to the case of two-dimensional steady gravity water waves with vorticity. Finally, we show how the same methods can be adapted to describe the asymptotic behaviour near singularities in the problem of steady three-dimensional axisymmetric free surface flows with gravity.

References:

[1] Buffoni, B.; Toland, J. F. Analytic theory of global bifurcation. An introduction. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, N.J., 2003.

[2] Constantin, A. Nonlinear water waves with applications to wave-current interactions and tsunamis. CBMS-NSF Regional Conference Series in Applied Mathematics, 81. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, P.A., 2011.

[3] Constantin, A.; Strauss, W. Exact steady periodic water waves with vorticity. Comm. Pure Appl. Math. 57 (2004), no. 4, 481--527.

[4] Varvaruca, E. On the existence of extreme waves and the Stokes conjecture with vorticity. J. Differential Equations 246 (2009), no. 10, 4043--4076.

[5] Varvaruca, E.; Weiss, G. S. A geometric approach to generalized Stokes conjectures. Acta Math. 206 (2011), no. 2, 363--403.

[6] Varvaruca, E.; Weiss, G. S. The Stokes conjecture for waves with vorticity. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), no. 6, 861--885.

[7] Varvaruca, E.; Weiss, G. S. Singularities of steady axisymmetric free surface flows with gravity, to appear in Comm. Pure Appl. Math., http://arxiv.org/abs/1210.3682.

FRBW01 9th January 2014
13:30 to 14:30
Regularity of Free Boundaries in Obstacle Type Problems - 3
The aim of these lectures is to give an introduction to the regularity theory of free boundaries related to the obstacle problem. Besides the classical obstacle problem, we will consider the problem on harmonic continuation of Newtonian potentials, the thin obstacle problem, and their parabolic counterparts (as much as the time permits).

Lecture 1. In this lecture, we will introduce the problems we will be working on and discuss initial regularity results for the solutions.

Lecture 2. In this lecture, we will discuss the optimal regularity of solutions and give proofs by using monotonicity formulas.

Lecture 3. In this lecture, we will consider the blowups of the solutions at free boundary points. We will then classify the blowups and thereby classify the free boundary points.

Lecture 4. In this lecture, we will show how to prove the regularity of the "regular set" and obtain a structural theorem on the singular set.

Suggested reading:

[1] Petrosyan, Arshak ; Shahgholian, Henrik; Uraltseva, Nina . Regularity of free boundaries in obstacle-type problems. Graduate Studies in Mathematics, 136. American Mathematical Society, Providence, RI, 2012. x+221 pp. ISBN: 978-0-8218-8794-3

[2] Caffarelli, L. A. The obstacle problem revisited. J. Fourier Anal. Appl. 4 (1998), no. 4-5, 383--402.

[3] Weiss, Georg S. A homogeneity improvement approach to the obstacle problem. Invent. Math. 138 (1999), no. 1, 23--50.

[4] Caffarelli, Luis A. ; Karp, Lavi ; Shahgholian, Henrik . Regularity of a free boundary with application to the Pompeiu problem. Ann. of Math. (2) 151 (2000), no. 1, 269--292.

[5] Caffarelli, Luis ; Petrosyan, Arshak ; Shahgholian, Henrik . Regularity of a free boundary in parabolic potential theory. J. Amer. Math. Soc. 17 (2004), no. 4, 827--869.

[6] Garofalo, Nicola ; Petrosyan, Arshak . Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem. Invent. Math. 177 (2009), no. 2, 415­461.

[7] Danielli, Donatella ; Garofalo, Nicola ; Petrosyan, Arshak ; To, Tung . Optimal regularity and the free boundary in the parabolic Signorini problem. arXiv:1306.5213

FRBW01 10th January 2014
09:00 to 10:00
N Garofalo Regularity of Free Boundaries in Obstacle Type Problems - 4
The aim of these lectures is to give an introduction to the regularity theory of free boundaries related to the obstacle problem. Besides the classical obstacle problem, we will consider the problem on harmonic continuation of Newtonian potentials, the thin obstacle problem, and their parabolic counterparts (as much as the time permits).

Lecture 1. In this lecture, we will introduce the problems we will be working on and discuss initial regularity results for the solutions.

Lecture 2. In this lecture, we will discuss the optimal regularity of solutions and give proofs by using monotonicity formulas.

Lecture 3. In this lecture, we will consider the blowups of the solutions at free boundary points. We will then classify the blowups and thereby classify the free boundary points.

Lecture 4. In this lecture, we will show how to prove the regularity of the "regular set" and obtain a structural theorem on the singular set.

Suggested reading:

[1] Petrosyan, Arshak ; Shahgholian, Henrik; Uraltseva, Nina . Regularity of free boundaries in obstacle-type problems. Graduate Studies in Mathematics, 136. American Mathematical Society, Providence, RI, 2012. x+221 pp. ISBN: 978-0-8218-8794-3

[2] Caffarelli, L. A. The obstacle problem revisited. J. Fourier Anal. Appl. 4 (1998), no. 4-5, 383--402.

[3] Weiss, Georg S. A homogeneity improvement approach to the obstacle problem. Invent. Math. 138 (1999), no. 1, 23--50.

[4] Caffarelli, Luis A. ; Karp, Lavi ; Shahgholian, Henrik . Regularity of a free boundary with application to the Pompeiu problem. Ann. of Math. (2) 151 (2000), no. 1, 269--292.

[5] Caffarelli, Luis ; Petrosyan, Arshak ; Shahgholian, Henrik . Regularity of a free boundary in parabolic potential theory. J. Amer. Math. Soc. 17 (2004), no. 4, 827--869.

[6] Garofalo, Nicola ; Petrosyan, Arshak . Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem. Invent. Math. 177 (2009), no. 2, 415­461.

[7] Danielli, Donatella ; Garofalo, Nicola ; Petrosyan, Arshak ; To, Tung . Optimal regularity and the free boundary in the parabolic Signorini problem. arXiv:1306.5213

FRBW01 10th January 2014
10:00 to 11:00
E Varvaruca Geometric approaches to water waves and free surface flows - 4
These lectures aim to present a new geometric approach to the asymptotic behaviour near singularities in some classical free-boundary problems in fluid dynamics. We start by introducing the problems and providing an outline of the methods that have been used to prove existence of solutions. We then present a modern proof, using monotonicity formulas and frequency formulas, of the famous Stokes conjecture from 1880, which asserts that at any stagnation point on the free surface of a two-dimensional steady irrotational gravity water wave, the wave profile necessarily has lateral tangents enclosing a symmetric angle of 120 degrees. (This result was first proved in the 1980s under restrictive assumptions and by somewhat ad-hoc methods.) We then explain how the methods extend to the case of two-dimensional steady gravity water waves with vorticity. Finally, we show how the same methods can be adapted to describe the asymptotic behaviour near singularities in the problem of steady three-dimensional axisymmetric free surface flows with gravity.

References:

[1] Buffoni, B.; Toland, J. F. Analytic theory of global bifurcation. An introduction. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, N.J., 2003.

[2] Constantin, A. Nonlinear water waves with applications to wave-current interactions and tsunamis. CBMS-NSF Regional Conference Series in Applied Mathematics, 81. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, P.A., 2011.

[3] Constantin, A.; Strauss, W. Exact steady periodic water waves with vorticity. Comm. Pure Appl. Math. 57 (2004), no. 4, 481--527.

[4] Varvaruca, E. On the existence of extreme waves and the Stokes conjecture with vorticity. J. Differential Equations 246 (2009), no. 10, 4043--4076.

[5] Varvaruca, E.; Weiss, G. S. A geometric approach to generalized Stokes conjectures. Acta Math. 206 (2011), no. 2, 363--403.

[6] Varvaruca, E.; Weiss, G. S. The Stokes conjecture for waves with vorticity. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), no. 6, 861--885.

[7] Varvaruca, E.; Weiss, G. S. Singularities of steady axisymmetric free surface flows with gravity, to appear in Comm. Pure Appl. Math., http://arxiv.org/abs/1210.3682.

FRB 15th January 2014
14:00 to 15:00
Optimal potentials for Schrödinger operators
We consider the Schr\"odinger operator $-\Delta+V(x)$ on $H^1_0(\Omega)$, where $\Omega$ is a given domain of ${\mathbb R}^d$. Our goal is to study some optimization problems where an optimal potential $V\ge0$ has to be determined in some suitable admissible classes and for some suitable optimization criteria, like the energy or the Dirichlet eigenvalues.
FRB 15th January 2014
15:15 to 16:15
Minimization of Dirichlet eigenvalues
We discuss some recent progress on minimization of eigenvalues of the Dirichlet Laplacian over regions in Euclidean space with geometric constraints.
FRB 22nd January 2014
14:00 to 15:00
Nonlinear eigenvalue problems for nonhomegenous differential operators
We establish some concentration properties for various classes of nonlinear eigenvalue problems with variable exponent. These striking phenomena are due to the combined effect of several terms, a crucial role being the presence of one or several unusual nonlinearities. The treatment is mainly variational and the problems are inspired by models in electrorheological fluids and image processing.
FRB 22nd January 2014
15:15 to 16:15
Nonlocal problems of p-laplacian type
We would like to present some results on the asymptotic behaviour of problems of the form $$ u_t -\nabla \cdot \{a(\int_\Omega |\nabla u(t,x)|^pdx) |\nabla u|^{p-2}\nabla u\} = f \hbox{ in } (0,+\infty)\times \Omega $$ together with boundary and initial conditions.
FRB 29th January 2014
14:00 to 15:00
A Volberg Singular Integrals and Geometric Measure Theory: towards a solution of David--Semmes problem.
The boundedness of the Riesz operator (whose kernel is the gradient of the fundamental solution for Laplacian in R^d) in L^2 with respect to d-1 dimensional Hausdorff measure must imply the rectifiability of this measure. This statement became known as David--Semmes problem. Guy David and Steven Semmes devoted two books to it. But it has been proved only for d=2, first by Mattila--Melnikov--Verdera for the case of homogeneous set, and later by Tolsa in a non-homogeneous situation. The non-homogeneous situation for d=2 also involves relations between beta numbers of Peter Jones and Menger's curvature. However, Menger's curvature is ``cruelly missing" (by the expression of Guy David) in dimensions d>2. In a recent work of Nazarov--Tolsa--Volberg the conjecture of David and Semmes has been validated. The proof (which does not involve Menger's curvature) gives a new and completely different proofs of the abovementioned results also in the case d=2. It is a long and not-so-easy paper. The result can be cast in the language of the existence of bounded harmonic vector fields in certain (infinitely connected) domains. In fact, our result is a certain co-dimension 1 claim. In higher co-dimensions the problem (which we will explain) rests open.
FRB 29th January 2014
15:15 to 16:15
On the Mathematical Analysis of Thick Fluids
In chemical engineering models, shear-thickening or dilatant fluids may converge to the limit class of incompressible fluids with a maximum admissible shear rate, the so-called thick fluids. These non-Newtonian fluids may be obtained, in particular, as the power limit of Ostwald-deWaele fluids, and may be formulated as a new class of evolution variational inequalities. We discuss the existence, uniqueness and continuous dependence of solutions, as well as the asymptotic stabilization in time towards steady state solutions.
FRB 5th February 2014
14:00 to 15:00
Energy minimizing maps with free boundaries
I am going to present recent results joint with J.Andersson, H.Shahgholian and Georg Weiss. We study the regularity problem for a singular elliptic system of Euler equations corresponding to energy functional with the Lipschitz integrand. It is proved that the set of "regular" free boundary points is localy a C^{1+\betha} surface. In proving this result we need an array of technical tools including monotonicity formulas, quadratic growth of solutions and an epiperimetric inequality for the balanced energy functional.
FRB 5th February 2014
15:15 to 16:15
C Elliott Surface PDEs and Interfaces
FRB 6th February 2014
14:00 to 15:45
Free Boundary Problems in Transonic Flow: Introudction I
FRB 6th February 2014
16:00 to 17:00
Free Boundary Problems in Transonic Flow: Introudction II
FRB 10th February 2014
13:00 to 14:00
Optimal partitions for first eigenvalue; numerical approximations and related problems
FRB 12th February 2014
14:00 to 15:00
Flatness implies smoothness for the porous medium equation
We call a solution to the porous medium equation flat on a unit size space time cylinder if it is sandwiched between two close and parallel traveling wave solutions of velocity 1. Those solutions are smooth in the sense that the pressure is a smooth function on the closure of its support, and inside a smaller space time cylinder, with nonvanishing gradient at the boundary of the support. The proof follows Caffarellis strategy of improving the flatness for rescaled solutions. An important part consists in a linearization of equation and geometry.
FRB 12th February 2014
15:15 to 16:15
On regularity properties of solutions to hysteresis-type problems.
We consider equations with the simplest hysteresis operator at the right-hand side. Such equations describe the so-called processes "with memory" in which various substances interact according to the hysteresis law. The main feature of this problem is that the operator at the right-hand side is a multivalued. We present some results concerning the optimal regularity of solutions. Our arguments are based on quadratic growth estimates for solutions near the free boundary. The talk is based on joint work with Nina Uraltseva.
FRB 18th February 2014
13:00 to 14:00
T Ranner Computational methods for an optimal partition problem on surfaces
FRB 25th February 2014
13:00 to 14:30
D Bucur Shape optimization of spectral functionals: a short introduction.
FRB 26th February 2014
14:00 to 15:00
S Luckhaus Free energies, discrete Hamiltonians and two scale convergence in statistical mechanics.
FRB 26th February 2014
15:15 to 16:15
Multidimensional Shock Waves and Free Boundary Problems
In this talk we will discuss several longstanding, fundamental shock problems in mathematical fluid mechanics and related free boundary problems for nonlinear partial differential equations of mixed elliptic-hyperbolic type. These shock problems include the supersonic flow onto solid wedges (the Prandtl-Meyer problem), shock reflection-diffraction by concave cornered wedges (the von Neumann's conjectures), and shock diffraction by convex cornered wedges (the Lighthill problem).
FRB 5th March 2014
14:00 to 15:00
First and second variation of domain functionals and applications to problems with Robin boundary conditions.
We discuss the first and second domain variation for functionals related to elliptic boundary and eigenvalue problems with Robin boundary conditions. A characterization of optimal domains is obtained. Special attention is given to perturbations of the ball where the first variation vanishes. The sign of the second variation depends on the eigenvalues of a Steklov problem. A stability result for nearly circular domains follows as a byproduct. The talk is based on a project in collaboration with Alfred Wagner.
FRB 5th March 2014
15:15 to 16:15
From Boltzmann to Euler: Hilbert's 6th problem revisited.
This talk addresses the hydrodynamic limit of the Boltzmann equation, namely the compressible Euler equations of gas dynamics. An exact summation of the Chapman-Enskog expansion originally given by Gorban and Karlin is the key to the analysis. An appraisal of the role of viscosity and capillarity in the limiting process is then given where the analogy is drawn to the limit of the Korteweg-de Vries-Burgers equations as a small parameter tends to zero.
FRB 6th March 2014
13:00 to 14:00
Nonlinear Fractional Diffusion Equations. Numerics and Free Boundaries
FRB 11th March 2014
13:00 to 14:00
Nondegeneracy of solution to degenerate obstacle problem.
FRB 13th March 2014
14:00 to 15:00
Mathematical Modeling of the Cellular Cytoskeleton.
FRB 13th March 2014
15:15 to 16:15
A Henrot An overdetermined Free Boundary Problem with non constant boundary conditions
FRB 18th March 2014
13:00 to 14:00
H Mikayelyan Regularity of the Mumford-Shah minimizers at the crack-tip
FRB 19th March 2014
14:00 to 15:00
Contact-point behaviour of the free boundary for the porous-medium equation
FRB 19th March 2014
15:15 to 16:15
C Venkataraman Modelling and simulation of cell motility with surface finite elements
We propose a general framework for the modelling and simulation of cell motility. The cell membrane dynamics is governed by a geometric evolution law accounting for its mechanical properties. For the polarisation of the cell we postulate a reaction diffusion system for species located on the moving cell membrane. Protrusion is than achieved by back-coupling these surface quantities to the geometric equation for the membrane position. The numerical method to approximate the general model is based on surface finite elements for both the geometric equation and the surface equations. We demonstrate the versatility of this approach to describe the motion of different cells in two and three dimensions. We also discuss the problem of fitting to experimental data and to this end present a method for identifying parameters in the model.
FRB 2nd April 2014
14:00 to 15:00
A nonstandard PDE system of viscous Cahn-Hilliard type related to a model for phase segregation.
The talk deals with a diffusion model of phase-field type, leading to a parabolic system of two partial differential equations, interpreted as balances of microforces and microenergy, for two unknowns: the problem's order parameter and the chemical potential; each equation includes a viscosity term; Neumann homogeneous boundary conditions and initial conditions complement the field equations. The analysis of this system has been made the subject of a joint research program with G. Gilardi, P. Podio-Guidugli and J. Sprekels. The related model aims at describing two-species phase segregation on an atomic lattice under the presence of diffusion: the initial and boundary value problem will be considered and the existence-uniqueness of a global-in-time solution will be discussed along with other related results.
FRB 2nd April 2014
15:15 to 16:15
The longest shortest fence and the stability of floating trees
Over 50 years ago Polya stated the following problem. Given a plane convex set K (a piece of land), find the shortest curve (or fence) that bisects this set into two subsets of equal area. Is it true that this curve is never longer than the diameter of the circular disc of same area as K? Under the additional assumption that K is centrosymmetric (i.e, K = -K) he gave a simple proof that this is indeed the case. Without this assumption the question is much harder to answer positively. This is joint work with L. Esposito, V.Ferone, C. Nitsch and C. Trombetti. By the way, a result of N. Fusco and A. Pratelli states, that if the fences are restricted to be straight line segments, the answer is negative. In that case the longest shortest fence is attained for the Auerbach triangle and not for the disc.
FRB 8th April 2014
13:00 to 14:00
The eigenvalue problem for the infinity Laplacian
The eigenvalue problem for the infinity Laplacian arises as the limit as $p\to \infty$ of the eigenvalue problem for the p-Laplacian and it is given by the free boundary problem $$ max (\lambda u-|\nabla u|,\Delta_\infty u ) = 0 $$ I will briefly discuss how this limit can be obtained, some known result for the case $p=\infty$ and some related open problems.
FRB 9th April 2014
14:00 to 15:00
Non-univalent solutions of the Polubarinova-Galin equation
The Polubarinova-Galin equation describes the time evolution of the conformal map onto a Hele-Shaw blob of fluid, which expands by injection of fluid at one point. As the evolution goes on different parts of the fluid region may collide with each other. This can be handled by turning to a weak formulation of the problem, but in the talk I will discuss another way, namely by keeping the fluid region simply connected by letting it go up onto a Riemann surface. In this way one can let the solution go on forever as a simply connected solution, but the construction of the Riemann surface is by no means trivial, because it cannot constructed in advance. It has to be updated all the time, as new collisions occurs in the fluid region. The talk describes work in progress, in collaboration with Yu-Lin Lin (KTH).
FRB 9th April 2014
15:15 to 16:15
Traffic on networks: modeling and analysis
We are interested in the traffic of cars on a network, like for instance a city. We describe traffic using hamilton-Jacobi equations, and propose new general junction conditions that can be fully characterized. In the framework of viscosity solutions for discontinuous hamiltonians,we also prove a comparison principle. With this powerful tool in hands, we show how to homogenize the traffic on a whole network like a city.
FRB 15th April 2014
13:00 to 14:00
A Minne Interior regularity of solutions to elliptic fully nonlinear free boundary problems.
We will consider $W^{2,n}(B_{1})$ solutions to the fully nonlinear elliptic problem \begin{equation*} \begin{cases} F(D^{2}u,x)=f(x) & \text{a.e. in }B_{1}\cap\Omega,\\ |D^{2}u|\le K & \text{a.e. in }B_{1}\backslash\Omega, \end{cases} \end{equation*} where $\Omega$ is an unknown open set and $K$ is a given constant. For $F$ convex and some regularity assumptions on $F$ and $f$, $C^{1,1}$ regularity of $u$ is proven in $B_{1/2}$.
FRB 16th April 2014
14:00 to 15:00
M Feldman Shock Reflection, von Neumann conjectures, and free boundary problems
We discuss shock reflection problem for compressible gas dynamics, and von Neumann conjectures on transition between regular and Mach reflections. Then we will talk about recent results on existence of regular reflection solutions for potential flow equation up to the detachment angle, and discuss some techniques. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear equation of mixed elliptic-hyperbolic type. Open problems will also be discussed. The talk is based on the joint work with Gui-Qiang Chen.
FRB 16th April 2014
15:15 to 16:15
N Garofalo The Signorini problem for the heat equation: regularity of the solution and of the free boundary
Over the past decade the lower-dimensional, or Signorini, problem has received a great deal of attention, especially after the 2004 breakthrough result of Athanasopoulos and Caffarelli on the optimal C^{1,1/2} smoothness of the solution up to the thin manifold. However, until recently, there has been no significant progress on the parabolic counterpart of such optimal regularity and on the regularity of the free boundary. In this lecture I will discuss recent joint work with D.Danielli, A. Petrosyan and T. To in which we give a comprehensive treatment of the parabolic Signorini problem based on a generalization of Almgren's monotonicity of the frequency. This includes the proof of the optimal regularity of solutions, classification of free boundary points, the regularity of the regular set and the structure of the singular set.
FRB 23rd April 2014
14:00 to 15:00
Phase field modelling of drops in electric fields
The modelling of drops under the action of electric fields is of practical importance in connection with microdevices where small amounts of fluid need to be manipulated. Nevertheless, the associated free boundary problems involve various difficulties: presence of moving contact lines, changes in the drop's topology, instabilities of various types,... Some of these can be overcome by introducing a phase field formulation of the problem. We introduce it in a thermodynamically consistent way and present analysis of existence and uniqueness of solutions for the resulting model as well as numerical implementation for various situations of interest.
FRB 23rd April 2014
15:15 to 16:15
Does p matter in the p-obstacle problem?
In this talk I shall discuss the growth, from the free boundary,for solutions to the p-obstacle problem. I shall unveil a surprising behavior for the p-elliptic case, and also discuss some strange behavior for the p-parabolic case. A "healthy" exercise, in the latter case, is to find two different growth rates, for Barenblatt solution, at different free boundary points. This is joint work with John Andersson and Erik Lindgren.
FRB 24th April 2014
11:00 to 12:30
M Hintermueller Numerical Methods for (Quasi)Variational Inequalities - Part I
Motivated by the obstacle problem as well as by optimization problems with partial differential equations subject to pointwise constraints on the control, the state or its derivative, semismooth Newton methods and Moreau-Yosida based path-following techniques will be discussed. Besides the convergence analysis in function space, mesh independence properties of the iterations are presented and numerical analysis aspects, such as the optimal link between the Moreau-Yosida parameter and the mesh-width of discretization as well as adaptive finite element methods, will be addressed. Quasi-variational inequalities involving the $p$-Laplacian and constraints on the gradient of the state will be briefly studied, too. Finally, the potential of the introduced methodology is highlighted by means of various applications ranging from phase-separation processes to problems in mathematical image processing.
FRB 24th April 2014
14:30 to 16:00
M Hintermueller Numerical Methods for (Quasi)Variational Inequalities - Part II
Motivated by the obstacle problem as well as by optimization problems with partial differential equations subject to pointwise constraints on the control, the state or its derivative, semismooth Newton methods and Moreau-Yosida based path-following techniques will be discussed. Besides the convergence analysis in function space, mesh independence properties of the iterations are presented and numerical analysis aspects, such as the optimal link between the Moreau-Yosida parameter and the mesh-width of discretization as well as adaptive finite element methods, will be addressed. Quasi-variational inequalities involving the $p$-Laplacian and constraints on the gradient of the state will be briefly studied, too. Finally, the potential of the introduced methodology is highlighted by means of various applications ranging from phase-separation processes to problems in mathematical image processing.
FRB 29th April 2014
13:00 to 14:00
A Segatti Some observations on a fractional Cahn Hilliard equation and its relations with the fractional Porous Medium equation
I will present some recent results with G. Akagi and G. Schimperna on a fractional version of the Cahn Hilliard equation. In particular, I will discuss its (asymptotic) relation with the fractional Porous Medium equation.
FRB 30th April 2014
14:00 to 15:00
Free boundary regularity in the parabolic Signorini problem
FRB 30th April 2014
15:15 to 16:15
Nonlinear fractional diffusion equations. Two problems with free boundaries
FRB 7th May 2014
14:00 to 15:00
Harnack's inequality for the inhomogeneous $p(x)-$laplace equation
FRB 7th May 2014
15:15 to 16:15
The Calderón Problem For Schrödinger Operators
The problem of determining the electrical conductivity of a body by making voltage and current measurements on the object's surface has various applications in fields such as oil exploration and early detection of malignant breast tumour. This classical problem posed by Calderón remained open until the late '80s when it was finally solved in a breakthrough paper by Sylvester-Uhlmann. In the recent years, geometry has played an important role in this problem. The unexpected connection of this subject to fields such as dynamical systems, symplectic geometry, and Riemannian geometry has led to some interesting progress. This talk will be an overview of some of the recent results and an outline of the techniques used to treat this problem. The work described here is partially supported by NSF Grant No. DMS-0807502, Academy of Finland Fellowship 256378, Vetenskapsradet 2012-3782
FRB 13th May 2014
13:00 to 14:00
H Fritz Time-periodic solutions of advection-diffusion equations on moving hypersurfaces.
FRB 13th May 2014
15:00 to 16:30
D Serre Why people in hyperbolic conservation laws are interested in free boundary problems.
In hyperbolic equations, wave propagate with finite velocity. When the equation is quasilinear, this velocity depends on the state itself; this non-constancy leads to the formation of shock waves. Shock waves are inherently free boundaries. I shall present illustrate basic concepts in one space dimension and discuss a little bit the multi-dimensional setting, where stability is encoded in a Lopatinskii condition. I'll also show that in specific problems (steady or self-similar solutions), one is naturally led to deal with elliptic equations.
FRB 14th May 2014
14:00 to 15:00
Weak singularities in the multi-dimensional Riemann Problem
The Riemann Problem consists in looking at self-similar solutions of first order systems of conservation laws, like compressible gas dynamics. In one space dimension, it is solved by shocks and rarefaction waves. Both kinds have generalizations in several space variables, where the wave fronts are free boundaries. We study the rarefaction case in detail and show that the gradient jump at the front can be calculated explicitly when the outer state is constant. This jump depends upon the dimension $d$ and vanishes when $d=3$. This is a joint work with H. Freisthueler (Univ. Konstanz)
FRB 14th May 2014
15:15 to 16:15
Interface singularities for the Euler equations.
I will discuss the problem of interface singularities for the 3-D Euler equations. In the case of one-fluid interfaces, I will discuss the so-called "splash" and "splat" singularities, wherein the crest of a breaking wave crashes onto the trough, and hypersurfaces collide. In the case of two-fluid interfaces, I will explain why such singularities cannot form. This is joint work with D. Coutand.
FRB 20th May 2014
15:00 to 16:30
An Introduction to Free Boundary Problems in MHD
FRB 22nd May 2014
11:00 to 12:00
A Figalli A transportation approach to random matrices.
Optimal transport theory is an efficient tool to construct change of variables between probability densities. However, when it comes to the regularity of these maps, one cannot hope to obtain regularity estimates that are uniform with respect to the dimension except in some very special cases (for instance, between uniformly log-concave densities). In random matrix theory the densities involved (modeling the distribution of the eigenvalues) are pretty singular, so it seems hopeless to apply optimal transport theory in this context. However, ideas coming from optimal transport can still be used to construct approximate transport maps (i.e., maps which send a density onto another up to a small error) which enjoy regularity estimates that are uniform in the dimension. Such maps can then be used to show universality results for the distribution of eigenvalues in random matrices. The aim of this talk is to give a self-contained presentation of these results.
FRB 22nd May 2014
14:00 to 15:00
I Díaz Confinement in the framework of Schrödinger equations: a revision of the classical "particle in a box" example and further free boundary results.
One of the main modifications to the Classical Mechanics introduced by Quantum Mechanics is the impossibility to localize the state (position and velocity) in the dynamics of a particle (Heisenberg Principle). This fact is connected with the study of the support of solutions of the associated Schrödinger equation. Some of the more popular simplifications for the linear Schrödinger equation (attributed by him, in 1935, to George Gamow [1904-1968] and repeated in any text book in Quantum Mechanics) deals with the case of the stationary eigenvalue problem associated to several discontinuous potentials $V(x)$, which, among other things allows to illustrate the tunneling effect. Nevertheless, surprisingly enough, it seems that it was nor observed before in the literature (lecture by this author at Tours, 2012) that the confinement argument used in the case of "the infinite well potential" ($V(x)=V_{0}$ if $x\in \lbrack 0,L]$ and $ V(x)=+\infty $ if $x\notin \lbrack 0,L]$) leads to a serious mathematical mistake: the usual "popular" solution does not satisfy the global Schrödinger equation in $\mathbb{R}$ since a Dirac delta is generated at each point of the boundary of the box. A first goal of the lecture is to present a different confinement argument which requires the study of bifurcation diagrams associated to problems of the type \begin{equation*} \left\{ \begin{array}{lr} -\dfrac{d^{2}u}{dx^{2}}+V(x)u=\lambda u & \hbox{in }(0,L), \\ u=0 & \text{on }\partial (0,L), \end{array} \right. \end{equation*} $V(x)=\frac{V_{0}}{\left\vert u(x)\right\vert ^{1-m}}$, when $m\in \lbrack 0,1).$ By a suitable application of the "bifurcation from the infinity" method (Rabinowitz 1973) it is possible to show the existence of a numerable set of branches emanating (from the infinity) from the eigenvalue subspaces of the linear problem. Moreover, the exact multiplicity can be given by extending some previous joint results with J. Hernández, which leads to a complete description of each branch. In particular, in each branch, there exists a suitable value $\lambda ^{\ast }$ of the energy (the parameter $ \lambda $) such that if $\lambda \geq \lambda ^{\ast }$ then the solutions satisfy that $\dfrac{du}{dx}(0)=\dfrac{du}{dx}(L)=0$ and so the confinement argument does not develop any singularity on the boundary of the box. In a second part of the lecture, I will make mention (very briefly) to the question of the confinement when it is studied, by other type of methods (integral enery methods), for the Schrödinger equation with a singular nonlinear potential of the type \begin{equation*} i\frac{\partial u}{\partial t}+\Delta u=a|u|^{-(1-m)}u,\mbox{ in }(0,\infty )\times \mathbb{R}^{N}, \end{equation*} with $a\in C $ and $ 0<m<1 $ (a series of joint works with P. Bégout).
FRB 22nd May 2014
15:15 to 16:15
P Secchi Stability of the linearized MHD-Maxwell free interface problem
In the talk we consider the free boundary problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region, the flow is governed by the usual compressible MHD equations, while in the vacuum region we consider the Maxwell system for the electric and the magnetic fields, in order to investigate the well-posedness of the problem, in particular in relation with the electric field in vacuum. At the free interface, driven by the plasma velocity, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. Under suitable stability conditions satisfied at each point of the plasma-vacuum interface, we derive a basic a priori estimate for solutions to the linearized problem in the Sobolev space $H^1_{\tan}$ with conormal regularity. The proof follows by a suitable secondary symmetrization of the Maxwell equations in vacuum and the energy method. An interesting novelty is represented by the fact that the interface is characteristic with variable multiplicity, so that the problem requires a different number of boundary conditions, depending on the direction of the front velocity (plasma expansion into vacuum or viceversa). To overcome this difficulty, we recast the vacuum equations in terms of a new variable which makes the interface characteristic of constant multiplicity. In particular, we don't assume that plasma expands into vacuum. This is a joint work with D.Catania and M.D'Abbicco.
FRB 27th May 2014
10:00 to 10:55
H Berestycki Propagation and blocking for reaction-diffusion equations in non homogeneous media.
I will discuss bi-stable reaction-diffusion equations in cylinders with varying cross-sections motivated by biology and medicine. The aim is to understand the effect of the non-homogenous medium on propagation or blocking of advancing waves. The role played by the geometry of the domain of propagation is of particular interest for these models. I will report on joint work with Juliette Bouhours and Guillemette Chapuisat.
FRB 27th May 2014
11:15 to 12:10
E Varvaruca Global Bifurcation of Steady Gravity Water Waves with Critical Layers
I will present some recent results on the problem of two-dimensional travelling water waves propagating under the influence of gravity in a flow of constant vorticity over a flat bed. By means of a conformal mapping and an application of Riemann-Hilbert theory, the free-boundary problem is equivalently reformulated as a one-dimensional pseudodifferential equation which involves a modified Hilbert transform and, moreover, has a variational structure. Using the new formulation, existence is established, by means of real-analytic global bifurcation theory, of a family of solutions which includes waves of large amplitude, even in the presence of critical layers in the flow. This is joint work with Adrian Constantin and Walter Strauss.
FRB 27th May 2014
13:30 to 14:25
A Karakhanyan TBA
FRB 27th May 2014
14:30 to 15:25
Free boundary problems in random environments.
I will discuss about a class of free boundary problems involving elliptic operators with random, discontinuous coefficients. I will also comment on how the ideas and tools designed for the study of such problems reveal improvement of smoothness of solutions to classical equations presenting "non-physical" free boundaries.
FRB 27th May 2014
16:00 to 17:00
What Makes a Surface "Optimal" : Rothschild Distinguished Visiting Fellow Lecture
Area minimizing surfaces, a solid liquid interphase, the surface across which a strategy must switch in a game have some sort of "optimality" that often reflects in its stability, regularity,propagation properties. In the last few decades, starting with the pioneering work of de Giorgi on minimal surfaces, we have attained considerable understanding of the underlying mechanisms that produce these effects. I will try to provide an overall view of these developments.
FRB 4th June 2014
14:00 to 15:00
J Carrillo Regularity of local minimizers of the interaction energy via obstacle problems.
Local minimizers of the interaction energy present a very rich structure. We show that if the repulsion at the origin is strong enough but integrable, then the local minimizers are in fact regular, at least bounded probability densities, and they satisfy an implicit obstacle problem. This is the key to establish also uniqueness of global minimizers upto translations in some particular case. This is a work in collaboration with M. Delgadino and A. Mellet.
FRB 11th June 2014
14:00 to 15:00
Nonlinear elliptic equations with absorption.
We describe some recent results on boundary value problems for equations of the form $-Lu+f(x,u)=0$, in a domain $D$ in $R^N$. Here $L$ is the Laplacian or a more general second order elliptic equation and $f$ is positive in $D\times R_+$, monotone increasing with respect to the second variable.The boundary data is given by positive measures, possibly unbounded.
FRB 11th June 2014
15:15 to 16:15
On incompressible two-phase flows with phase transitions
A thermodynamically consistent model for incompressible two-phase fluid flows with phase transitions driven by temperature is introduced and analyzed. Concentrating on the case of equal densities, we establish well-posedness and study the qualitative behavior of solutions. In particular, we characterize all equilibria and study their stability properties. The entropy turns out to be an important quantity in the stability analysis.
FRB 18th June 2014
14:00 to 16:00
Phase transition problems involving minimal surfaces
FRB 19th June 2014
14:00 to 15:00
A classical Perron method for existence of smooth solutions to boundary value & obstacle problems for boundary-degenerate elliptic operators via holomorphic map
We prove existence of solutions to boundary value problems and obstacle problems for degenerate-elliptic, linear, second-order partial differential operators with partial Dirichlet boundary conditions using a new version of the Perron method. The elliptic operators considered have a degeneracy along a portion of the domain boundary which is similar to the degeneracy of a model linear operator identified by Daskalopoulos and Hamilton (1998) in their study of the porous medium equation or the degeneracy of the Heston operator (Heston, 1993) in mathematical finance. Our Perron method relies on weak and strong maximum principles for degenerate-elliptic operators, concepts of continuous subsolutions and supersolutions for boundary value and obstacle problems for degenerate-elliptic operators, and maximum and comparison principle estimates previously developed by us.
FRB 19th June 2014
15:15 to 16:15
Modelling the Growth of Tyndall Stars
A "liquid snowflake" or "Tyndall star" is a small, thin, star-shaped region of liquid water produced when ice is melted by absorption of light. In the early stages of its growth, the liquid inclusion appears as a small, approximately axially symmetric, lens-like object. We investigate some models, based on having a highly anisotropic kinetic-undercooling condition on the ice/water boundary, to try to understand the morphology of such a "Tyndall shape".
FRBW04 23rd June 2014
10:00 to 10:45
R Goldstein Plenary Lecture 1: How a Volvox embryo turns itself inside out
During the growth of daughter colonies of the multicellular alga Volvox the spherical embryos must turn themselves inside out to complete their development. This process of 'inversion' has many features in common with gastrulation, the process by which an initially convex spherical shell of animal cells develops an invagination, leading to the formation of a gastric system. In both cases it is understood that cell shape changes play a major role in guiding the process, but quantification of the dynamics, and formulation of a mathematical description of the process, have been lacking. In this talk I will describe advances my group has made recently on both fronts. Using the technique of SPIM (selective plane illumination microscopy) we have obtained the first real-time three-dimensional time-lapse movies of inversion in Volvox, using several species displaying distinct morphological events. The beginnings of an elastic theory of these processes will also be descri bed.
FRBW04 23rd June 2014
11:30 to 12:15
B Wagner Plenary Lecture 2: Unstaedy non-uniform base states and their stability
In this talk we consider several pattern forming systems, ranging from phase separation of polymer blends, self-assembly of crystalline films to dewetting of polymer films. These systems all have unstaedy non-uniform base states. We develop asymptotic techniques to analyse their associated linear stability problems and derive expressions for predicting the dominant wave-length of the pattern.
FRBW04 23rd June 2014
13:30 to 14:15
Plenary Lecture 3: Free Boundaries and Fluid Mixing at the Micro Level
Turbulent mixing often occurs with immiscible fluids or with miscible fluids over rapid time scales, so that the flow is locally inhomogeneous at a micro level for periods of interest. We start with a review of problems in which such flows arise.

The flow regions in which the mixing occurs can generally be identified reliably; examples will be given. The challenge for current research is to describe the microscopic and inhomogeneous mixture in a statistical sense.

Full resolution of the flows is generally out of the question and will remain so for decades. Thus we are interested in statistical properties of the flow that are stable and appear to converge under mesh refinement, with sufficient detail in the statistical description (for example a pdf or cdf (cumulative distribution function) to support reaction processes in flows of engineering interest. A first step, generally insufficient, is to compute means and variances of fluctuating processes.

This goal is still in the future. Partial results leading in this direction will be presented. We formulate a notion of stochastic convergence and present numerical algorithms which appear to be convergent in this metric. We introduce theoretical ideas related to convergence based on the renormalization group, including the important notion that the solution, at the LES level of resolution of necessity considered here, is not unique. In other words, the usual standard of convergence under mesh refinement is not sufficient to guarantee a simulation in agreement with experimental data. We discuss methods to mitigate this serious obstacle to scientific progress. Basically, experiments are essential to select the correct non-unique solution and the algorithm and its adjustable parameters to reach this goal. While flows of interest are commonly at high Reynolds numbers outside the regime of relevant experiments, the expansion parameter is 1/Reynolds number. In terms of this parameter, the perturbation from experiment to applications is small, within normally accepted ranges for perturbative extensions of validation regimes. "cambridge.14.abs" 37L, 2201C

FRBW04 23rd June 2014
14:50 to 15:20
Some Free Boundary Problems for Flows at high Re
This talk will describe some models and open problems for (i) disturbance waves in annular 2-phase flow and (ii) bubble collapse.
FRBW04 23rd June 2014
14:50 to 15:20
Dynamics of an Euler beam with unilateral constraints
We study the vibrations of an elastic beam between rigid obstacles. The non penetrability condition leads to a description of the dynamics as a hyperbolic fourth order variational inequality. For this free boundary problem we construct a sequence of approximate solutions by combining some classical space discretizations with time-stepping schemes especially suited to unilateral contact problems for discrete mechanical systems. We prove the stability and the convergence of these numerical methods and we obtain an existence result for our original problem under very general assumptions on the geometry of the obstacles.
FRBW04 23rd June 2014
15:20 to 15:50
M Fontelos Shapes of charged drops in an electric field and Rayleigh jets
When a viscous drop is electrically charged or subject to an external electric field, it may undergo deformations and, occasionally produce thin jets (called Rayleigh jets) that are of practical interest. We study the drop's shapes and related bifurcation problems, as well as the possible mechanisms for jet formation.
FRBW04 23rd June 2014
15:20 to 15:50
Nondegeneracy in the Obstacle Problem with a Degenerate Force Term
In this talk I present the proof of the optimal nondegeneracy of the solution $u$ of the obstacle problem $\triangle u=f\chi_{\{u>0\}}$ in a bounded domain $D\subset\mathbb{R}^{n}$, where we only require $f$ to have a nondegeneracy of the type $f(x)\geq\lambda\vert (x_1,\cdots,x_p)\vert^{\alpha}$ for some $\lambda>0$, $1\leq p\leq n$ (an integer) and $\alpha>0$. We prove optimal uniform $(2+\alpha)$-th order and nonuniform quadratic nondegeneracy, more precisely we prove that there exists $C>0$ (depending only on $n$, $p$ and $\alpha$) such that for $x$ a free boundary point and $r>0$ small enough we have $\sup_{\partial B_r(x)}u\geq C\lambda (r^{2+\alpha}+\vert(x_1,\cdots,x_p)\vert^{\alpha}r^{2})$. I also present the proof of the optimal growth with the assumption $\vert f(x)\vert\leq\Lambda\vert (x_1,\cdots,x_p)\vert^{\alpha}$ for some $\Lambda\geq 0$ and the porosity of the free boundary.

Preprint: http://www.newton.ac.uk/preprints/NI14045.pdf

FRBW04 23rd June 2014
16:00 to 16:30
J Oliver On contact-line dynamics with mass transfer
We investigate the effect of mass transfer on the evolution of a thin (two-dimensional) partially wetting drop. While the effects of viscous dissipation, capillarity, slip and uniform mass transfer are taken into account, other effects, such as gravity, surface tension gradients, vapour transport and heat transport, are neglected in favour of mathematical tractability. Our matched-asymptotic analysis reveals that the leading-order outer formulation and contact-line law that is selected in the small-slip limit depends delicately on both the sign and size of the mass transfer flux, leading in particular to novel generalisations of Tanner's law. We analyse the resulting evolution of the drop and report good agreement with numerical simulations. Co-Authors: Matthew Saxton (University of Oxford), Jonathan Whiteley (University of Oxford), Dominic Vella (University of Oxford); John King (University of Nottingham).
FRBW04 23rd June 2014
16:00 to 16:30
Spreading profile and nonlinear Stefan problems
I will report some recent progresses (in joint works with Z. Guo, B. Lou, H. Matsuzawa, H. Matano, K. Wang, M. Zhou etc.) on the study of a general nonlinear Stefan problem, used as a model for the understanding of a variety of spreading phenomena, where the unknown function u(t,x) represents the density of concentration of a certain (chemical or biological) species at time t and space location x, with the free boundary standing for the spreading front. Such spreading problems are usually modeled by the corresponding Cauchy problem, which has attracted extensive research starting from the well-known 1937 paper of Kolmogorov-Petrovski-Piskunov. We will discuss the similarity and differences of the long-time behavior of these two types of mathematical models by closely examining their spreading profiles.
FRBW04 23rd June 2014
16:30 to 17:00
Surfactants in two-phase flow
Co-authors: Harald Garcke (Universitat Regensburg), Bjorn Stinner (University of Warwick)

Emulsification is an important industrial process that involves mixing two or more fluids that normally are unmixable. More precisely, in the process of emulsification, it is desirable to have stable dispersions of one fluid in another. Common examples of emulsions are milk, fire extinguishers and hand cream. The mixture is thermodynamically unstable and will progressively revert back to their unmixed states over time. Surface active agents (or surfactants) are often added to increase the stability of the mixture and hence there is great interest, especially in industrial applications, to understand the influence of surfactants on the dynamics of multi-fluid systems.

In this talk, I will outline the derivation of two new models on soluble surfactants in two phase flow. The first is a sharp interface model that describes the interfaces with moving hypersurfaces, and the second is a phase field model. Both models are thermodynamically consistent and generalise existing models in the literature. I will then discuss the relationship between two models, which is supported with some numerical simulations.

FRBW04 23rd June 2014
16:30 to 17:00
A model for the quasi-static crack growth in hydraulic fracture
We present a variational model for the quasi-static crack growth in hydraulic fracture in the framework of the energy formulation of rate-independent processes. The cracks are assumed to lie on a prescribed plane and to satisfy a very weak regularity assumption.
FRBW04 24th June 2014
09:00 to 09:45
Plenary Lecture 4
FRBW04 24th June 2014
10:20 to 10:50
Multiscale analyses of tissue growth and front propagation
The derivation of continuum models which represent underlying discrete or microscale phenomena is emerging as an important part of mathematical biology: integration between subcellular, cellular and tissue-level behaviour is crucial to understanding tissue growth and mechanics. I will consider the application of a multiscale method to two problems on this theme.

Firstly a new macroscale description of nutrient-limited tissue growth, which is formulated as a microscale moving-boundary problem within a porous medium, is introduced. A multiscale homogenisation method is employed to enable explicit accommodation of the influence of the underlying microscale tissue structure, and its evolution, on the macroscale dynamics.

A challenging consideration in continuum models of tissue is the accommodation of (spatially-discrete) cell-signalling events, a feature of which being the progression of moving fronts of cell-signalling activity across a lattice. New (continuum) analyses of monotone waves in a discrete diffusion equation are presented, and extended to modulated fronts exhibited in cell signalling models.

FRBW04 24th June 2014
10:20 to 10:50
Weak shock diffraction
Co-author: Allen Tesdall (CUNY)

We study the diffraction of a weak, self-similar shock in two space dimensions near a point where its shock strength approaches zero and the shock turns continuously into an expansion wavefront. For example, this happened when a weak shock hits a semi-infinite screen. The local asymptotic solution satisfies the unsteady transonic small disturbance equation. We also consider a related half-space problem where a shock whose strength approaches zero reflects off a ``soft'' boundary. Numerical solutions show a complex reflection pattern similar to one that occurs in the Guderley Mach reflection of weak shocks.

FRBW04 24th June 2014
10:50 to 11:20
Dielectric Boundary in Biomolecular Solvation
A dielectric boundary in a biomolecular system is a solute-solvent (e.g., protein-water) interface that defines the dielectric coefficient to be one value in the solute region and another in solvent. The inhomogeneous dielectric medium gives rise to an effective dielectric boundary force that is crucial to the biomolecular conformation and dynamics. This lecture presents a precise definition and explicit formula of the dielectric boundary force based on the classical Poisson-Boltzmann theory of electrostatic interactions. These results are used to study the motion of a cylindrical dielectric boundary driven by the competition between the surface tension, dielectric boundary force, and solvent viscous force. Implications of the mathematical findings to biomolecular conformational stabilities are discussed.
FRBW04 24th June 2014
10:50 to 11:20
Shock diffraction by convex cornered wedges
Co-author: Prof. Gui-Qiang G. Chen (University of Oxford)

In this talk, I would like to present one of our current research projects, that is on the mathematical analysis of shock diffraction by convex cornered wedges. The existence of the regular configuration is established up to the critical wedge angle, which should be the criterion of the transition between the regular configuration and the Mach configuration. This work is joint with Prof. G.-Q. Chen.

FRBW04 24th June 2014
11:30 to 12:00
Numerical investigations of long range segregation systems
We investigate numerical approximation for a class of elliptic and parabolic competition-diffusion systems of long range segregation models for two and more competing species. Also we study the long term behavior for parabolic system. We prove that as the competition rate goes to infinity the solution converges, along with suitable sequences, to a spatially long range segregated state satisfying some differential inequalities. Moreover, we provide numerical simulations for parabolic and elliptic cases.
FRBW04 24th June 2014
11:30 to 12:00
The Signorini problem, fractional Laplacians and the language of semigroups
The Signorini problem can be equivalently formulated as a thin obstacle problem for an elastic membrane. The resulting free boundary problem turns out to be equivalent to the obstacle problem for the fractional Laplacian on the whole space. We will show how to understand this problem under the light of the language of semigroups that I developed in my PhD thesis (2010). In particular, we are able to consider different kinds of Signorini problems that are equivalent to obstacle problems for fractional powers of operators different than the Laplacian on the whole space. Boundary conditions of different kinds (Dirichlet, Neumann, periodic) and radial solutions can also be treated with this unified language. Another advantage is that this language avoids the use of the Fourier transform. The basic regularity results (Harnack inequalities, Schauder estimates) for these fractional nonlocal operators can be studied by means of the generalization of the Caffarelli--Silvestre extensio n problem that I proved in my PhD thesis. It turns out that the solution for the extension problem can be written in terms of the heat semigroup.
FRBW04 24th June 2014
12:00 to 12:30
C Venkataraman Moving boundary problems in cell biology
We discuss the mathematical formulation and numerical solution of free and moving boundary problems that arise in the study of cell motility. We consider the dynamics of the cell membrane and species that reside on the cell membrane. We also present novel finite element methods for the simulation of the model equations and report on numerical results.
FRBW04 24th June 2014
12:00 to 12:30
B Fang On Stability of Steady Transonic Shocks in Supersonic Flow around a Wedge
In this talk we are concerned with the stability of steady transonic shocks in supersonic flow around a wedge. When a uniform supersonic flow comes against a straight wedge with small vertex angle, a shock-front attached to the wedge appears. It is indicated in the book "Supersonic Flow and Shock Waves" by Courant and Friedrichs that there are two admissible shock solutions satisfying both Rankine-Hugoniot conditions and the entropy condition. The weaker shock solution may be transonic, while the stronger one must be transonic. In this talk, we shall present stability results for 2-D and M-D flows.

This talk is based on the joint works with Prof. G.-Q. Chen, and Prof. S.X. Chen.

FRBW04 24th June 2014
13:30 to 13:35
J King & C Elliot Welcome and Introduction
FRBW04 24th June 2014
13:35 to 13:45
Free Boundary Problems in Biology and Medicine
FRBW04 24th June 2014
13:45 to 14:10
How the Geometry of the Cell Boundary Couples Cellular Blebbing and Actin Based Protrusions
FRBW04 24th June 2014
14:10 to 14:35
Mathematical Modelling of Tissue Growth in a Perfusion Bioreactor
FRBW04 24th June 2014
14:35 to 15:00
J Lowengrub Feedback, Lineages and Cancer Therapy
FRBW04 24th June 2014
15:20 to 15:45
Free Boundary Problems in Models for Bacterial Biofilms
FRBW04 24th June 2014
15:45 to 16:10
Modelling the Spatial Temporal Dynamics of Small Monomeric G Proteins
FRBW04 24th June 2014
16:10 to 17:00
Open Discussion
FRBW04 25th June 2014
09:00 to 09:45
Plenary Lecture 5: Obstacle problems in geometric evolutions
I will consider geometric evolutions of fronts in the presence of obstacles. I will discuss in particular the mean curvature flow and the Willmore flow, presenting some new results and some open questions.
FRBW04 25th June 2014
10:20 to 11:05
P Markowich Plenary Lecture 6: On Boltzmann-type and Free Boundary Models for Price Formation
We present a theory for a price formation free boundary model formulated by Lasry and Lions. Also we introduce a mesoscopic Boltzmann-type model for price formation in economic markets of single goods, which gives the Lasry-Lions model in the limit of large trading frequency.
FRBW04 25th June 2014
11:15 to 11:45
Symmetries of Boundary Value Problems: Definitions, Algorithms and Applications to Physically Motivated Problems
One may note that the symmetry-based methods were not widely used for solving boundary-value problems (BVPs). To the best of our knowledge, the first rigorous definition of Lie's invariance for BVPs was formulated by George Bluman in early 1970s and applied to some classical BVPs. However, Bluman's definition cannot be directly applied to BVPs of more general form, for example, to those involving boundary conditions on the moving surfaces, which are described by unknown functions. In our recent papers, a new definition of Lie's invariance of BVP with a wide range of boundary conditions (including those at infinity and moving surfaces) was formulated. Moreover, an algorithm of the group classification for the given class of BVPs was worked out. The definition and algorithm were applied to some classes of nonlinear two-dimensional and multidimensional BVPs of Stefan type with the aim to show their efficiency. In particular, the group classification problem for these classes of BVPs was solved, reductions to BVPs of lower dimensionality were constructed and examples of exact solutions with physical meaning were found. Very recently, the definition and algorithms were extended on the case of conditional invariance for BVPs and applied to some nonlinear BVPs. This research was supported by a Marie Curie International Incoming Fellowship within the 7th European Community Framework Programme.
FRBW04 25th June 2014
11:15 to 11:45
Generalized Neumann solutions for the two-phase fractional Lam\'e-Clapeyron-Stefan problems
FRBW04 25th June 2014
11:45 to 12:15
J Gwinner Free Boundary Value Problems in Contact Mechanics
In this talk we report on recent progress in the solution of various non-smooth free boundary value problems that arise in contact mechanics, including unilateral contact with Tresca friction and non-monotone adhesion/delamination problems. We analyse these problems and employ for their numerical treatment optimization and regularization techniques combined with fi nite element discretisation. This presentation is based on the recent papers [1, 2] and the recent PhD Thesis [3] of N. Ovcharova. [1] J.Gwinner, hp-FEM convergence for unilateral contact problems with Tresca friction in plane linear elastostatics, J. Comput. Appl. Math. 254 (2013) 175 -184 [2] N. Ovcharova and J.Gwinner, A study of regularization techniques of nondi erentiable optimization in view of application to hemivariational inequalities, J. Optim Theory Appl., DOI 10.1007/s10957-014-0521-y (2014) [3] N. Ovcharova, Regularization Methods and Finite Element Approximation of Some Hemivariational Inequalities with Applications to Nonmonotone Contact Problems, Ph.D. Thesis, Universitaet der Bundeswehr Muenchen, 2012.
FRBW04 25th June 2014
11:45 to 12:15
Convergence of optimal control problems governed by second kind parabolic variational inequalities
We consider a family of optimal control problems where the control variable is given by a free boundary condition of Neumann type. This family is governed by parabolic variational inequalities of the second kind. We prove the strong convergence of the optimal controls and state systems associated to this family to a similar optimal control problem.
FRBW04 25th June 2014
13:30 to 14:15
K Deckelnick Plenary Lecture 7: Double obstacle phase field approach for an elliptic inverse problem with discontinuous coefficients
We consider the inverse problem of recovering interfaces where the diffusion coefficient in an elliptic PDE has jump discontinuities. We employ a least squares approach together with a perimeter regularization. A suitable relaxation of the perimeter leads to a sequence of Cahn--Hilliard type functionals for which we obtain a $\Gamma$--convergence result. Using a finite element discretization of the elliptic PDE and a suitable adjoint problem we derive an iterative method in order to approximate discrete critical points. We prove convergence of the iteration and present results of numerical tests. This is joint work with C.M. Elliott (Warwick) and V. Styles (Sussex).
FRBW04 25th June 2014
14:50 to 15:20
T Ranner Computational methods for an optimal partition problem on surfaces
In this talk I will explore computational techniques for solving a free boundary problem posed on a surface. The problem is to divide a surface into regions such that the sum of the first eigenvalue of the Dirichlet Laplace-Beltrami operator over each region is minimized. Different relaxations of this problem will be explored. Each takes the form of large system of partial differential equations which will be solved using algorithms designed for high performance computing techniques.
FRBW04 25th June 2014
14:50 to 15:20
Optimal Regularity and the Free Boundary in the Parabolic Signorini Problem
Co-authors: Nicola Garofalo (University of Padova), Arshak Petrosyan (Purdue University), Tung To (Purdue University)

In this talk we will give an overview of our comprehensive treatment of the parabolic Signorini problem, based on a generalization of Almgren's monotonicity of the frequency. In particular, we will discuss the optimal regularity of solutions, the classi fication of free boundary points, the regularity of the regular set, and the structure of the singular set.

FRBW04 25th June 2014
15:20 to 15:50
Multi-material structural topology optimization using phase field methods
Co-authors: Luise Blank (University of Regensburg), Hassan Farshbaf-Shaker (WIAS Berlin), Harald Garcke (University of Regensburg)

A phase field approach for structural topology optimization which allows for topology changes and multiple materials is analyzed. First order optimality conditions are rigorously derived and several numerical results for mean compliance problems are presented.

FRBW04 25th June 2014
15:20 to 15:50
D Bucur Isoperimetric inequalities and free discontinuity problems
The Faber-Krahn inequality for the first eigenvalue of the Laplace operator with Robin boundary conditions was recently proved by D. Daners in the context of Lipschitz sets. This talk introduces a new approach to deal with this type of isoperimetric inequalities and is based on recent advances on the regularity theory for general free discontinuity problems, in particular on a new monotonicity formula jointly obtained with S. Luckhaus. As a consequence, we not only extend isoperimetric inequalities to arbitrary sets, but also obtain a range of new ones (joint work with A. Giacomini).
FRBW04 25th June 2014
16:00 to 16:30
A stable discretization for the simulation of two-phase flow
Co-authors: Harald Garcke (University of Regensburg), Michael Hinze (University of Hamburg)

In [Abels, Garcke, Gruen, M3AS, 22, 2012] a model for the simulation of variable density two-phase flow using the diffuse interface approach is presented. Due to an additional transport term in the Navier-Stokes system this model enjoys consistency with thermodynamics. To preserve this important property in the numerical simulation we propose a time discretization that uses a carefully designed weak formulation and delivers a time discrete variant of the continuous energy equality. The spatial discretization is enriched by a suitable post processing step to ensure the energy inequality to hold in the fully discrete setting. An a-posteriori adaptive procedure is proposed that deals with the errors both in the Navier-Stokes and the Cahn-Hilliard system.

FRBW04 25th June 2014
16:00 to 16:30
KA Lee Curvature Flows with Free Boundaries
In this talk, we are going to discuss alpha- Gauss Curvature Flows with free boundaries. For a=1n, a flat spot on convex manifold will persists for some time, which is similar as the waiting time in Porous Medium Equations. Naturally, the interface between strictly convex region and flat spot will form a free boundary that will moves along the time. We will show the relation between optimal regularity and bounded nondegenerate speed of free boundaries. The optimal regularity and the smoothness of the free boundary will be discussed. We will also consider the dynamics and regularity of free boundary of the manifold with a concave region.
FRBW04 25th June 2014
16:30 to 17:00
L Banas Numerical approximation of phase-field models for multiphase flow
Flow of mixtures of incompressible fluids with complex fluid interactions (surface tensions, contact angles) can be described by a system of (multicomponent) Cahn-Hilliard-Navier-Stokes equations. We propose finite element based numerical approximations of some non-smooth phase-field models for mixtures of incompressible fluids with variable densities and viscosities. We discuss theoretical and practical issues related to the proposed numerical approximations and present computational experiments to demonstrate main features of the considered models.
FRBW04 25th June 2014
16:30 to 17:00
A Karakhanyan The Proximity of Free Boundary to Wedges and Planes
The talk is about the geometry of free boundary striking the centre of the half ball B_1^+ where a wedge like boundary data is prescribed near the origin. It will be seen that the effect of the wedge-like data dooms the asymptotic shape of the free boundary and the homogeneous global solutions turn out to have wedge-like free boundary too. A delicate analysis involving monotonicity formula techniques is employed to furnish the main results, which are from a joint paper with H. Shahgholian (to appear in the Transactions of the AMS).
FRBW04 26th June 2014
09:00 to 09:45
J Ball Plenary Lecture 8: Interfaces arising from solid phase transformations
Solid phase transformations give rise to a variety of unusual kinds of interfaces between different phases, some only observed in recent experiments. The lecture will discuss ways of describing and predicting these, and related questions of nonlinear analysis.
FRBW04 26th June 2014
10:20 to 10:50
Free Boundary-Value Problems for Liquid Crystals
We discuss modelling strategies for free boundary-value problems arising in liquid crystal applications. We have recently modelled stable liquid crystal patterns in shallow cuboid-shaped nematic wells and have computed phase diagrams as a function of geometrical aspect ratio and anchoring strength on well boundaries. Recent work by Telo da Gama group (CFTC, Lisbon) shows that our computed patterns are also relevant to wedge-shaped nematic-filled wells at the isotropic-nematic transition temperature. At the isotropic-nematic transition temperature, the isotropic and nematic phases are separated by a free boundary, depending on the geometry and the surface effects at play. We discuss modelling approaches to this free-boundary problem and the role of such free boundaries in applications. This is joint work with Chong Luo, Samo Kralj and Samo Kralj.
FRBW04 26th June 2014
10:20 to 10:50
Free boundary problems in glacial hydrology
Many aspects of a glacier’s dynamics depend on the evolution of an interface between slowly creeping ice and fast flowing water. As ice can be modelled as a very viscous fluid, variations on free boundary Stokes flow arise in the modelling of such interfaces. In this talk I will focus on the shape of a channel that is cut through or under the ice by viscous dissipation of meltwater. Networks of these channels form a vital part of a glacier’s hydrological system, which carries meltwater from the surface to the glacier margin, where mixing with dense ocean water has a strong effect on melting at the ice face. The evolving cross section of the channel is related to the problem of a contracting or expanding bubble in two dimensional Stokes flow, which allows us to derive analytic results. I will also discuss the impact of a meltwater source, such as a channel, on the spatial distribution of melting at the ice face.
FRBW04 26th June 2014
10:50 to 11:20
On the production of high-porosity metallic solids
Co-author: Stephen Davis (Northwestern University)

High-porosity metallic solids are useful as lightweight materials in many engineering disciplines. However, batch processing techniques for producing such materials by solidification of a molten metallic foam are problematic, and it is not currently possible to control the porosity distribution of the final product {\it a priori}. The molten metallic foam is inherently unstable; the thin liquid bridges between bubbles drain rapidly and rupture due to intermolecular forces, leading to bubble coalescence and large-scale topological rearrangement. In this talk we will consider the competition between coalesecence and freezing in these dynamically evolving foams, examining in particular the coupling between the microscale hydrodynamics in the molten liquid films and the progression of a solidification front. The foam is modelled as a coupled system of free boundary problems involving both liquid/gas and liquid/solid interfaces, which we solve using a combination of asymptotic and numerical techniques. An understanding of these microscale processes motivates other protocols for solid foam production where the porosity distribution of the final product can be more effectively controlled.

FRBW04 26th June 2014
10:50 to 11:20
A classical Perron method for existence of smooth solutions to boundary value and obstacle problems for boundary-degenerate elliptic operators via holomorphic m
We prove existence of solutions to boundary value problems and obstacle problems for degenerate-elliptic, linear, second-order partial differential operators with partial Dirichlet boundary conditions using a new version of the Perron method. The elliptic operators considered have a degeneracy along a portion of the domain boundary which is similar to the degeneracy of a model linear operator identified by Daskalopoulos and Hamilton (1998) in their study of the porous medium equation or the degeneracy of the Heston operator (Heston, 1993) in mathematical fi nance. Our Perron method relies on weak and strong maximum principles for degenerate-elliptic operators, concepts of continuous subsolutions and supersolutions for boundary value and obstacle problems for degenerate-elliptic operators, and maximum and comparison principle estimates previously developed by us.
FRBW04 26th June 2014
11:30 to 12:00
A phase field approach for shape optimization in fluids
Co-author: Harald Garcke (University of Regensburg)

We consider the problem of shape optimization with a general objective functional using the Stokes equations as state constraints. By combining a porous medium and a phase field approach we obtain a well-posed problem in a diffuse interface setting. We discuss well-posedness and optimality conditions and relate the phase field problem to a perimeter penalized sharp interface problem.

FRBW04 26th June 2014
11:30 to 12:00
Existence and Qualitative Properties of Grounds States to the Choquard-Type Equations
Co-author: Jean Van Schaftingen (Louvain-la-Neuve, Belgium)

The Choquard equation, also known as the Hartree equation or nonlinear Schrodinger-Newton equation is a stationary nonlinear Schrodinger type equation where the nonlinearity is coupled with a nonlocal convolution term given by an attractive gravitational potential. We present sharp Liouville-type theorems on nonexistence of positive supersolutions of such equations in exterior domains and consider existence, positivity, symmetry and optimal decay properties of ground state solutions under various assumptions on the decay of the external potential and the shape of the nonlinearity. We also discuss the existence of semiclassical solutions to the equation.

FRBW04 26th June 2014
12:00 to 12:30
M Glicksman Deterministic Pattern Formation in Diffusion-Limited Systems
An interface evolving under local equilibrium develops gradients in the Gibbs-Thomson-Herring interface temperature distribution that provide tangential energy fluxes. The Leibniz-Reynolds transport theorem exposes a 4th-order, net-zero energy field (the 'Bias' field) that autonomously deposits and removes capillary-mediated thermal energy. Where energy is released locally, the freezing rate is persistently retarded, and where energy is removed, the rate is enhanced. These contravening dynamic field responses balance at points (roots) where the surface Laplacian of the chemical potential vanishes, inducing an inflection, or ‘curling’, of the interface. Interfacial inflection couples to the main transport fields producing pattern branching, folding, and complexity.

Precision noise-free numerical schemes, including integral equation sharp-interface solvers (J. Lowengrub, S. Li) and, recently, three noise-free phase-field simulations (A. Mullis, M. Zaeem, K. Reuther) independently confirm that pattern branching initiates at locations predicted using analytical methods for smooth, noise-free starting shapes in 2-D. A limit cycle may develop as the interface and its energy field co-evolve, synchronizing the inflection points to produce classical dendritic structures. Noise and stochastics play no direct role in the proposed deterministic mechanism of branching and pattern morphogenesis induced by persistent 'perturbations'.

FRBW04 26th June 2014
12:00 to 12:30
C-H Cheng Some moving boundary value problems consisting of viscous incompressible fluids moving and interacting with nonlinear elastic shells
In this talk, I will briefly talk about my past works on the interaction of incompressible Navier-Stokes with either the biofluid shells or the Koiter shells. The biofluid shells can be used to model the cell membranes, while the Koiter shells are used to model general elastic thin shells that have equilibrium state. The dynamics of both kinds of shells are described by nonlinear PDEs derived from energy functionals depending on geometric structures of the shells. The well-posedness of the problems concerning the coupling of the fluids and the shells are non-trivial, and we established the existence and uniqueness of solutions in Sobolev spaces.
FRBW04 26th June 2014
13:30 to 14:15
AL Bertozzi Plenary Lecture 9: Curvature flow on graphs for large data classification
In the continuum, close connections exist between mean curvature ow, the Allen-Cahn (AC) partial di erential equation, and the Merriman-Bence-Osher (MBO) threshold dynamics scheme. Graph analogues of these processes have recently seen a rise in popularity as relaxations of NP-complete combinatorial problems, which demands deeper theoretical underpinnings of the graph processes. We discuss several applications including supervised and unsupervised machine learning and community detection in social networks. We discuss connections to spectral graph theory and fast algorithms and some recent results for curvature flow on graphs and open problems.
FRBW04 26th June 2014
14:50 to 15:20
M Chipot Obstacle problems in unbounded domains and related issues
We will present a formulation of obstacle problems in unbounded domains when the energy method does not work i.e. when the force does not belong to H-1(O). We will report also on some new results for the pure Neumann problem in unbounded cylinders.
FRBW04 26th June 2014
14:50 to 15:20
Oscillating facets
Co-author: Piotr Rybka (The University of Warsaw)

We study a very singular one-dimensional parabolic problem with initial data in the energy space. We study facet creation and extinction of solutions caused by the evolution of facets.

FRBW04 26th June 2014
15:20 to 15:50
Pointwise estimates and regularity for the parabolic obstacle problem
Co-author: Régis Monneau (ENPC)

I will discuss pointwise regularity properties for parabolic equations. In particular, I will talk about a method that can be used to obtain a second order Taylor expansion. I will also discuss how the same method can be applied to the parabolic obstacle problem in order to study the regularity of the free boundary.

FRBW04 26th June 2014
15:20 to 15:50
C Heinemann Analysis of coupled phase-field models describing damage phenomena
Co-authors: Christiane Kraus (WIAS Berlin), Elisabetta Rocca (WIAS Berlin / University of Pavia)

In this talk we are going to investigate evolution inclusions describing damage processes in elastic media which are coupled with further physical phenomena such as heat conduction or phase separation. The main difficulty in the analysis of these PDE systems is the irreversibility condition for the damage progression. To handle this highly nonlinear behavior we develop an appropriate notion of weak solution consisting of a variational inequality and a full energy inequality. Results ensuring the existence of weak solutions will be presented. The talk is based on joint works with C. Kraus (WIAS) and E. Rocca (WIAS/University of Pavia).

FRBW04 26th June 2014
16:00 to 16:30
Free boundaries in fractional filtration equations
Co-authors: Arturo de Pablo (U. Carlos III de Madrid), Ana Rodríguez (U. Politécnica de Madrid), Juan Luis Vázquez (U. Autónoma de Madrid)

In this talk we will review some recent results on the Cauchy problem for the fractional filtration equation $\partial_t u+(-\Delta)^{\sigma/2}\varphi(u)$, $\sigma\in(0,2)$, where the nonlinearity $\varphi$ satisfies some not very restrictive conditions.

Solutions to these problems become instantaneously positive if the initial data are nonnegative. However, a free boundary emerges in certain cases in which there is a distinguished value, as in the fractional Stefan problem, $\varphi(u)=(u-1)_+$, or in the ‘mesa’ limit, $m\to\infty$, for the fractional porous media equation, $\varphi(u)=u^m$.

FRBW04 26th June 2014
16:00 to 16:30
S Mitchell Numerical and asymptotic solutions of vertical continuous casting with and without superheat
Co-author: Michael Vynnycky (Royal Institute of Technology (Sweden))

In a continuous casting process, such as the strip casting of copper, molten metal first passes through a water-cooled mould region, before being subjected to a high cooling rate further downstream. Consequently, the molten metal solidifies and the solidified metal is withdrawn at a uniform casting speed. Industrialists need to understand the factors influencing product quality and process productivity. Of key significance is the heat transfer that occurs during solidification, particularly the location of the interface between molten metal and solid.

The modelling of the continuous casting of metals is known to involve the complex interaction of non-isothermal fluid and solid mechanics. Typically, the flow in the molten metal is turbulent, and it is generally believed that a computational fluid dynamics (CFD) approach is necessary in order to correctly capture the heat transfer characteristics. However, we can show that an asymptotically reduced version of the CFD-based model, which neglects this turbulence, gives predictions for the pool depth, local temperature profiles and mould wall heat flux that agree very well with results of the original CFD model.

This reduced model can be described as a steady state 2D heat flow Stefan problem, with a degenerate initial condition and non-standard Neumann-type boundary condition. If we assume the incoming metal is at the melt temperature then we obtain a one-phase model but with potentially two stages, depending whether the metal is fully solidified before leaving the mould. However, in reality, the incoming temperature is greater than the melt temperature, termed as including superheat, and this leads to a two-phase model with a pre-solidification stage, where the second phase only first appears after a finite delay.

In this work we highlight some numerical challenges in solving the systems with and without superheat. The Keller box finite-difference scheme is used, along with a boundary immobilisation method.

FRBW04 26th June 2014
16:30 to 17:00
On a thermodynamically consistent Stefan problem with variable surface energy
A thermodynamically consistent two-phase Stefan problem with temperature dependent surface tension is studied. It is shown that this problem generates a local semiflow on a well-defined state manifold. Moreover, stability and instability results of equilibrium configurations will be presented. It will be pointed out that surface heat capacity has a striking effect on the stability behavior of multiple equilibria.
FRBW04 26th June 2014
16:30 to 17:00
Higher order convergent trial methods for Bernoulli's free boundary problem
Co-author: Helmut Harbrecht (University of Basel)

Free boundary problem is a partial differential equation to be solved in a domain, a part of whose boundary is unknown – the so-called free boundary. Beside the standard boundary conditions that are needed in order to solve the partial differential equation, an additional boundary condition is imposed at the free boundary. One aims thus to determine both, the free boundary and the solution of the partial differential equation.

This work is dedicated to the solution of the generalized exterior Bernoulli free boundary problem which is an important model problem for developing algorithms in a broad band of applications such as optimal design, fluid dynamics, electromagnentic shaping etc. For its solution the trial method, which is a fixed-point type iteration method, has been chosen.

The iterative scheme starts with an initial guess of the free boundary. Given one boundary condition at the free boundary, the boundary element method is applied to compute an approximation of the violated boundary data. The free boundary is then updated such that the violated boundary condition is satisfied at the new boundary. Taylor's expansion of the violated boundary data around the actual boundary yields the underlying equation, which is formulated as an optimization problem for the sought update function. When a target tolerance is achieved, the iterative procedure stops and the approximate solution of the free boundary problem is detected.

The efficiency of the trial method as well as its speed of convergence depends significantly on the update rule for the free boundary, and thus on the violated boundary condition. This talk focuses on the trial method with violated Dirichlet boundary data and on the development of higher order convergent versions of the trial method with the help of shape sensitivity analysis.

FRBW04 27th June 2014
09:00 to 09:45
Plenary Lecture 10: Absence of the interface splash singularity for the two-fluid Euler equations
An interface splash singularity occurs when a locally smooth fluid interface self-intersects. Such interface singularities occur for one-fluid interfaces in the Euler equations and other fluids models.

By means of elementary arguments in Lagrangian coordinates, we prove that such a singularity cannot occur in finite-time for a two-fluid interface evolved by either the incompressible Euler equations (with surface tension) or the Muskat equations. By assuming that such a singularity can occur, we find a sharp blow-up rate for the vorticity, and characterize the geometry of the evolving interface. This leads to a contradiction, showing that such a singularity can occur. This is joint work with D. Coutand.

FRBW04 27th June 2014
10:00 to 10:45
J Lowengrub Plenary Lecture 11: Numerical simulation of endocytosis
Co-authors: Sebastian Aland (UC Irvine/TU Dresden), Jun Allard (UC Irvine)

Many cell processes involve the formation of membrane vesicles from a larger membrane, including endocytosis, inter-organelle transport and virus entry. These events are typically orchestrated by curvature-inducing molecules attached to the membrane, such as clathrin and bar-domain proteins. Recent reports demonstrate that in some circumstances vesicles can form de novo in a few milliseconds, e.g., ultrafast endocytosis at the neurological synapse. Membrane dynamics at these scales (millisecond, nanometer) are dominated by hydrodynamic interactions, as the membrane pushes the intracellular and extracellular fluids around to accommodate curvature. To study this problem, we develop new diffuse interface models for the dynamics of inextensible vesicles in a viscous fluid with stiff, curvature-inducing molecules. A new feature of this work is the implementation of the local inextensibility condition by using a local Lagrange multiplier harmonically extended off the interface. To make the method even more robust, we develop a local relaxation scheme that dynamically corrects local stretching/compression errors thereby preventing their accumulation. This is critical to accurately capturing hydrodynamic effects during endocytosis. By varying the membrane coverage of curvature-inducing molecules, we find that there is a cri tical (smallest) neck radius and a critical (fastest) budding time.

FRBW04 27th June 2014
11:30 to 12:15
O Jensen Plenary Lecture 12: Slamming in flexible-channel flows
Co-author: Feng Xu (University of Nottingham)

Large-amplitude self-excited oscillations of high-Reynolds-number flow in a long flexible-walled channel can exhibit vigorous slamming motion, whereby the channel is almost completely occluded over a very short interval in space and time. Treating the flexible channel wall as an inertialess elastic membrane, this near-singular behaviour is exhibited in two-dimensional Navier-Stokes simulations and can be captured in a reduced one-dimensional PDE model (Stewart et al. J. Fluid Mech. 662:288, 2010). The properties of the rigid parts of the system, upstream and downstream of the membrane, play a major role in determining the onset of oscillations. In order to investigate the extreme flow structure that arises during a brief slamming event, we systematically reduce the PDE model to a third-order nonlinear algebraic-differential system, which identifies the likely dominant physical balances.

FRBW04 27th June 2014
13:30 to 14:15
M Feldman Plenary Lecture 13: Shock Reflection, von Neumann conjectures, and free boundary problems
Co-author: Gui-Qiang Chen (Oxford)

We discuss shock reflection problem for compressible gas dynamics, and von Neumann conjectures on transition between regular and Mach reflections. Then we will talk about recent results on existence of regular reflection solutions for potential flow equation up to the detachment angle, and discuss some techniques. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear equation of mixed elliptic-hyperbolic type. Open problems will also be discussed.

FRBW04 27th June 2014
15:00 to 15:45
Plenary Lecture 14: Free boundary problems for mechanical models of tumor growth
Mathematical models of tumor growth, now commonly used, present several levels of complexity, both in terms of the biomedical ingredients and the mathematical description. The simplest ones contain competition for space using purely fluid mechanical concepts. Another possible ingredient is the supply of nutrients. The models can describe the tissue either at the level of cell densities, or at the scale of the solid tumor, in this latter case by means of a free boundary problem. We first formulate a free boundary model of Hele-Shaw type, a variant including growth terms, starting from the description at the cell level and passing to a certain singular limit which leads to a Hele-Shaw type problem. A detailed mathematical analysis of this purely mechanical model is performed. Indeed, we are able to prove strong convergence in passing to the limit, with various uniform gradient estimates; we also prove uniqueness for the limit problem. At variance with the classical Hele-Shaw problem, here the geometric motion governed by the pressure is not sufficient to completely describe the dynamics. Using this theory as a basis, we go on to consider a more complex model including nutrients. Here, technical difficulties appear, that reduce the generality and detail of the description. We prove uniqueness for the system, a main mathematical difficulty. Joint work with Benoit Perthame, Paris, and Fernando Quiros, Madrid
FRBW04 27th June 2014
15:45 to 16:00
Closing Remarks
FRB 2nd July 2014
14:00 to 15:00
Interactions of a gasdynamic type. Two significant multidimensional examples of a constructive and classifying approach
We associate each of the two mentioned examples of interaction with a pair of interactive gasdynamic elements: shock-turbulence or, respectively, wave-wave. The nature of each exemplified interaction essentially depends on the presence of a shock discontinuity: which contributes as an interactive element or, respectively, as a precursory structuring element.

The first example constructs and describes, in presence of a minimal nonlinearity [in the form of a nonlinear subconscious (following P.D. Lax and A.Majda)], a significant deterministic substructure reflecting the shock-turbulence interaction. This example has essentially two objectives: (a) structuring [via identifying a gasdynamic inner coherence] an explicit, closed, and optimal form for the interaction solution, and (b) offering [via identifying some Lorentz type arguments of pseudo-relativistic criticity] an exhaustively classifying characterization of this mentioned solution.

The second example uses, in an anisentropic context [which reflects the structuring presence of a precursory shock wave], two genuinely nonlinear, geometrical approaches [of a Burnat type, respectively of a Martin type] to construct two analogous classes of solutions - of a wave-wave regular interaction type. These two approaches are coincident in a very restrictive frame [isentropic context; two independent variables]. We finally parallel the two anisentropic approaches by using various comparisons between the mentioned analogous classes: to identify some significant consonances and, concurrently, some highly nontrivial contrasts: a classifying aspect.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons