Theory of Water Waves: Summer School
Wednesday 6th August 2014 to Friday 8th August 2014
09:15 to 09:45  Registration  
09:45 to 10:00  Welcome from John Toland (INI Director) and Opening  
10:00 to 12:30 
D Ambrose (Drexel University) Initial Value Problem and Vortex Sheets: Analysis and Computation Session: 1 In this lecture, we will discuss the irrotational water wave problem. We will place the problem in the larger context of vortex sheets; the vortex sheet is the interface between two irrotational fluids, allowing for a jump in the tangential components of velocity across the interface. We will present the equations of motion in both 2D and 3D, for the water wave problem both with and without surface tension. We will present the numerical method of Hou, Lowengrub, and Shelley (HLS) for computing solutions of the initial value problem in 2D, and how the HLS ideas can be used to prove shorttime wellposedness of the initial value problem. The corresponding numerical method and wellposedness proofs for the threedimensional problem will also be discussed. If time allows, we will go beyond the initial value problem, and discuss how the vortex sheet formulation with the HLS ideas can be extended to treat other problems, such as the traveling wave problem. This talk includes joint work with Nader Masmoudi, Michael Siegel, Svetlana Tlupova, and possibly others. 

12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00  Forward looking / open problems session  
15:00 to 15:30  Afternoon Tea  Served at INI  
15:30 to 16:30  Forward looking / open problems session continued  
16:45 to 17:45  Wine Reception  Served at INI 
10:00 to 12:30 
S Wu (University of Michigan) Well Posedness and Singularities of Water Waves Session: 2 A class of water wave problems concerns the dynamics of the free interface separating an inviscid, incompressible and irrotational fluid (water), under the influence of gravity, from a zerodensity region (air). In these lectures I will present some recent methods and ideas developed concerning the local and global wellposedness of these problems, where the fluid region has either no fixed boundary, or has a fixed vertical wall. The emphasis will be on the understanding of the mathematical structure that leads to the results on global existence of small and smooth waves, local existence of arbitrary smooth waves, the persistence of water waves with angled crests, and in the case where there is a vertical wall, the interaction of the free interface with the wall. These lectures will be accessible to graduate students and postdoctors. 

12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
A Nachbin (IMPA  Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro) Conformal Mapping and Complex Topography Session: 3 Water waves propagating over nonsmooth, large amplitude, disordered topography leads to novel asymptotic theory both at the level of equations (i.e. reduced models) as well as at the level of solutions (i.e. effective behavior). In the reduced modeling of twodimensional flows, conformal mapping plays an important role. This lecture will introduce the use of conformal mapping together with nonlinear potential theory. The SchwarzChristoffel Toolbox (by T. Driscoll) will also be introduced showing how to extract quantitative information from the conformal mapping in a specific flow domain. This is useful for computational applications. As time permits recent research examples will be presented such as pulse shaped waves over a disordered (random) topography or in branching channels 

15:00 to 15:30  Afternoon Tea  served at INI  
15:30 to 16:30 
A Nachbin (IMPA  Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro) Conformal Mapping and Complex Topography Session: 3 Continued 

19:30 to 22:00  BBQ at the Møller Centre 
10:00 to 12:30 
O Bokhove (University of Leeds) Variational Water Waves: on Continuum and Discrete Modelling, and Experimental Validation Session: 4 1. Some variational mechanics will be introduced first before embarking on formulating continuum models for water waves in fluids. A forceddissipative nonlinear oscillator will be used as an example, inspired by wave motion in a laboratory HeleShaw cell for (breaking) water waves and beach morphodynamics. Subsequently, variational principles for 3D water waves by Miles and Luke will be derived using constraints. 2. The simpler, depthaveraged, spatially 2D, shallow water analogue of Miles' variational principle will be discretized in space and subsequently in time, using space and time finite elements, to yield a compatible geometric algebraic variational principle. Likewise, Miles’ variational principle will be discretized in space and subsequently in time, using space and time finite elements, to yield a compatible geometric algebraic variational principle. The latter interim variational principle with continuous time will be essential to find the time discretization, ensuring that there is an underlying discrete boundary element structure and that the mesh movement is soundly integrated in the variational structure to ensure numerical stability. The simple oscillator will serve as an illustrative example for the discontinuous Galerkin finite element time discretization developed. The similarities between these discrete 2D shallow water and 3D water wave models at the fre e surface will be discussed. 3. The resulting discretizations will be validated against wave tank data from the Maritime Research Institute Netherlands (MARIN) and inhouse HeleShaw wave tank experiments (shown live). 4. Time permitting, extensions of the above results will be discussed. These concern water wave models with a vertical component of vorticity, experiments of a boresoliton rogue wave, modelling of breaking waves, or a waveenergy device using geometric roguewave focussing. 

12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
T Bridges (University of Surrey) Modulation of Water Waves Session: 5 The lectures will provide an introduction to the theory of "modulation" and its role in the derivation of model equations, such as the KdV equation, Boussinesq equation, KP equation, and Whitham modulation equations, and their role in the theory of water waves. The classical theory of modulation, such as Whitham modulation theory, will be reviewed, and a new approach will be introduced. The new approach is based on modulation of background flow. Methodology that is key to the theory is symmetry and conservation laws, relative equilibria, Hamiltonian and Lagrangian structures, multiple scale perturbation theory, and elementary differential geometry. By basing the theory on modulation of relative equilibria, new settings are discovered for the emergence of KdV and other modulation equations. For example, it is shown that the KdV equation can be a valid model for deep water as well as shallow water. The lectures are introductory, and no prior knowledge is assumed. 

15:00 to 15:30  Afternoon Tea  Served at INI  
15:30 to 16:30 
T Bridges (University of Surrey) Modulation of Water Waves Session: 5 Continued 