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Timetable (TWWW02)

Theory of Water Waves (Spitalfields Day)

Wednesday 23rd July 2014

Wednesday 23rd July 2014
10:30 to 11:10 Registration and Tea & Coffee
11:10 to 11:15 Welcome from John Toland (INI Director)
11:15 to 12:00 G Schneider (Universität Stuttgart)
Validity and non-validity of the NLS approximation for the water wave problem - Recent developments and open problems
We consider the 2D water wave problem in case of finite depth with and without surface tension. We are interested in the validity of the NLS approximation for the description of surface water waves. After giving an overview about positive results we explain that in case of small surface tension there are situations where the NLS approximation fails to describe the water wave problem correctly.
12:00 to 14:00 Lunch - Sandwich lunch at INI
14:00 to 14:45 Interface singularities for the Euler equations
In fluid dynamics, a "splash" singularity occurs when a locally smooth interface self-intersects in finite-time. It is now well-known that solutions to the water waves equations (and a host of other one-phase fluid interface models) has a finite-time splash singularity. By means of elementary arguments, we prove that such a singularity cannot occur in finite-time for vortex sheet evolution (or two-fluid interfaces). This means that the evolving interface must lose regularity prior to self-intersection. We give a proof by contradiction: we assume that such a singularity does indeed occur in finite-time. Based on this assumption, we find precise blow-up rates for the components of the velocity gradient which, in turn, allows us to characterize the geometry of the evolving interface just prior to self-intersection. The constraints on the geometry then lead to an impossible outcome, giving the contradiction. This is joint work D. Coutand.
14:45 to 15:30 E Varvaruca (University of Reading)
Singularities of steady free surface water flows
We present some recent results, based on a geometric analysis approach, that provide a characterization of all possible singularities in two related free-boundary problems in hydrodynamics: that of steady two-dimensional gravity water waves and that of steady three-dimensional axisymmetric water flows under gravity. In the 2D problem, we outline a modern proof, using a blow-up analysis based on a monotonicity formula and a frequency formula, of the Stokes conjecture from 1880, which asserts that at any stagnation point on the free surface of a 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 some restrictive assumptions and by somewhat ad-hoc methods.) The new approach extends easily to the case when the effects of vorticity in the flow are included. Moreover, we explain how the methods can be adapted to the 3D axisymmetric problem, where several different types of singularities are possible, depending on whether one is dealing with a stagnation point, a point on the axis of symmetry, or both (in the case of the origin). For example, in the case of the origin, there are only two possible types of singular asymptotic behaviour: one is a conical singularity called ``Garabedian corner flow", and the other is a flat degenerate point; while in the case of points on the axis of symmetry different from the origin, cusps are the only possible singularities. These results were obtained in joint works with Georg Weiss (Dusseldorf).
15:30 to 16:15 Afternoon Tea
16:15 to 17:00 Three-dimensional water waves
The existence of solitary-wave solutions to the three-dimensional water-wave problem with is predicted by the Kadomtsev-Petviashvili (KP) equation for strong surface tension and Davey-Stewartson (DS) equation for weak surface tension.The term solitary wave describes any solution which has a pulse-like profile in its direction of propagation, and these model equations admit three types of solitary waves. A line solitary wave is spatially homogeneous in the direction transverse to its direction of propagation, while a periodically modulated solitary wave is periodic in the transverse direction. A fully localised solitary wave on the other hand decays to zero in all spatial directions. In this talk I outline mathematical results which confirm the existence of all three types of solitary wave for the full gravity-capillary water-wave problem in its usual formulation as a free-boundary problem for the Euler equations. Both strong and weak surface tension are treated. The line solitary waves are found by establishing the existence of a low-dimensional invariant manifold containing homoclinic orbits. The periodically modulated solitary waves are created when a line solitary wave undergoes a dimension-breaking bifurcation in which it spontaneously loses its spatial homogeneity in the transverse direction; an infinite-dimensional version of the Lyapunov centre theorem is the main ingredient in the existence theorem. The fully localised solitary waves are obtained by finding critical points of a variational functional.
17:00 to 18:00 Welcome Wine Reception
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons