Theory of Water Waves (Spitalfields Day)
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 nonvalidity 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.

INI 1  
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 selfintersects in finitetime. It is now wellknown that solutions to the water waves equations (and a host of other onephase fluid interface models) has a finitetime
splash singularity.
By means of elementary arguments, we prove that such a
singularity cannot occur in finitetime for vortex sheet evolution (or twofluid interfaces). This means that the evolving interface must lose regularity prior to
selfintersection.
We give a proof by contradiction: we assume that such a singularity does indeed occur in finitetime. Based on this assumption, we find precise blowup rates for the components of the velocity gradient which, in turn, allows us to characterize
the geometry of the evolving interface just prior to selfintersection. The constraints on the geometry then lead to an impossible outcome, giving the contradiction. This is joint work D. Coutand.

INI 1  
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 freeboundary problems in hydrodynamics: that of steady twodimensional gravity water waves and that of steady threedimensional axisymmetric water flows under gravity. In the 2D problem, we outline a modern proof, using a blowup 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
adhoc 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).

INI 1  
15:30 to 16:15  Afternoon Tea  
16:15 to 17:00 
Threedimensional water waves
The existence of solitarywave solutions to the threedimensional
waterwave problem with is predicted by the KadomtsevPetviashvili (KP)
equation for strong surface tension and DaveyStewartson (DS) equation
for weak surface tension.The term solitary wave describes any solution which
has a pulselike 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 gravitycapillary waterwave
problem in its usual formulation as a freeboundary 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 lowdimensional
invariant manifold containing homoclinic orbits. The periodically modulated solitary
waves are created when a line solitary wave undergoes a dimensionbreaking
bifurcation in which it spontaneously loses its spatial homogeneity in the transverse
direction; an infinitedimensional 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.

INI 1  
17:00 to 18:00  Welcome Wine Reception 