On Thursday morning 24 July, Mark Cooker gave a talk on violent wave motion on a short time scale. He began the talk by deriving a “pressure-impulse theory”, based on Euler’s equations of motion in an impact setting. The analysis is reduced to a BVP for the pressure, which is harmonic in the interior and satisfies mixed Dirichlet-Neumann conditions on the boundary. Explicit solutions were obtained and discussed. As part of the talk he gave a demonstration of the implication of the theory of impact by perturbing a glass of water. See the previous blog post for references.
On Thursday afternoon, 24 July, Nathan Totz, gave a talk on validity results for the NLS equation in 2D and 3D. The first half of the talk reviewed the validity proof of NLS in deep water, co-authored with Sijue Wu. It was a summary of their 2012 paper which can be found here. The second half discussed the validity of the hyperbolic NLS in deep water. The methodology included the use of Lagrangian coordinates, Clifford analysis, quaternionic representation of the Lagrangian position coordinates, normal form transformations, and energy estimates, leading to results in Sobolev spaces, including a priori bounds on the difference between the true and approximate solutions. In both the 2D and 3D case surface tension was neglected. A preprint by Totz on the 3D case can be found here.
Mark Groves spoke about 3D localized solutions of the water wave problem. There were two classes of waves of interest, Bond number greater than 1/3 (case A) and Bond number less than 1/3 (case B). In case A the solutions are truly localised, without oscillating tails, and decay algebraically to zero at infinity. In case B, oscillations occur and the decay is exponential to zero at infinity; an example is in the figure above. The appropriate simplified model in case A is steady KP-I, and in case B it is steady elliptic Davey-Stewartson. The main part of the talk was to show how the existence of these waves could be proved for the full water wave problem. The range of mathematical tools used were Luke’s Lagrangian variational principle, Lyapunov Centre Theorem in infinite dimensions, Legendre transform, Hamiltonian formulation, concentration compactness, and the fact that non-local terms are quasi-local. A key step was to show that the Lagrangian for the full water wave problem could be written in two parts where the first part is the Lagrangian for KP (as in de Brouard & Saut (1997)) or Davey-Stewartson (as in Cipolatti (1993)) plus a remainder. A byproduct of the proof is validity of steady KP and steady Davey-Stewartson. A recent paper on this theory is Buffoni, Groves, Sun & Wahlen (2013).
Eugen Varvaruca spoke on singularities of steady free surface flows. He starting by reviewing the history of the classic case of the largest periodic Stokes wave from Stokes conjecture to the proof of Amick, Fraenkel & Toland (1982). Then he reviewed the generalisation of this conjecture to include vorticity which culminated in Varvaruca (2009). The main part of the talk was on three-dimensional axisymmetric gravity waves without swirl. Stagnation points as well as points on the axis of symmetry were analyzed. The latter can have downward-pointing cusps. At stagnation points on the axis of symmetry the unique blowup profile found was the Garabedian pointed bubble. An asymptotic form of the velocity field is found. Concentration compactness is used. A recent paper on this is Varvaruca & Weiss (2014).
Steve Shkoller spoke about dynamic interface singularities in the Euler equations. He started by reviewing previous work on the “splash singularity”, showing how the result of Coutand & Shkoller (2014) extends to 3D, using different methods, a proof of Castro, Cordoba, Fefferman, Gancedo, Gomez-Serrano (2012). The main part of the talk was devoted to the case where there are two fluids. He started by summarizing the recent result of Fefferman, Ionescu & Lie (2013) which proved that splash singularities could not occur in this case. Then Steve presented a new proof of this result using different methods. The principal tool was to derive an ODE for the tangential derivative of the tangential component of the velocity difference. This ODE could then be analyzed exactly. A preprint on this work can be found here. The talk also included a range of videos of large breaking waves, an example is the Teahupoo video.
At the end of Guido Schneider‘s talk, Victor Shrira spoke about recent work of his on validity of NLS from a different perspective. He mentioned the recent paper On the highest non-breaking wave in a group: fully nonlinear water wave breathers versus weakly nonlinear theory. This paper was interested in the highest possible wave in a wave group. The question was addressed by comparing breather solutions of NLS with their counterparts in the full Euler equations. Breathers, the key basic elements of wave field evolution withing the NLS framework, are computed and extended to the full Euler equations. Direct numerical simulations of the full Euler equations were computed using both time-dependent conformal mapping and the HOSM approach. To create fully nonlinear counterparts to NLS breathers, the initial conditions for fully nonlinear simulations were taken in the form of a train of Stokes waves modulated in accordance with the exact breather solution of the NLS equation. The fact that the breather-like structures survive in the fully nonlinear simulations and behave qualitatively like the NLS framework reveals that the underlying physics of the self-focusing effect remains qualitatively the same.
The first talk of the Spitalfields Day was by Guido Schneider. He presented an introduction to the approximation scheme used to prove validity in finite depth. The full solution of the irrotational Euler equations was expressed in terms of an NLS component and a remainder. Then a PDE was derived for the remainder term, and then analyzed. Key tools were normal form transformations, energy estimates, and the Cauchy-Kowalevskaya theorem. The non-validity result was for water waves with surface tension and spatially periodic boundaries. There a 3-wave resonance was shown to nullify validity of NLS. Some discussion of validity of the 3-wave interaction was also presented, based on the preprint here. A preprint on the proof of validity for NLS can be found here.