The nonlinear shallow water equations and the Green-Naghdi equations are the most commonly used models to describe coastal flows. A natural question is therefore to investigate their behavior at the shoreline, i.e. when the water depth vanishes. For the nonlinear shallow water equations, this problem is closely related to the vacuum problem for compressible Euler equations, recently solved by Jang-Masmoudi and Coutand-Shkoller. For the Green-Naghdi equation, the analysis is of a different nature due to the presence of linear and nonlinear dispersive terms. We will show in this talk how to address this problem.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on "Recent progress on surface and internal waves models" (2017)

Using constrained minimisation and a decomposition in Fourier space, we prove that the Kadomtsev-Petviashvili (KPI) equation modified with the exact dispersion relation from the gravity-capillary water-wave problem admits a family of small solitary solutions, approximating these of the standard KPI equation. The KPI equation, as well as its fully dispersive counterpart, describes gravity-capillary waves with strong surface tension. This is joint work with Mark Groves, Saarbrücken

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on "Recent progress on surface and internal waves models" (2017)

We will motivate and study a model for the propagation of surface gravity waves, which can be viewed as a fully nonlinear bi-directional Whitham equation. This model belongs to a family of systems of Green-Naghdi type with modified frequency dispersion. We will discuss the well-posedness of such systems, as well as the existence of solitary waves. The talk will be based on a work in collaboration with Samer Israwi and Raafat Talhouk (Beirut) and another in collaboration with Dag Nilsson and Erik Wahlén (Lund)

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on "Recent progress on surface and internal waves models" (2017)

We consider the classical gravity-capillary water-wave problem in its usual formulation as a three-dimensional free-boundary problem for the Euler equations for a perfect fluid. A solitary wave is a solution representing a wave which moves in a fixed direction with constant speed and without change of shape; it is fully localised if its profile decays to the undisturbed state of the water in every horizontal direction. The existence of fully localised solitary waves has been predicted on the basis of simpler model equations, namely the Kadomtsev-Petviashvili (KP) equation in the case of strong surface tension and the Davey-Stewartson (DS) system in the case of weak surface tension. In this talk we confirm the existence of such waves as solutions to the full water-wave problem and give rigorous justification for the use of the model equations.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on "Recent progress on surface and internal waves models" (2017)

In this talk we will review some long time existence results for the abcd-Boussinesq systems. We will discuss both the Sobolev and the nonlocalized, bore-type initial data cases. The main idea in order to get a priori estimates is to symmetrize the family of systems of equations verified by the frequencies of magnitude 2^{j} of the unknowns for each j¡Ý0. For the bore-type case, an additional decomposition of the initial data into low-high frequencies is needed in order to tackle the infinite-energy aspect of these kind of data.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on "Recent progress on surface and internal waves models" (2017)

We justify rigorously an Isobe-Kakinuma model for water waves as a higher order shallow water approximation in the case of a flat bottom. It is known that the full water wave equations are approximated by the shallow water equations with an error of order $delta^2$, where $delta$ is a small nondimensional parameter defined as the ratio of the typical wavelength to the mean depth. The Green-Naghdi equations are known as higher order approximate equations to the water wave equations with an error of order $delta^4$. In this talk I report that the Isobe-Kakinuma model is a much higher approximation to the water wave equations with an error of order $delta^6$.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on "Recent progress on surface and internal waves models" (2017)

We will discuss modified scattering properties, for small Solutions and/or in the vicinity of a solitary waves for model dispersive equations in dimension one. We will mainly focus on the modified Korteweg de Vries equation and the cubic Nonlinear Schrodinger equation with potential. Joint works with P. Germain and F. Pusateri.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on "Recent progress on surface and internal waves models" (2017)

The Euler-Korteweg equations are a modification of the Euler equations that takes into account capillary effects. In the general case they form a quasi-linear system that can be recast as a degenerate Schr ̈odinger type equation. Local well-posedness (in subcritical Sobolev spaces) was obtained by Benzoni-Danchin-Descombes in any space dimension, however, except in some special case (semi-linear with particular pressure) no global well- posedness is known. We prove here that under a natural stability condition on the pressure, global well-posedness holds in dimension d ¡Ý 3 for small irrotational initial data. The proof is based on a modified energy estimate, standard dispersive properties if d ¡Ý 5, and a careful study of the nonlinear structure of the quadratic terms in dimension 3 and 4 involving the theory of space time resonance.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on "Recent progress on surface and internal waves models" (2017)

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on "Recent progress on surface and internal waves models" (2017)

Large amplitude internal waves in a three-layer flow confined between two rigid walls will be examined in this talk. The mathematical model under consideration arises as a particular case of the multi-layer model proposed by Choi (2000) and is an extension of the two-layer MCC (Miyata-Choi-Camassa) model. The model can be derived without imposing any smallness assumption on the wave amplitudes and is well-suited to describe internal waves within a strongly nonlinear regime. We will investigate its solitary-wave solutions and unveil some of their properties by carrying out a critical point analysis of the underlying dynamical system.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on "Recent progress on surface and internal waves models" (2017)

We establish the existence of solitary waves for two classes of two-layers systems modeling the propagation of internal waves. More precisely we consider the Boussinesq-Full dispersion system and the Intermediate Long Wave (ILW) system together with its Benjamin-Ono (B0) limit. This is work in progress with Jaime Angulo Pava (USP)

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on "Recent progress on surface and internal waves models" (2017)