We prove generalized Gaussian bounds for the large time behavior of finite difference approximations of the transport equation. The goal of the talk is to review the connection with the central limit theorem in probability theory, some history of the problem and to present the general methodology leading to sharp estimates for the Green's function. The talk is based on a joint work with Grégory Faye.
We shall discuss how standard models in oceanography can be rigorously justified as asymptotic models, here in the shallow water regime, focusing on the general strategy and mathematical tools which are involved. Most of the discussion will be devoted to the justification of the Saint-Venant system for homogeneous potential flows with a free surface. Yet going through this somewhat standard...
In the context of nearshore wave energy facilities, we have tackled, with David Lannes and Lisl Weynans, the interaction of waves with a floating structure immersed in a 2D fluid. Some difficulties come from the presence of several surfaces: the surface of the sea and the contact surface between the structure and the fluid. The horizontal plane is decomposed into two regions: the exterior...
In this work, a novel numerical algorithm is introduced for the study of nonlinear interactions between free-surface shallow-water flows and a partly immersed floating object. The object's motion may be either prescribed, or computed as a response to the hydrodynamic forcing. Away from the object, the nonlinear hyperbolic shallow-water equations are used, while the description of the flow...
Les interactions océan-atmosphère (OA) jouent un rôle important dans de nombreux phénomènes, tels que cyclones tropicaux ou dynamique du climat. La représentation de ces interactions au sein d’un système de modélisation OA consiste principalement à évaluer les flux échangés entre les deux milieux, et à les imposer à l’interface air-mer.
Cet exposé abordera quelques questions mathématiques...
I will show here how the coupling between numerical modelling of glaciers and landslides associated with the analysis of the generated seismic waves makes it possible to recover unique information on the source processes. These waves provide a unique tool to constrain the models by providing detailed information on the dynamics of landslides and iceberg calving. This approach allows...
I will show here how the coupling between numerical modelling of glaciers and landslides associated with the analysis of the generated seismic waves makes it possible to recover unique information on the source processes. These waves provide a unique tool to constrain the models by providing detailed information on the dynamics of landslides and iceberg calving. This approach allows...
We study the well-posedness of initial boundary value problems for the linear Schrödinger equations on a half space. The boundary data lie in a (allegedly optimal) Bourgain type Sobolev space, which allows to include Neuman and transparent boundary conditions in the analysis. Strichartz estimates (in $L^2$) are obtained thanks to an explicit solution formula. In the case of Dirichlet boundary...
We explore the boundaries of damping estimates by comparing and contrasting two closely related models of combustion, the Majda and ZND models. We show that singularities form in the unweighted Lipschitz norm in finite time on both sides of the shock for both models, extending classical results of John and Liu to suitable variable coefficient systems. On the other hand, we show some energy...
We shall present some results on the existence of smooth branches of travelling waves for the 2D nonlinear Schrödinger equation parametrized by the speed. In the limit of small speed (joint works with E. Pacherie), the travelling wave has two well separated vortices and we prove that these are the only minimizers of the energy for fixed momentum. In the limit where the speed is close to the...
We shall discuss how standard models in oceanography can be rigorously justified as asymptotic models, here in the shallow water regime, focusing on the general strategy and mathematical tools which are involved. Most of the discussion will be devoted to the justification of the Saint-Venant system for homogeneous potential flows with a free surface. Yet going through this somewhat standard...
The goal of this talk is to study the stability of finite differences schemes for scalar hyperbolic initial boundary value problem. It is based on the GKS theory (introduced by Gustafsson, Kreiss and Sundström) and it deals with the Kreiss-Lopantinskii determinant. We will use complex analysis and geometric consideration to find zeros of this determinant and conclude on the stability of the scheme.
The classical PML approach is first applied to the linearised Korteweg-de Vries equation. These equations are not always stable, the main obstruction being the classical condition found in the literature on PMLs that we recover in our analysis. We introduce two alternative strategies to design absorbing boundary conditions. We start from studying hyperbolic relaxation of the Korteweg-de Vries...
