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About global existence and asymptotic behavior for two dimensional gravity water waves

Thomas Alazard (2012/2013)

Séminaire Laurent Schwartz — EDP et applications

The main result of this talk is a global existence theorem for the water waves equation with smooth, small, and decaying at infinity Cauchy data. We obtain moreover an asymptotic description in physical coordinates of the solution, which shows that modified scattering holds.The proof is based on a bootstrap argument involving L 2 and L estimates. The L 2 bounds are proved in the paper [5]. They rely on a normal forms paradifferential method allowing one to obtain energy estimates on the Eulerian formulation...

Accurate numerical discretizations of non-conservative hyperbolic systems

Ulrik Skre Fjordholm, Siddhartha Mishra (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We present an alternative framework for designing efficient numerical schemes for non-conservative hyperbolic systems. This approach is based on the design of entropy conservative discretizations and suitable numerical diffusion operators that mimic the effect of underlying viscous mechanisms. This approach is illustrated by considering two model non-conservative systems: Lagrangian gas dynamics in non-conservative form and a form of isothermal Euler equations. Numerical experiments demonstrating...

Accurate numerical discretizations of non-conservative hyperbolic systems

Ulrik Skre Fjordholm, Siddhartha Mishra (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

We present an alternative framework for designing efficient numerical schemes for non-conservative hyperbolic systems. This approach is based on the design of entropy conservative discretizations and suitable numerical diffusion operators that mimic the effect of underlying viscous mechanisms. This approach is illustrated by considering two model non-conservative systems: Lagrangian gas dynamics in non-conservative form and a form of isothermal Euler equations. Numerical experiments demonstrating...

Control of underwater vehicles in inviscid fluids

Rodrigo Lecaros, Lionel Rosier (2014)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we investigate the controllability of an underwater vehicle immersed in an infinite volume of an inviscid fluid whose flow is assumed to be irrotational. Taking as control input the flow of the fluid through a part of the boundary of the rigid body, we obtain a finite-dimensional system similar to Kirchhoff laws in which the control input appears through both linear terms (with time derivative) and bilinear terms. Applying Coron’s return method, we establish some local controllability...

Controllability of 3D incompressible Euler equations by a finite-dimensional external force

Hayk Nersisyan (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we study the control system associated with the incompressible 3D Euler system. We show that the velocity field and pressure of the fluid are exactly controllable in projections by the same finite-dimensional control. Moreover, the velocity is approximately controllable. We also prove that 3D Euler system is not exactly controllable by a finite-dimensional external force.

Derivation and mathematical analysis of a nonlocal model for large amplitude internal waves

David Lannes (2008/2009)

Séminaire Équations aux dérivées partielles

This note is devoted to the study of a bi-fluid generalization of the nonlinear shallow-water equations. It describes the evolution of the interface between two fluids of different densities. In the case of a two-dimensional interface, this systems contains unexpected nonlocal terms (that are of course not present in the usual one-fluid shallow water equations). We show here how to derive this systems from the two-fluid Euler equations and then show that it is locally well-posed.

Dissipative Euler flows and Onsager's conjecture

Camillo De Lellis, László Székelyhidi (2014)

Journal of the European Mathematical Society

Building upon the techniques introduced in [15], for any θ < 1 10 we construct periodic weak solutions of the incompressible Euler equations which dissipate the total kinetic energy and are Hölder-continuous with exponent θ . A famous conjecture of Onsager states the existence of such dissipative solutions with any Hölder exponent θ < 1 3 . Our theorem is the first result in this direction.

Équations de transport à coefficient dont le gradient est donné par une intégrale singulière

François Bouchut, Gianluca Crippa (2007/2008)

Séminaire Équations aux dérivées partielles

Nous rappelons tout d’abord l’approche maintenant classique de renormalisation pour établir l’unicité des solutions faibles des équations de transport linéaires, en mentionnant les résultats récents qui s’y rattachent. Ensuite, nous montrons comment l’approche alternative introduite par Crippa et DeLellis estimant directement le flot lagrangien permet d’obtenir des résultats nouveaux. Nous établissons l’existence et l’unicité du flot associé à une équation de transport dont le coefficient a un gradient...

Global in Time Stability of Steady Shocks in Nozzles

Jeffrey Rauch, Chunjing Xie, Zhouping Xin (2011/2012)

Séminaire Laurent Schwartz — EDP et applications

We prove global dynamical stability of steady transonic shock solutions in divergent quasi-one-dimensional nozzles. One of the key improvements compared with previous results is that we assume neither the smallness of the slope of the nozzle nor the weakness of the shock strength. A key ingredient of the proof are the derivation a exponentially decaying energy estimates for a linearized problem.

Interface model coupling via prescribed local flux balance

Annalisa Ambroso, Christophe Chalons, Frédéric Coquel, Thomas Galié (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

This paper deals with the non-conservative coupling of two one-dimensional barotropic Euler systems at an interface at x = 0. The closure pressure laws differ in the domains x &lt; 0 and x &gt; 0, and a Dirac source term concentrated at x = 0 models singular pressure losses. We propose two numerical methods. The first one relies on ghost state reconstructions at the interface while the second is based on a suitable relaxation framework. Both methods satisfy a well-balanced property for stationary...

Lower and upper bounds for the Rayleigh conductivity of a perforated plate

S. Laurens, S. Tordeux, A. Bendali, M. Fares, P. R. Kotiuga (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Lower and upper bounds for the Rayleigh conductivity of a perforation in a thick plate are usually derived from intuitive approximations and by physical reasoning. This paper addresses a mathematical justification of these approaches. As a byproduct of the rigorous handling of these issues, some improvements to previous bounds for axisymmetric holes are given as well as new estimates for tilted perforations. The main techniques are a proper use of the Dirichlet and Kelvin variational principlesin...

On power series solutions for the Euler equation, and the Behr–Nečas–Wu initial datum

Carlo Morosi, Mario Pernici, Livio Pizzocchero (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider the Euler equation for an incompressible fluid on a three dimensional torus, and the construction of its solution as a power series in time. We point out some general facts on this subject, from convergence issues for the power series to the role of symmetries of the initial datum. We then turn the attention to a paper by Behr, Nečas and Wu, ESAIM: M2AN 35 (2001) 229–238; here, the authors chose a very simple Fourier polynomial as an initial datum for the Euler equation and analyzed...

On some implicit and semi-implicit staggered schemes for the shallow water and Euler equations

R. Herbin, W. Kheriji, J.-C. Latché (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper, we propose implicit and semi-implicit in time finite volume schemes for the barotropic Euler equations (hence, as a particular case, for the shallow water equations) and for the full Euler equations, based on staggered discretizations. For structured meshes, we use the MAC finite volume scheme, and, for general mixed quadrangular/hexahedral and simplicial meshes, we use the discrete unknowns of the Rannacher−Turek or Crouzeix−Raviart finite elements. We first show that a solution...

Relaxation of the incompressible porous media equation

László Székelyhidi Jr (2012)

Annales scientifiques de l'École Normale Supérieure

It was shown recently by Córdoba, Faraco and Gancedo in [1] that the 2D porous media equation admits weak solutions with compact support in time. The proof, based on the convex integration framework developed for the incompressible Euler equations in [4], uses ideas from the theory of laminates, in particular T 4 configurations. In this note we calculate the explicit relaxation of IPM, thus avoiding T 4 configurations. We then use this to construct weak solutions to the unstable interface problem (the...

Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid

Olivier Glass, Franck Sueur, Takéo Takahashi (2012)

Annales scientifiques de l'École Normale Supérieure

We consider the motion of a rigid body immersed in an incompressible perfect fluid which occupies a three-dimensional bounded domain. For such a system the Cauchy problem is well-posed locally in time if the initial velocity of the fluid is in the Hölder space C 1 , r . In this paper we prove that the smoothness of the motion of the rigid body may be only limited by the smoothness of the boundaries (of the body and of the domain). In particular for analytic boundaries the motion of the rigid body is analytic...

Sur l’équation de Prandtl

David Gérard-Varet, Emmanuel Dormy (2008/2009)

Séminaire Équations aux dérivées partielles

L’objet de cette note est le problème de Cauchy pour l’équation de Prandtl, dans des espaces de régularité Sobolev. Nous discutons de façon synthétique des résultats récents [4], établissant le caractère fortement linéairement mal posé de ce problème.

Sur un problème de stabilité posé en optique géométrique non linéaire surcritique

Christophe Cheverry (2008/2009)

Séminaire Équations aux dérivées partielles

Cet exposé s’intéresse à un modèle réaliste issu de la mécanique des fluides. L’objectif est de montrer qu’il est possible de traiter dans un tel cadre des problèmes d’instabilité soulevés par la propagation de singularités qualifiées de surcritiques. D’abord, nous introduisons le modèle (équations de type Navier-Stokes) et ses motivations (questions liées à la propagation d’oscillations en régime turbulent). Ensuite, nous présentons deux résultats (relatifs au caractère bien posé d’un problème...

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