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82 results on '"Frédéric Rousset"'

15. Stability of equilibria uniformly in the inviscid limit for the Navier-Stokes-Poisson system

Author

Frédéric Rousset and Changzhen Sun

Subjects
Physics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Mathematics::Analysis of PDEs, Conservative vector field, 01 natural sciences, Stability (probability), Physics::Fluid Dynamics, 010101 applied mathematics, Viscosity, Mathematics - Analysis of PDEs, Inviscid flow, Compressibility, Limit (mathematics), 0101 mathematics, Constant (mathematics), Mathematical Physics, Analysis, Energy (signal processing)
Abstract

We prove a stability result of constant equilibria for the three-dimensional Navier-Stokes-Poisson system uniform in the inviscid limit. We allow the initial density to be close to a constant and the potential part of the initial velocity to be small independently of the rescaled viscosity parameter $\varepsilon$ while the incompressible part of the initial velocity is assumed to be small compared to $\varepsilon$. We then get a unique global smooth solution. We also prove a uniform in $\varepsilon$ time decay rate for these solutions. Our approach allows to combine the parabolic energy estimates that are efficient for the viscous equation at $\varepsilon$ fixed and the dispersive techniques (dispersive estimates and normal form transformation) that are useful for the inviscid irrotational system., Comment: Accepted version. In the current version, some typos are fixed and a sketchy proof of the case for the general pressure and the general viscosity is added in Section 6

Published
2021
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16. Uniform regularity for the compressible Navier-Stokes system with low Mach number in domains with boundaries

Author

Nader Masmoudi, Frédéric Rousset, Changzhen Sun, New York University [Abu Dhabi], NYU System (NYU), New York University [New York] (NYU), Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), ANR-18-CE40-0027,SingFlows,Ecoulements avec singularités : couches limites, filaments de vortex, interaction vague-structure(2018), and ANR-18-CE40-0020,ODA,Ondes déterministes et aléatoires(2018)

Subjects
SLIP, Boundary layer, Uniform regularity, Fast oscillation, Applied Mathematics, General Mathematics, INCOMPRESSIBLE LIMIT, EULER EQUATIONS, SINGULAR LIMITS, [MATH]Mathematics [math], Low Mach number limit
Abstract

We establish uniform with respect to the Mach number regularity estimates for the isentropic compressible Navier-Stokes system in smooth domains with Navier-slip condition on the boundary in the general case of ill-prepared initial data. To match the boundary layer effects due to the fast oscillations and the ill-prepared initial data assumption, we prove uniform estimates in an anisotropic functional framework with only one normal derivative close to the boundary. This allows to prove the local existence of a strong solution on a time interval independent of the Mach number and to justify the incompressible limit through a simple compactness argument.; Nous établissons des estimations de régularité uniformes par rapport au nombre de Mach pour le système de Navier-Stokes compressible isentropique dans les domaines réguliers avec condition de Navier au bord dans le cas général de données initiales mal préparées. Pour être cohérent avec les effets de couche limite dus aux oscillations rapides et à l'hypothèse de données initiales mal préparées, nous prouvons des estimations uniformes dans un cadre fonctionnel anisotrope avec une seule dérivée normale proche du bord. Ceci permet de prouver l'existence locale d'une solution forte sur un intervalle de temps indépendant du nombre de Mach et de justifier la limite incompressible par un argument de compacité simple.

Published
2022
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17. On the linearized Vlasov-Poisson system on the whole space around stable homogeneous equilibria

Author

Toan T. Nguyen, Daniel Han-Kwan, Frédéric Rousset, Centre de Mathématiques Laurent Schwartz (CMLS), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), ANR-19-CE40-0004,SALVE,Singularités dans des limites asymptotiques d'équations de Vlasov(2019), ANR-18-CE40-0027,SingFlows,Ecoulements avec singularités : couches limites, filaments de vortex, interaction vague-structure(2018), and ANR-18-CE40-0020,ODA,Ondes déterministes et aléatoires(2018)

Subjects
Physics, 010102 general mathematics, Complex system, Statistical and Nonlinear Physics, Space (mathematics), 01 natural sciences, 010101 applied mathematics, Mathematics - Analysis of PDEs, Homogeneous, Physical space, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Poisson system, Mathematical Physics, Analysis of PDEs (math.AP), Mathematical physics
Abstract

We study the linearized Vlasov–Poisson system around suitably stable homogeneous equilibria on $${\mathbb {R}}^d\times {\mathbb {R}}^d$$ (for any $$d \ge 1$$ ) and establish dispersive $$L^\infty $$ decay estimates in the physical space.

Published
2021
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19. Error estimates of finite difference schemes for the Korteweg–de Vries equation

Author

Clémentine Courtès, Frédéric Lagoutière, and Frédéric Rousset

Subjects
Pure mathematics, Applied Mathematics, General Mathematics, Numerical analysis, Finite difference, Order (ring theory), 010103 numerical & computational mathematics, 01 natural sciences, 010101 applied mathematics, Sobolev space, Computational Mathematics, Rate of convergence, Convergence (routing), Initial value problem, 0101 mathematics, Korteweg–de Vries equation, Mathematics
Abstract

This article deals with the numerical analysis of the Cauchy problem for the Korteweg–de Vries equation with a finite difference scheme. We consider the explicit Rusanov scheme for the hyperbolic flux term and a 4-point $\theta $-scheme for the dispersive term. We prove the convergence under a hyperbolic Courant–Friedrichs–Lewy condition when $\theta \geq \frac{1}{2}$ and under an ‘Airy’ Courant–Friedrichs–Lewy condition when $\theta

Published
2018
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20. The nonlinear Schrödinger equation with a potential

Author

Fabio Pusateri, Frédéric Rousset, and Pierre Germain

Subjects
Physics, Multilinear map, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Phase (waves), 01 natural sciences, Nonlinear system, symbols.namesake, Matrix (mathematics), Fourier transform, 0103 physical sciences, Bound state, symbols, 010307 mathematical physics, Scattering theory, 0101 mathematics, Nonlinear Schrödinger equation, Mathematical Physics, Analysis
Abstract

We consider the cubic nonlinear Schrodinger equation with a potential in one space dimension. Under the assumptions that the potential is generic, sufficiently localized, with no bound states, we obtain the long-time asymptotic behavior of small solutions. In particular, we prove that, as time goes to infinity, solutions exhibit nonlinear phase corrections that depend on the scattering matrix associated to the potential. The proof of our result is based on the use of the distorted Fourier transform – the so-called Weyl–Kodaira–Titchmarsh theory – a precise understanding of the “nonlinear spectral measure” associated to the equation, and nonlinear stationary phase arguments and multilinear estimates in this distorted setting.

Published
2018
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21. Asymptotic stability of solitons for mKdV

Author

Pierre Germain, Frédéric Rousset, and Fabio Pusateri

Subjects
Scattering, General Mathematics, 010102 general mathematics, Mathematical analysis, Perturbation (astronomy), 01 natural sciences, Polynomial decay, Nonlinear Sciences::Exactly Solvable and Integrable Systems, Exponential stability, 0103 physical sciences, Initial value problem, 010307 mathematical physics, 0101 mathematics, Nonlinear Sciences::Pattern Formation and Solitons, Real line, Mathematics
Abstract

We prove a full asymptotic stability result for solitary wave solutions of the mKdV equation. We consider small perturbations of solitary waves with polynomial decay at infinity and prove that solutions of the Cauchy problem evolving from such data tend uniformly, on the real line, to another solitary wave as time goes to infinity. We describe precisely the asymptotics of the perturbation behind the solitary wave showing that it satisfies a nonlinearly modified scattering behavior. This latter part of our result relies on a precise study of the asymptotic behavior of small solutions of the mKdV equation.

Published
2016
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22. Asymptotic stability of equilibria for screened Vlasov-Poisson systems via pointwise dispersive estimates

Author

Daniel Han-Kwan, Frédéric Rousset, Toan T. Nguyen, Centre de Mathématiques Laurent Schwartz (CMLS), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), Pennsylvania State University (Penn State), and Penn State System

Subjects
General Physics and Astronomy, Poisson distribution, Space (mathematics), 01 natural sciences, symbols.namesake, Mathematics - Analysis of PDEs, Exponential stability, 0103 physical sciences, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Landau damping, 0101 mathematics, Mathematical Physics, Mathematics, Pointwise, Smoothness (probability theory), Partial differential equation, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Lipschitz continuity, symbols, 010307 mathematical physics, Geometry and Topology, Analysis, Analysis of PDEs (math.AP)
Abstract

We revisit the proof of Landau damping near stable homogenous equilibria of Vlasov-Poisson systems with screened interactions in the whole space $\mathbb{R}^d$ (for $d\geq3$) that was first established by Bedrossian, Masmoudi and Mouhot. Our proof follows a Lagrangian approach and relies on precise pointwise in time dispersive estimates in the physical space for the linearized problem that should be of independent interest. This allows to cut down the smoothness of the initial data required in Bedrossian at al. (roughly, we only need Lipschitz regularity). Moreover, the time decay estimates we prove are essentially sharp, being the same as those for free transport, up to a logarithmic correction., Comment: 25 pages, minor typos fixed

Published
2019
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23. Long time estimates for the Vlasov-Maxwell system in the non-relativistic limit

Author

Daniel Han-Kwan, Toan T. Nguyen, Frédéric Rousset, Centre de Mathématiques Laurent Schwartz (CMLS), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Pennsylvania State University (Penn State), Penn State System, Laboratoire de Mathématiques d'Orsay (LMO), and Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Physics, Polynomial, 010102 general mathematics, Mathematical analysis, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Complex system, FOS: Physical sciences, Inverse, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), 01 natural sciences, Speed of light (cellular automaton), Stability (probability), 010101 applied mathematics, Sobolev space, Arbitrarily large, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Limit (mathematics), 0101 mathematics, Mathematical Physics, Analysis of PDEs (math.AP)
Abstract

International audience; In this paper, we study the Vlasov-Maxwell system in the non-relativistic limit, that is in the regime where the speed of light is a very large parameter. We consider data lying in the vicinity of homogeneous equilibria that are stable in the sense of Penrose (for the Vlasov-Poisson system), and prove Sobolev stability estimates that are valid for times which are polynomial in terms of the speed of light and of the inverse of size of initial perturbations. We build a kind of higher-order Vlasov-Darwin approximation which allows us to reach arbitrarily large powers of the speed of light.

Published
2018
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24. Quasineutral limit for Vlasov-Poisson with Penrose stable data

Author

Daniel Han-Kwan, Frédéric Rousset, Centre de Mathématiques Laurent Schwartz (CMLS), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Han-Kwan, Daniel, and Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)

Subjects
Physics, General Mathematics, 010102 general mathematics, Mathematical analysis, Dirac (software), Vlasov equation, Poisson distribution, 01 natural sciences, Stability (probability), 010101 applied mathematics, Sobolev space, symbols.namesake, Mathematics - Analysis of PDEs, Distribution (mathematics), Physics::Plasma Physics, Physics::Space Physics, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Limit (mathematics), [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Variable (mathematics)
Abstract

International audience; We study the quasineutral limit of a Vlasov-Poisson system that describes the dynamics of ions in a plasma. We handle data with Sobolev regularity under the sharp assumption that the profile of the initial data in the velocity variable satisfies a Penrose stability condition. As a by-product of our analysis, we obtain a well-posedness theory for the limit equation (which is a Vlasov equation with Dirac distribution as interaction kernel) for such data.

Published
2016
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25. Global Regularity for Some Oldroyd-B Type Models

Author

Frédéric Rousset, Tarek M. Elgindi, Courant Institute of Mathematical Sciences [New York] ( CIMS ), New York University [New York], Université Paris-Sud 11 - Faculté des Sciences ( UP11 UFR Sciences ), Université Paris-Sud - Paris 11 ( UP11 ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Courant Institute of Mathematical Sciences [New York] (CIMS), New York University [New York] (NYU), NYU System (NYU)-NYU System (NYU), Université Paris-Sud - Paris 11 - Faculté des Sciences (UP11 UFR Sciences), Université Paris-Sud - Paris 11 (UP11), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), DMS-1211806 and DMS-0807347, National Science Foundation, AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)

Subjects
Pure mathematics, flows, Applied Mathematics, General Mathematics, Mathematics::Analysis of PDEs, Structure (category theory), Type (model theory), Vorticity, viscoelastic fluids, Physics::Fluid Dynamics, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Transformation (function), Dimension (vector space), well-posedness, equations, hydrodynamics, viscosity, Besov-spaces, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Oldroyd-B model, Euler-Bousinesq system, Analysis of PDEs (math.AP), Mathematics
Abstract

We investigate some critical models for visco-elastic flows of Oldroyd-B type in dimension 2. We use a transformation that exploits the Oldroyd-B structure to prove an L∞ bound on the vorticity which allows us to prove global regularity for our systems.© 2014 Wiley Periodicals, Inc.

Published
2015
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26. Linear inviscid damping and enhanced viscous dissipation of shear flows by using the conjugate operator method

Author

Avy Soffer, Frédéric Rousset, Emmanuel Grenier, and Toan T. Nguyen

Subjects
010102 general mathematics, Mathematical analysis, 01 natural sciences, Physics::Fluid Dynamics, Viscosity, symbols.namesake, Operator (computer programming), Shear (geology), Inviscid flow, 0103 physical sciences, symbols, Euler's formula, 010307 mathematical physics, 0101 mathematics, Hamiltonian (quantum mechanics), Analysis, Schrödinger's cat, Conjugate, Mathematics
Abstract

We study the large time behavior of solutions to two-dimensional Euler and Navier-Stokes equations linearized about shear flows of the mixing layer type in the unbounded channel T × R . Under a simple spectral stability assumption on a self-adjoint operator, we prove a local form of the linear inviscid damping that is uniform with respect to small viscosity. We also prove a local form of the enhanced viscous dissipation that takes place at times of order ν − 1 / 3 , ν being the small viscosity. To prove these results, we use a Hamiltonian approach, following the conjugate operator method developed in the study of Schrodinger operators, combined with a hypocoercivity argument to handle the viscous case.

Published
2020
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27. Inviscid limit for free-surface Navier-Stokes equations

Author

Frédéric Rousset

Subjects
010101 applied mathematics, Physics, Inviscid flow, Free surface, 010102 general mathematics, Mathematical analysis, General Medicine, Limit (mathematics), 0101 mathematics, Navier–Stokes equations, 01 natural sciences
Published
2014
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28. Trigonometric integrators for quasilinear wave equations

Author

Jianfeng Lu, Ludwig J. Gauckler, Frédéric Rousset, Jeremy L. Marzuola, and Katharina Schratz

Subjects
65M15, 65P10, 65L70, 65M20, Algebra and Number Theory, Discretization, Applied Mathematics, Numerical analysis, Semiclassical physics, 010103 numerical & computational mathematics, Numerical Analysis (math.NA), Wave equation, Exponential integrator, 01 natural sciences, 010101 applied mathematics, Computational Mathematics, Convergence (routing), FOS: Mathematics, Applied mathematics, Mathematics - Numerical Analysis, 0101 mathematics, Trigonometry, Spectral method, Mathematics
Abstract

Trigonometric time integrators are introduced as a class of explicit numerical methods for quasilinear wave equations. Second-order convergence for the semi-discretization in time with these integrators is shown for a sufficiently regular exact solution. The time integrators are also combined with a Fourier spectral method into a fully discrete scheme, for which error bounds are provided without requiring any CFL-type coupling of the discretization parameters. The proofs of the error bounds are based on energy techniques and on the semiclassical G\aa rding inequality., Comment: 33 pages, 2 figures, comments welcome!! Version 3 has updates, typo corrections and simplifications due to referee reports Version 1 contains an extra example removed to shorten the paper overall

Published
2017
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29. Uniform regularity and vanishing viscosity limit for the free surface Navier-Stokes equations

Author

Nader Masmoudi, Frédéric Rousset, Courant Institute of Mathematical Sciences [New York] (CIMS), New York University [New York] (NYU), NYU System (NYU)-NYU System (NYU), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), 1211806, Directorate for Mathematical and Physical Sciences, Courant Institute of Mathematical Sciences [New York] ( CIMS ), New York University [New York], Laboratoire de Mathématiques d'Orsay ( LM-Orsay ), Université Paris-Sud - Paris 11 ( UP11 ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)

Subjects
Mechanical Engineering, 010102 general mathematics, Mathematical analysis, Mathematics::Analysis of PDEs, Navier–Stokes existence and smoothness, Non-dimensionalization and scaling of the Navier–Stokes equations, 16. Peace & justice, 01 natural sciences, Euler equations, 010101 applied mathematics, Sobolev space, Physics::Fluid Dynamics, symbols.namesake, Mathematics - Analysis of PDEs, Mathematics (miscellaneous), [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Inviscid flow, Hagen–Poiseuille flow from the Navier–Stokes equations, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Reynolds-averaged Navier–Stokes equations, Navier–Stokes equations, Analysis, Mathematics
Abstract

International audience; We study the inviscid limit of the free boundary Navier-Stokes equations. We prove the existence of solutions on a uniform time interval by using a suitable functional framework based on Sobolev conormal spaces. This allows us to use a strong compactness argument to justify the inviscid limit. Our approach does not rely on the justification of asymptotic expansions. In particular, we get a new existence result for the Euler equations with free surface from the one for Navier-Stokes.

Published
2017
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30. Long wave limit for Schrodinger maps

Author

Frédéric Rousset and Pierre Germain

Subjects
Mathematics - Differential Geometry, Applied Mathematics, General Mathematics, 010102 general mathematics, Type (model theory), Submanifold, 01 natural sciences, symbols.namesake, Mathematics - Analysis of PDEs, Differential Geometry (math.DG), General theory, FOS: Mathematics, Tangent space, symbols, Antiferromagnetism, Limit (mathematics), Mathematics::Differential Geometry, 0101 mathematics, Korteweg–de Vries equation, 35Q53, 35Q41, Mathematics::Symplectic Geometry, Schrödinger's cat, Analysis of PDEs (math.AP), Mathematical physics, Mathematics
Abstract

We study long wave limits for general Schrodinger maps systems into Kahler manifolds with a constraining potential vanishing on a Lagrangian submanifold. We obtain KdV type systems set on the tangent space of the submanifold. Our general theory is applied to study the long wave limit of the Gross-Pitaevskii equation, and of the Landau-Lifshitz systems for ferromagnetic and antiferromagnetic chains., 67 pages

Published
2016

31. On a Constrained 2-D Navier-Stokes Equation

Author

Emanuele Caglioti, Mario Pulvirenti, Frédéric Rousset, Dipartimento di Matematica 'Guido Castelnuovo' [Roma I] (Sapienza University of Rome), Università degli Studi di Roma 'La Sapienza' = Sapienza University [Rome] (UNIROMA), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Dipartimento di Matematica 'Guido Castelnuovo' [Roma I] ( Sapienza University of Rome ), Università degli Studi di Roma 'La Sapienza' [Rome], Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire Jean Alexandre Dieudonné ( JAD ), Université Nice Sophia Antipolis ( UNS ), Université Côte d'Azur ( UCA ) -Université Côte d'Azur ( UCA ) -Centre National de la Recherche Scientifique ( CNRS ), Università degli Studi di Roma 'La Sapienza' = Sapienza University [Rome], AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), and Université Nice Sophia Antipolis (... - 2019) (UNS)

Subjects
perfect fluids dynamics, statistical-mechanics description, Mathematics::Analysis of PDEs, 01 natural sciences, Physics::Fluid Dynamics, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Analysis of PDEs, Planar, Sobolev inqualities, 0103 physical sciences, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Lagrangian coherent structures, Navier stokes, 0101 mathematics, 010306 general physics, Mathematical Physics, Physics, 010102 general mathematics, Dynamics (mechanics), Statistical and Nonlinear Physics, Moment of inertia, Classical mechanics, 35Q30, 76D03, 76D05, Constrained equation, 2-dimensional Euler equations, equilibrium state, Analysis of PDEs (math.AP)
Abstract

International audience; The planar Navier-Stokes equation exhibits, in absence of external forces, a trivial asymptotics in time. Nevertheless the appearence of coherent structures suggests non-trivial intermediate asymptotics which should be explained in terms of the equation itself. Motivated by the separation of the different time scales observed in the dynamics of the Navier-Stokes equation, we study the well-posedness and asymptotic behaviour of a constrained equation which neglects the variation of the energy and moment of inertia.

Published
2009
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32. Stability of oscillating boundary layers in rotating fluids

Author

Nader Masmoudi, Frédéric Rousset, Courant Institute of Mathematical Sciences [New York] ( CIMS ), New York University [New York], Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Courant Institute of Mathematical Sciences [New York] (CIMS), New York University [New York] (NYU), NYU System (NYU)-NYU System (NYU), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Collin, Maryse, Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
Nonlinear instability, regularity, General Mathematics, Fluid layer, Geometry, integrability, 01 natural sciences, Non linear stability, Physics::Fluid Dynamics, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], perturbations, Navier-stokes equations, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, 10. No inequality, Mathematics, initial data, 010102 general mathematics, nonlinear instability, 010101 applied mathematics, flow, viscosity, limits, Humanities, 3d euler
Abstract

International audience; We prove the linear and non-linear stability of oscillating Ekman boundary layers for rotating fluids in the so-called ill-prepared case under a spectral hypothesis. Here, we deal with the case where the viscosity and the Rossby number are both equal to epsilon. This study generalizes the study of [23] where a smallness condition was imposed and the study of [26] where the well-prepared case was treated.

Published
2008
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33. Landau damping in Sobolev spaces for the Vlasov-HMF model

Author

Frédéric Rousset, Erwan Faou, Invariant Preserving SOlvers (IPSO), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), European Project: 279389,EC:FP7:ERC,ERC-2011-StG_20101014,GEOPARDI(2011), Invariant Preserving SOlvers ( IPSO ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique ( Inria ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques d'Orsay ( LM-Orsay ), Université Paris-Sud - Paris 11 ( UP11 ) -Centre National de la Recherche Scientifique ( CNRS ), European Project : 279389,EC:FP7:ERC,ERC-2011-StG_20101014,GEOPARDI ( 2011 ), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-AGROCAMPUS OUEST, Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Inria Rennes – Bretagne Atlantique, Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), and Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)

Subjects
Stability criterion, 01 natural sciences, Landau damping, Scattering, symbols.namesake, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Mathematics (miscellaneous), 0103 physical sciences, 35Q83 35P25, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Vlasov equations, Mathematical physics, Physics, Mechanical Engineering, 010102 general mathematics, Sobolev space, Nonlinear system, HMF model, Homogeneous, symbols, 010307 mathematical physics, Hamiltonian (quantum mechanics), Analysis
Abstract

International audience; We consider the Vlasov-HMF (Hamiltonian Mean-Field) model. We consider solutions starting in a small Sobolev neighborhood of a spatially homogeneous state satisfying a linearized stability criterion (Penrose criterion). We prove that these solutions exhibit a scattering behavior to a modified state, which implies a nonlinear Landau damping effect with polynomial rate of damping.

Published
2016
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34. Stability of Large Amplitude Ekman-Hartmann Boundary Layers in MHD : The Case of Ill-Prepared Data

Author

Frédéric Rousset

Subjects
Solenoidal vector field, Mathematical analysis, Boundary (topology), Statistical and Nonlinear Physics, Mixed boundary condition, Physics::Fluid Dynamics, symbols.namesake, Classical mechanics, Dirichlet boundary condition, No-slip condition, symbols, Vector field, Boundary value problem, Magnetohydrodynamics, Mathematical Physics, Mathematics
Abstract

In this paper, we study an incompressible highly rotating fluid submitted to a high magnetic field between two planes with a Dirichlet boundary condition. We investigate the nonlinear stability of Ekman-Hartmann boundary layers under a spectral assumption for general initial data; this means that the data can be chosen as an arbitrary (but smooth enough) three-dimensional divergence free vector field independent of the small parameter.

Published
2005
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35. Characteristic boundary layers in real vanishing viscosity limits

Author

Frédéric Rousset

Subjects
Applied Mathematics, Mathematical analysis, Mixed boundary condition, Boundary layer thickness, Initial boundary value problems, Robin boundary condition, Physics::Fluid Dynamics, symbols.namesake, Asymptotic expansions, Inviscid flow, Dirichlet boundary condition, Dissipative hyperbolic-parabolic systems, Neumann boundary condition, symbols, No-slip condition, Boundary value problem, Stability, Analysis, Mathematics
Abstract

In this paper, we study the vanishing viscosity limit of initial boundary value problems for one-dimensional mixed hyperbolic–parabolic systems when the boundary is characteristic for both the viscous and the inviscid systems: in particular, we assume that an eigenvalue of the inviscid system vanishes uniformly. We prove the stability of boundary layers expansions in small time (i.e before shocks for the inviscid system) as long as the amplitude of the boundary layers remains sufficiently small. In particular, by using Lagrangian coordinates, we apply our result to physical systems like gasdynamics and magnetohydrodynamics with homogeneous Dirichlet condition for the velocity at the boundary.

Published
2005
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36. Viscous perturbations of marginally stable Euler flow and finite-time Melnikov theory

Author

Frédéric Rousset, Emmanuel Grenier, Björn Sandstede, and Christopher K. R. T. Jones

Subjects
Applied Mathematics, General Physics and Astronomy, Statistical and Nonlinear Physics, Mechanics, Viscous liquid, Vorticity, Euler equations, Physics::Fluid Dynamics, symbols.namesake, Classical mechanics, Inviscid flow, Barotropic fluid, Euler's formula, symbols, Two-dimensional flow, Barotropic vorticity equation, Mathematical Physics, Mathematics
Abstract

The effect of small viscous dissipation on Lagrangian transport in two-dimensional vorticity conserving fluid flows motivates this work. If the inviscid equation admits a base flow in which different fluid regions are divided by separatrices, then transport between these regions is afforded by the splitting of separatrices caused by viscous dissipation. Finite-time Melnikov theory allows us to measure the splitting distance of separatrices provided the perturbed velocity field of the viscous fluid flow stays sufficiently close to vorticity-conserving base flow over sufficiently long time intervals. In this paper, we derive the necessary long-term estimates of solutions to Euler’s equation and to the barotropic vorticity equation upon adding viscous perturbations and forcing. We discover that a certain stability condition on the unperturbed flow is sufficient to guarantee these long time estimates.

Published
2004
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37. Stability of Large Ekman Boundary Layers in Rotating Fluids

Author

Frédéric Rousset

Subjects
Mechanical Engineering, Mathematical analysis, Zero (complex analysis), Boundary (topology), Geometry, Stability (probability), Physics::Geophysics, Physics::Fluid Dynamics, Rossby number, Nonlinear system, Mathematics (miscellaneous), Ekman number, Balanced flow, Physics::Atmospheric and Oceanic Physics, Analysis, Linear stability, Mathematics
Abstract

The aim of this paper is to investigate the stability of Ekman boundary layers for rotating fluids when the Ekman number and the Rossby number go to zero. More precisely, we prove that spectral stability implies linear and nonlinear stabilities of approximate solutions. In particular, we replace the smallness condition obtained with energy methods in [5] by a weaker spectral condition which is sharp.

Published
2004
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38. Large mixed Ekman–Hartmann boundary layers in magnetohydrodynamics

Author

Frédéric Rousset

Subjects
Partial differential equation, Applied Mathematics, Mathematical analysis, General Physics and Astronomy, Reynolds number, Boundary (topology), Statistical and Nonlinear Physics, Physics::Fluid Dynamics, Boundary layer, symbols.namesake, Nonlinear system, Classical mechanics, Flow (mathematics), symbols, Magnetohydrodynamics, Arch, Mathematical Physics, Mathematics
Abstract

In this paper, we study the nonlinear stability of Ekman–Hartmann type boundary layers in a rotating magnetohydrodynamics flow under a sharp spectral assumption. This generalizes the result of Desjardins et al (1999 Nonlinearity 12 181–99) obtained under a smallness assumption on a Reynolds number and the result of Rousset (2003 Arch. Rat. Mech. Anal. in press) about the stability of Ekman layers.

Published
2003
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39. Stability of small amplitude boundary layers for mixed hyperbolic-parabolic systems

Author

Frédéric Rousset

Subjects
Applied Mathematics, General Mathematics, Mathematical analysis, Boundary (topology), Mixed boundary condition, Robin boundary condition, symbols.namesake, Inviscid flow, Dirichlet boundary condition, Neumann boundary condition, symbols, No-slip condition, Boundary value problem, Mathematics
Abstract

We consider an initial boundary value problem for a symmetrizable mixed hyperbolic-parabolic system of conservation laws with a small viscosity ε \varepsilon , u t ε + F ( u ε ) x = ε ( B ( u ε ) u x ε ) x . u^\varepsilon _t+F(u^\varepsilon )_x =\varepsilon (B(u^\varepsilon ) u^\varepsilon _x )_x . When the boundary is noncharacteristic for both the viscous and the inviscid system, and the boundary condition dissipative, we show that u ε u^\varepsilon converges to a solution of the inviscid system before the formation of shocks if the amplitude of the boundary layer is sufficiently small. This generalizes previous results obtained for B B invertible and the linear study of Serre and Zumbrun obtained for a pure Dirichlet’s boundary condition.

Published
2003
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40. Nonlinear Stability of Semidiscrete Shock Waves

Author

Sylvie Benzoni-Gavage, Pierre Huot, and Frédéric Rousset

Subjects
Shock wave, Conservation law, Partial differential equation, Dynamical systems theory, Laplace transform, Differential equation, Applied Mathematics, Mathematical analysis, Computational Mathematics, symbols.namesake, Linearization, Green's function, symbols, Analysis, Mathematics
Abstract

The orbital stability of possibly large semidiscrete shock waves is considered. These waves are traveling wave solutions of discrete in space and continuous in time systems of conservation laws, which constitute a class of lattice dynamical systems (LDSs). The underlying lattice $\Delta x \mathbb{Z}$ is by nature not invariant by change of frame. Thus semidiscrete shock waves cannot really be transformed into stationary waves, unlike other kinds of approximate shock waves (e.g., viscous or relaxation shocks). This implies that the linearization of the LDS about a given semidiscrete shock wave yields a nonautonomous linear LDS, which cannot be tackled by means of Laplace transform in time. However, viewing the LDS as a finite-difference PDE and performing afterall the change of frame, the profile becomes a stationary solution of the transformed equation. Then, linearizing about the profile, we get an evolution finite-difference PDE in which the spatial operator L, a delayed and advanced differential operat...

Published
2003
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41. Multi-solitons and related solutions for the water-waves system

Author

Nikolay Tzvetkov, Frédéric Rousset, Mei Ming, Academy of Mathematics and Systems Science [Beijing], Laboratoire de Mathématiques d'Orsay ( LM-Orsay ), Université Paris-Sud - Paris 11 ( UP11 ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Analyse, Géométrie et Modélisation ( AGM ), Université de Cergy Pontoise ( UCP ), Université Paris-Seine-Université Paris-Seine-Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Analyse, Géométrie et Modélisation (AGM - UMR 8088), Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)

Subjects
Work (thermodynamics), media_common.quotation_subject, 35Q35, 35B36, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], multi-solitons, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Remainder, Nonlinear Sciences::Pattern Formation and Solitons, media_common, Variable (mathematics), Mathematics, Cauchy problem, Applied Mathematics, Mathematical analysis, water waves, Decoupling (cosmology), Infinity, Computational Mathematics, Nonlinear system, Nonlinear Sciences::Exactly Solvable and Integrable Systems, solitary waves, Line (geometry), Analysis, Analysis of PDEs (math.AP)
Abstract

The main result of this work is the construction of multi-solitons solutions, that is, solutions that are time asymptotics to a sum of decoupling solitary waves for the full water-waves system with surface tension. Our approach uses the construction of a precise approximate solution that is controlled by using spectral information for each solitary wave and a bootstrap argument for the control of the remainder. For this stage, we need only few properties about the nonlinear Cauchy problem, namely, local well-posedness for very smooth data. We also use a similar construction to refine our previous result [Invent. Math., 184 (2011), pp. 257--388] about the nonlinear instability of one-dimensional solitary waves in the two-dimensional model: we prove the existence of semiglobal solutions that depend nontrivially of the transverse variable and that tend to the line solitary wave as time goes to infinity.

Published
2015
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42. Generic types and transitions in hyperbolic initial–boundary-value problems

Author

Kevin Zumbrun, Frédéric Rousset, Sylvie Benzoni-Gavage, and Denis Serre

Subjects
Constant coefficients, General Mathematics, Hyperbolic function, Mathematical analysis, Free boundary problem, Boundary value problem, Hyperbolic partial differential equation, Robin boundary condition, Mathematics, Hyperbolic equilibrium point, Inverse hyperbolic function
Abstract

The stability of linear initial–boundary-value problems for hyperbolic systems (with constant coefficients) is linked to the zeros of the so-called Lopatinskii determinant. Depending on the location of these zeros, problems may be either unstable, strongly stable or weakly stable. The first two classes are known to be ‘open’, in the sense that the instability or the strong stability persists under a small change of coefficients in the differential operator and/or in the boundary condition.Here we show that a third open class exists, which we call ‘weakly stable of real type’. Many examples of physical or mathematical interest depend on one or more parameters, and the determination of the stability class as a function of these parameters usually needs an involved computation. We simplify it by characterizing the transitions from one open class to another one. These boundaries are easier to determine since they must solve some overdetermined algebraic system.Applications to the wave equation, linear elasticity, shock waves and phase boundaries in fluid mechanics are given.

Published
2002
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44. Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries II

Author

Daniel Han-Kwan, David Gérard-Varet, Frédéric Rousset, Institut de Mathématiques de Jussieu (IMJ), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Centre de Mathématiques Laurent Schwartz (CMLS), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques d'Orsay (LM-Orsay), and Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Physics, Work (thermodynamics), General Mathematics, Astrophysics::High Energy Astrophysical Phenomena, 010102 general mathematics, Mathematical analysis, Boundary (topology), 01 natural sciences, Isothermal process, 010101 applied mathematics, symbols.namesake, Mathematics - Analysis of PDEs, Euler's formula, symbols, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Astrophysics::Solar and Stellar Astrophysics, Supersonic speed, Outflow, Limit (mathematics), 0101 mathematics, Outflow boundary, Astrophysics::Galaxy Astrophysics, Analysis of PDEs (math.AP)
Abstract

International audience; In this paper, we study the quasineutral limit of the isothermal Euler-Poisson equation for ions, in a domain with boundary. This is a follow-up to our previous work \cite{GVHKR}, devoted to no-penetration as well as subsonic outflow boundary conditions. We focus here on the case of supersonic outflow velocities. The structure of the boundary layers and the stabilization mechanism are different.

Published
2014
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45. Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries

Author

David Gérard-Varet, Frédéric Rousset, Daniel Han-Kwan, Institut de Mathématiques de Jussieu ( IMJ ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ), Département de Mathématiques et Applications - ENS Paris ( DMA ), École normale supérieure - Paris ( ENS Paris ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Mathématiques de Jussieu (IMJ), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), and École normale supérieure - Paris (ENS Paris)

Subjects
General Mathematics, Boundary (topology), Electron, boundary layers, 01 natural sciences, linearized modulated energy, Domain (mathematical analysis), symbols.namesake, Mathematics - Analysis of PDEs, quasineutral limit, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Physics::Plasma Physics, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Limit (mathematics), 0101 mathematics, Mathematics, Partial differential equation, 010102 general mathematics, Mathematical analysis, Isothermal Euler-Poisson system, Plasma, 010101 applied mathematics, Massless particle, Euler's formula, symbols, 76N15, 76N25, 35Q35, Analysis of PDEs (math.AP)
Abstract

We study the quasineutral limit of the isothermal Euler-Poisson system describing a plasma made of ions and massless electrons. The analysis is achieved in a domain of $\R^3$ and thus extends former results by Cordier and Grenier [Comm. Partial Differential Equations, 25 (2000), pp.~1099--1113], who dealt with the same problem in a one-dimensional domain without boundary., Comment: 32 pages

Published
2013

46. Stability and instability of the KdV solitary wave under the KP-I flow

Author

Frédéric Rousset, Nikolay Tzvetkov, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Analyse, Géométrie et Modélisation (AGM - UMR 8088), Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Analyse, Géométrie et Modélisation ( AGM ), Université de Cergy Pontoise ( UCP ), Université Paris-Seine-Université Paris-Seine-Centre National de la Recherche Scientifique ( CNRS ), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
Physics, media_common.quotation_subject, 010102 general mathematics, Mathematics::Analysis of PDEs, Statistical and Nonlinear Physics, Infinity, 01 natural sciences, Stability (probability), Instability, 010101 applied mathematics, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Nonlinear Sciences::Exactly Solvable and Integrable Systems, transverse nonlinear instability, Flow (mathematics), equations, solitons, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Soliton, 0101 mathematics, Korteweg–de Vries equation, Nonlinear Sciences::Pattern Formation and Solitons, Mathematical Physics, media_common, Mathematical physics
Abstract

We consider the KP-I and gKP-I equations in $\mathbb{R}\times (\mathbb{R}/2\pi \mathbb{Z})$. We prove that the KdV soliton with subcritical speed $0c^*$, in the spirit of the work by Duyckaerts and Merle \cite{DM}, we sharpen our previous instability result and construct a global solution which is different from the solitary wave and its translates and which converges to the solitary wave as time goes to infinity. This last result also holds for the gKP-I equation.

Published
2012
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47. Uniform Regularity for the Navier-Stokes Equation with Navier Boundary Condition

Author

Nader Masmoudi, Frédéric Rousset, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)

Subjects
domain, Mathematics::Analysis of PDEs, Directional derivative, system, 01 natural sciences, Physics::Fluid Dynamics, symbols.namesake, Mathematics (miscellaneous), Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], FOS: Mathematics, Uniform boundedness, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, 0101 mathematics, Mathematics, Mechanical Engineering, initial data, 010102 general mathematics, Mathematical analysis, existence, Euler system, stability, 16. Peace & justice, Euler equations, 010101 applied mathematics, Sobolev space, Compact space, Vanishing viscosity limit, Bounded function, rotating fluids, symbols, layers, Analysis, Analysis of PDEs (math.AP), incompressible limit
Abstract

International audience; We prove that there exists an interval of time which is uniform in the vanishing viscosity limit and for which the Navier-Stokes equation with the Navier boundary condition has a strong solution. This solution is uniformly bounded in a conormal Sobolev space and has only one normal derivative bounded in L (a). This allows us to obtain the vanishing viscosity limit to the incompressible Euler system from a strong compactness argument.

Published
2012
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48. Oscillatory integral estimates and global well-posedness for the 2D Boussinesq equation

Author

Frédéric Rousset, Nikolay Tzvetkov, Luiz Gustavo Farah, Departamento de Matemática [Minas Gerais] (DM - UFMG), Universidade Federal de Minas Gerais, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Analyse, Géométrie et Modélisation (AGM - UMR 8088), Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY), Departamento de Matemática [Minas Gerais] ( DM - UFMG ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Analyse, Géométrie et Modélisation ( AGM ), Université de Cergy Pontoise ( UCP ), Université Paris-Seine-Université Paris-Seine-Centre National de la Recherche Scientifique ( CNRS ), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
Small data, 2D Boussinesq equation, General Mathematics, 010102 general mathematics, Mathematical analysis, 01 natural sciences, multidimensional oscillatory integral, global well-posedness, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], 0103 physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 010307 mathematical physics, 0101 mathematics, Oscillatory integral, Well posedness, Mathematics
Abstract

International audience; We prove a multidimensional oscillatory integral estimate using a variant ofthestationaryphasemethod. Asaconsequence, weobtainglobalwell-posednessfor small data for the 2D Boussinesq equation u (tt) + (u (xx) + u (2) - u) (xx) - u (yy) = 0.

Published
2012
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49. Global well-posedness for the Euler-Boussinesq system with axisymmetric data

Author

Taoufik Hmidi, Frédéric Rousset, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)

Subjects
Axisymmetric flows, 010102 general mathematics, Mathematical analysis, Rotational symmetry, Characteristic equation, Mathematics::Analysis of PDEs, Global well-posedness, 01 natural sciences, Euler equations, 010101 applied mathematics, Harmonic analysis, Physics::Fluid Dynamics, symbols.namesake, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], FOS: Mathematics, Euler's formula, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Well posedness, Analysis, Analysis of PDEs (math.AP), Mathematics
Abstract

In this paper we prove the global well-posedness for the three-dimensional Euler-Boussinesq system with axisymmetric initial data without swirl. This system couples the Euler equation with a transport-diffusion equation governing the temperature., Comment: 39 pages

Published
2011
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50. Global well-posedness for Euler-Boussinesq system with critical dissipation

Author

Sahbi Keraani, Frédéric Rousset, Taoufik Hmidi, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille, Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire Paul Painlevé - UMR 8524 ( LPP ), and Université de Lille-Centre National de la Recherche Scientifique ( CNRS )

Subjects
Mathematics::Analysis of PDEs, 01 natural sciences, 76D03 (35B33 35Q35 76D05), Physics::Fluid Dynamics, symbols.namesake, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Fractional diffusion, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, 76D03, 35B33, 35Q35, 76D05, Dissipation, Euler equations, 010101 applied mathematics, symbols, Euler's formula, Compressibility, Convection–diffusion equation, Analysis, Well posedness, Analysis of PDEs (math.AP)
Abstract

In this paper we study a fractional diffusion Boussinesq model which couples the incompressible Euler equation for the velocity and a transport equation with fractional diffusion for the temperature. We prove global well-posedness results., Comment: 24 pages

Published
2011
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