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Solutions globales ( - ∞ < t < + ∞ ) des systèmes paraboliques de lois de conservation

Denis Serre (1998)

Annales de l'institut Fourier

Nous considérons ici des solutions particulières des systèmes paraboliques de lois de conservation dans le domaine x > 0 ou bien pour x ∈ ℝ : ∂ t u + ∂ x f ( u ) = ∂ x 2 u . Nous faisons l’hypothèse que le système réduit ∂ t u + ∂ x f ( u ) = 0 est hyperbolique. Notre but est la description de l’interaction d’ondes simples, mono-dimensionnelles, le plus souvent deux ondes exactement. L’une d’elle, au moins, est une onde de choc (pour le système réduit) visqueuse (pour le système parabolique). Il y a donc a priori un champ caractéristique vraiment non linéaire....

Some new results in multiphase geometrical optics

Olof Runborg (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In order to accommodate solutions with multiple phases, corresponding to crossing rays, we formulate geometrical optics for the scalar wave equation as a kinetic transport equation set in phase space. If the maximum number of phases is finite and known a priori we can recover the exact multiphase solution from an associated system of moment equations, closed by an assumption on the form of the density function in the kinetic equation. We consider two different closure assumptions based on delta...

Some regularizing methods for transport equations and the regularity of solutions to scalar conservation laws

Pierre-Emmanuel Jabin (2008/2009)

Séminaire Équations aux dérivées partielles

We study several regularizing methods, stationary phase or averaging lemmas for instance. Depending on the regularity assumptions that are made, we show that they can either be derived one from the other or that they lead to different results. Those are applied to Scalar Conservation Laws to precise and better explain the regularity of their solutions.

Some remarks on multidimensional systems of conservation laws

Alberto Bressan (2004)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

This note is concerned with the Cauchy problem for hyperbolic systems of conservation laws in several space dimensions. We first discuss an example of ill-posedness, for a special system having a radial symmetry property. Some conjectures are formulated, on the compactness of the set of flow maps generated by vector fields with bounded variation.

Stabilité L 1 d’ondes progressives de lois de conservation scalaires

Denis Serre (1998/1999)

Séminaire Équations aux dérivées partielles

A powerfull method has been developped in [2] for the study of L 1 -stability of travelling waves in conservation laws or more generally in equations which display L 1 -contractivity, maximum principle and mass conservation. We recall shortly the general procedure. We also show that it partly applies to the waves of a model of radiating gas. These waves have first been studied by Kawashima and Nishibata [5,6] in a different framework. Therefore, shock fronts for this model are stable under mild perturbations....

Stability of periodic stationary solutions of scalar conservation laws with space-periodic flux

Anne-Laure Dalibard (2011)

Journal of the European Mathematical Society

This article investigates the long-time behaviour of parabolic scalar conservation laws of the type ∂ t u + div y A ( y , u ) - Δ y u = 0 , where y ∈ ℝ N and the flux A is periodic in y . More specifically, we consider the case when the initial data is an L 1 disturbance of a stationary periodic solution. We show, under polynomial growth assumptions on the flux, that the difference between u and the stationary solution behaves in L 1 norm like a self-similar profile for large times. The proof uses a time and space change of variables which is...

Stable upwind schemes for the magnetic induction equation

Franz G. Fuchs, Kenneth H. Karlsen, Siddharta Mishra, Nils H. Risebro (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider the magnetic induction equation for the evolution of a magnetic field in a plasma where the velocity is given. The aim is to design a numerical scheme which also handles the divergence constraint in a suitable manner. We design and analyze an upwind scheme based on the symmetrized version of the equations in the non-conservative form. The scheme is shown to converge to a weak solution of the equations. Furthermore, the discrete divergence produced by the scheme is shown to be...

Stochastic averaging lemmas for kinetic equations

Pierre-Louis Lions, Benoît Perthame, Panagiotis E. Souganidis (2011/2012)

Séminaire Laurent Schwartz — EDP et applications

We develop a class of averaging lemmas for stochastic kinetic equations. The velocity is multiplied by a white noise which produces a remarkable change in time scale.Compared to the deterministic case and as far as we work in L 2 , the nature of regularity on averages is not changed in this stochastic kinetic equation and stays in the range of fractional Sobolev spaces at the price of an additional expectation. However all the exponents are changed; either time decay rates are slower (when the right...

Sur la stabilité des couches limites de viscosité

Denis Serre (2001)

Annales de l’institut Fourier

Pour un système parabolique de lois de conservation, nous considérons le problème mixte, dans le domaine x > 0 . Pour une condition de Dirichlet, le système admet en général des solutions stationnaires U ( x ) , qui tendent vers une limite en + ∞ . Ce sont les profils des couches limites, dans l’approximation du second ordre, pour le système hyperbolique du premier ordre sous-jacent. La stabilité de cette couche limite est liée à la stabilité linéaire asymptotique de U . On étudie celle-ci au moyen d’une fonction d’Evans,...

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