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

François BouchutGianluca Crippa

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...

Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient

Francois BouchutFrancois JamesSimona Mancini — 2005

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

The Cauchy problem for a multidimensional linear transport equation with discontinuous coefficient is investigated. Provided the coefficient satisfies a one-sided Lipschitz condition, existence, uniqueness and weak stability of solutions are obtained for either the conservative backward problem or the advective forward problem by duality. Specific uniqueness criteria are introduced for the backward conservation equation since weak solutions are not unique. A main point is the introduction of a generalized...

A Roe-type scheme for two-phase shallow granular flows over variable topography

Marica PelantiFrançois BouchutAnne Mangeney — 2008

ESAIM: Mathematical Modelling and Numerical Analysis

We study a depth-averaged model of gravity-driven flows made of solid grains and fluid, moving over variable basal surface. In particular, we are interested in applications to geophysical flows such as avalanches and debris flows, which typically contain both solid material and interstitial fluid. The model system consists of mass and momentum balance equations for the solid and fluid components, coupled together by both conservative and non-conservative terms involving the derivatives of the...

An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment

François BouchutTomás Morales de Luna — 2008

ESAIM: Mathematical Modelling and Numerical Analysis

We consider the system of partial differential equations governing the one-dimensional flow of two superposed immiscible layers of shallow water. The difficulty in this system comes from the coupling terms involving some derivatives of the unknowns that make the system nonconservative, and eventually nonhyperbolic. Due to these terms, a numerical scheme obtained by performing an arbitrary scheme to each layer, and using time-splitting or other similar techniques leads to instabilities in...

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