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

Pierre-Emmanuel Jabin

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.

Regularity in kinetic formulations via averaging lemmas

Pierre-Emmanuel JabinBenoît Perthame — 2002

ESAIM: Control, Optimisation and Calculus of Variations

We present a new class of averaging lemmas directly motivated by the question of regularity for different nonlinear equations or variational problems which admit a kinetic formulation. In particular they improve the known regularity for systems like γ = 3 in isentropic gas dynamics or in some variational problems arising in thin micromagnetic films. They also allow to obtain directly the best known regularizing effect in multidimensional scalar conservation laws. The new ingredient here is to use velocity...

Regularity in kinetic formulations via averaging lemmas

Pierre-Emmanuel JabinBenoît Perthame — 2010

ESAIM: Control, Optimisation and Calculus of Variations

We present a new class of averaging lemmas directly motivated by the question of regularity for different nonlinear equations or variational problems which admit a kinetic formulation. In particular they improve the known regularity for systems like γ = 3 in isentropic gas dynamics or in some variational problems arising in thin micromagnetic films. They also allow to obtain directly the best known regularizing effect in multidimensional scalar conservation laws. The new ingredient here is to...

Global stability of steady solutions for a model in virus dynamics

Hermano FridPierre-Emmanuel JabinBenoît Perthame — 2003

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

We consider a simple model for the immune system in which virus are able to undergo mutations and are in competition with leukocytes. These mutations are related to several other concepts which have been proposed in the literature like those of shape or of virulence – a continuous notion. For a given species, the system admits a globally attractive critical point. We prove that mutations do not affect this picture for small perturbations and under strong structural assumptions. Based on numerical...

Line-energy Ginzburg-Landau models : zero-energy states

Pierre-Emmanuel JabinFelix OttoBenoÎt Perthame — 2002

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider a class of two-dimensional Ginzburg-Landau problems which are characterized by energy density concentrations on a one-dimensional set. In this paper, we investigate the states of vanishing energy. We classify these zero-energy states in the whole space: They are either constant or a vortex. A bounded domain can sustain a zero-energy state only if the domain is a disk and the state a vortex. Our proof is based on specific entropies which lead to a kinetic formulation, and on a careful...

Global Stability of Steady Solutions for a Model in Virus Dynamics

Hermano FridPierre-Emmanuel JabinBenoît Perthame — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a simple model for the immune system in which virus are able to undergo mutations and are in competition with leukocytes. These mutations are related to several other concepts which have been proposed in the literature like those of shape or of virulence – a continuous notion. For a given species, the system admits a globally attractive critical point. We prove that mutations do not affect this picture for small perturbations and under strong structural assumptions. Based on numerical...

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