THE CAUCHY PROBLEM FOR QUASI-LINEAR PARABOLIC SYSTEMS REVISITED

Isabelle Gallagher; Ayman Moussa

doi:10.5802/jep.319

THE CAUCHY PROBLEM FOR QUASI-LINEAR PARABOLIC SYSTEMS REVISITED

Titre traduit de la contribution: Le problème de Cauchy pour les systèmes paraboliques quasi-linéaires: une nouvelle approche

Isabelle Gallagher
, Ayman Moussa

École Polytechnique

PSL research University & IPSL
Laboratoire de Probabilités et Modèles Aléatoires

Résultats de recherche: Contribution à un journal › Article › Revue par des pairs

Résumé

We study a class of parabolic quasilinear systems, in which the diffusion matrix is not uniformly elliptic, but satisfies the Petrovskii condition (positivity of eigenvalues’ real part). Local well-posedness is known since the work of Amann in the 90s, by a semi-group method. We first revisit these results in the context of Sobolev spaces modelled on L² and then explore the endpoint Besov case B_p,^d/p₁ . We also exemplify our method on the SKT system, showing the existence of local, non-negative, strong solutions.

Titre traduit de la contribution	Le problème de Cauchy pour les systèmes paraboliques quasi-linéaires: une nouvelle approche
langue originale	Anglais
Pages (de - à)	1633-1676
Nombre de pages	44
journal	Journal de l'Ecole Polytechnique - Mathematiques
Volume	12
Les DOIs	https://doi.org/10.5802/jep.319
état	Publié - 1 janv. 2025

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10.5802/jep.319

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title = "THE CAUCHY PROBLEM FOR QUASI-LINEAR PARABOLIC SYSTEMS REVISITED",

abstract = "We study a class of parabolic quasilinear systems, in which the diffusion matrix is not uniformly elliptic, but satisfies the Petrovskii condition (positivity of eigenvalues{\textquoteright} real part). Local well-posedness is known since the work of Amann in the 90s, by a semi-group method. We first revisit these results in the context of Sobolev spaces modelled on L2 and then explore the endpoint Besov case Bp,d/p1 . We also exemplify our method on the SKT system, showing the existence of local, non-negative, strong solutions.",

keywords = "Littlewood-Paley, Petrovskii{\textquoteright}s condition, Quasi-linear parabolic systems, paraproduct",

author = "Isabelle Gallagher and Ayman Moussa",

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N2 - We study a class of parabolic quasilinear systems, in which the diffusion matrix is not uniformly elliptic, but satisfies the Petrovskii condition (positivity of eigenvalues’ real part). Local well-posedness is known since the work of Amann in the 90s, by a semi-group method. We first revisit these results in the context of Sobolev spaces modelled on L2 and then explore the endpoint Besov case Bp,d/p1 . We also exemplify our method on the SKT system, showing the existence of local, non-negative, strong solutions.

AB - We study a class of parabolic quasilinear systems, in which the diffusion matrix is not uniformly elliptic, but satisfies the Petrovskii condition (positivity of eigenvalues’ real part). Local well-posedness is known since the work of Amann in the 90s, by a semi-group method. We first revisit these results in the context of Sobolev spaces modelled on L2 and then explore the endpoint Besov case Bp,d/p1 . We also exemplify our method on the SKT system, showing the existence of local, non-negative, strong solutions.

KW - Littlewood-Paley

KW - Petrovskii’s condition

KW - Quasi-linear parabolic systems

KW - paraproduct

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JO - Journal de l'Ecole Polytechnique - Mathematiques

JF - Journal de l'Ecole Polytechnique - Mathematiques

ER -

THE CAUCHY PROBLEM FOR QUASI-LINEAR PARABOLIC SYSTEMS REVISITED

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