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

Isabelle Gallagher
, Ayman Moussa

École Polytechnique

PSL research University & IPSL
UFR de mathématiques
Laboratoire de Probabilités et Modèles Aléatoires

Research output: Contribution to journal › Article › peer-review

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’ 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.

Translated title of the contribution	Le problème de Cauchy pour les systèmes paraboliques quasi-linéaires: une nouvelle approche
Original language	English
Pages (from-to)	1633-1676
Number of pages	44
Journal	Journal de l'Ecole Polytechnique - Mathematiques
Volume	12
DOIs	https://doi.org/10.5802/jep.319
Publication status	Published - 1 Jan 2025

Keywords

Littlewood-Paley
Petrovskii’s condition
Quasi-linear parabolic systems
paraproduct

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10.5802/jep.319

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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",

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AU - Moussa, Ayman

N1 - Publisher Copyright: © Les auteurs, 2025.

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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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SP - 1633

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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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Keywords

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