Well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity

Haspot, Boris

Annales de l'Institut Fourier, Tome 62 (2012) no. 5, pp. 1717-1763.

Ce papier est dédié à l’étude de Cauchy pour le système de Navier-Stokes non homogène dans ${ℝ}^N$ avec $N\ge 2$. Nous adressons la question du caractère bien posé pour des données initiales grandes et petites ayant une régularité critique dans des espaces de Besov aussi proches que possible de ceux utilisés par Cannone, Meyer et Planchon pour Navier Stokes incompressible (où $u_0\in B_{p,r}^{\frac{N}{p}-1}$ avec $1\le p<+\infty ,1\le r\le +\infty$). Cela améliore l’analyse classique où la vitesse initiale $u_0$ est supposée appartenir à $B_{p,1}^{\frac{N}{p}-1}$ de telle manière que la vitesse $u$ reste Lipschitz. Notre résultat utilise de nouvelles estimées pour l’équation de transport introduites par Bahouri, Chemin et Danchin lorsque la vitesse $u$ n’est pas nécessairement Lipschitz mais seulement log Lipschitz. De plus, cela donne une première réponse de résultat au problème des solutions autosimilaires.

This paper is dedicated to the study of the initial value problem for density dependent incompressible viscous fluids in ${ℝ}^N$ with $N\ge 2$. We address the question of well-posedness for large and small initial data having critical Besov regularity in functional spaces as close as possible to the ones imposed in the incompressible Navier Stokes system by Cannone, Meyer and Planchon (where $u_0\in B_{p,r}^{\frac{N}{p}-1}$ with $1\le p<+\infty ,1\le r\le +\infty$). This improves the classical analysis where $u_0$ is considered belonging in $B_{p,1}^{\frac{N}{p}-1}$ such that the velocity $u$ remains Lipschitz. Our result relies on a new a priori estimate for transport equation introduce by Bahouri, Chemin and Danchin when the velocity $u$ is not necessary Lipschitz but only log Lipschitz. Furthermore it gives a first kind of answer to the problem of self-similar solution.

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