On optimal Hölder regularity of solutions to the equation Δu + b ⋅ ∇u = 0 in two dimensions

Nam Q. Le

doi:10.1016/j.crma.2017.02.005

Partial differential equations

On optimal Hölder regularity of solutions to the equation Δu + b ⋅ ∇u = 0 in two dimensions

Presented by: Haïm Brézis

Nam Q. Le ¹

¹ Department of Mathematics, Indiana University, Bloomington, IN 47405, USA

Comptes Rendus. Mathématique, Volume 355 (2017) no. 4, pp. 439-446.

Abstract
Résumé

We show that for an $L^{2}$ drift b in two dimensions, if the Hardy norm of $div b$ is small, then the weak solutions to $Δ u + b \cdot \nabla u = 0$ have the same optimal Hölder regularity as in the case of divergence-free drift, that is, $u \in C_{loc}^{α}$ for all $α \in (0, 1)$ .

Nous démontrons que, pour une dérive $b \in L_{loc}^{2} (R^{2}; R^{2})$ , si la norme de Hardy de $div b$ est petite, alors les solutions faibles de $Δ u + b \cdot \nabla u = 0$ (en dimension deux) ont la même régularité Hölder que dans le cas de la dérive incompressible, c'est-à-dire que $u \in C_{loc}^{α}$ pour tout $α \in (0, 1)$ .

Received: 2016-09-05
Accepted: 2017-02-22
Published online: 2017-03-06

DOI: 10.1016/j.crma.2017.02.005

Author's affiliations:

Nam Q. Le ¹

¹ Department of Mathematics, Indiana University, Bloomington, IN 47405, USA

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     author = {Nam Q. Le},
     title = {On optimal {H\"older} regularity of solutions to the equation {\ensuremath{\Delta}\protect\emph{u}\,+\,\protect\emph{b}\,\ensuremath{\cdot}\,\ensuremath{\nabla}\protect\emph{u}\,=\,0} in two dimensions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {439--446},
     publisher = {Elsevier},
     volume = {355},
     number = {4},
     year = {2017},
     doi = {10.1016/j.crma.2017.02.005},
     language = {en},
}

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Nam Q. Le. On optimal Hölder regularity of solutions to the equation Δu + b ⋅ ∇u = 0 in two dimensions. Comptes Rendus. Mathématique, Volume 355 (2017) no. 4, pp. 439-446. doi : 10.1016/j.crma.2017.02.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.02.005/

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