Stokes Resolvent Estimates in Spaces of Bounded Functions

Abe, Ken; Giga, Yoshikazu; Hieber, Matthias

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 3, pp. 537-559.

L'équation de Stokes sur un ouvert $\Omega \subset {𝐑}^n$ a été bien étudiée dans le cadre de $L^p$ pour $1<p<\infty$ et pour une grande classe d'ouverts réguliers. La situation est bien différente pour le cas $p=\infty$, car la projection de Leray n'est pas bornée dans ce cas. Il a été démontré par les premier et second auteurs de cet article que l'opérateur de Stokes engendre tout de même un semigroupe holomorphe sur des espaces de fonctions bornées pour une grande classe d'ouverts. Cet article présente une nouvelle approche et des nouvelles estimations a priori de type $L^{\infty }$ pour l'équation de Stokes. Celles-ci impliquent en particulier que l'opérateur de Stokes engendre un semigroupe holomorphe d'angle $\pi /2$ sur $L_{\sigma }^{\infty }\left(\Omega \right)$ (pas fortement continu) ou $C_{0,\sigma }\left(\Omega \right)$ pour une grande classe d'ouverts $\Omega$. L'approche est inspirée par la méthode de Masuda-Stewart. D'autre part, il est démontré que la méthode s'applique aussi à d'autres conditions de bord, par exemple aux conditions de Robin.

The Stokes equation on a domain $\Omega \subset {𝐑}^n$ is well understood in the $L^p$-setting for a large class of domains including bounded and exterior domains with smooth boundaries provided $1<p<\infty$. The situation is very different for the case $p=\infty$ since in this case the Helmholtz projection does not act as a bounded operator anymore. Nevertheless it was recently proved by the first and the second author of this paper by a contradiction argument that the Stokes operator generates an analytic semigroup on spaces of bounded functions for a large class of domains. This paper presents a new approach as well as new a priori $L^{\infty }$-type estimates to the Stokes equation. They imply in particular that the Stokes operator generates a $C_0$-analytic semigroup of angle $\pi /2$ on $C_{0,\sigma }\left(\Omega \right)$, or a non-$C_0$-analytic semigroup on $L_{\sigma }^{\infty }\left(\Omega \right)$ for a large class of domains. The approach presented is inspired by the so called Masuda-Stewart technique for elliptic operators. It is shown furthermore that the method presented applies also to different types of boundary conditions as, e.g., Robin boundary conditions.

Classification : 35Q35, 35K90
Keywords: Analytic semigroups, bounded function spaces, resolvent estimates.
Mot clés : Semi-groupes holomorphes, espace des fonctions bornées, estimation de la résolvante.

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