On a parabolic logarithmic Sobolev inequality

Hassan Ibrahim (1) , Régis Monneau (1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique

In order to extend the blow-up criterion of solutions to the Euler equations, Kozono and Taniuchi have proved a logarithmic Sobolev inequality by means of isotropic (elliptic) $BMO$ norm. In this paper, we show a parabolic version of the Kozono-Taniuchi inequality by means of anisotropic (parabolic) $BMO$ norm. More precisely we give an upper bound for the $L^{\infty}$ norm of a function in terms of its parabolic $BMO$ norm, up to a logarithmic correction involving its norm in some Sobolev space. As an application, we also explain how to apply this inequality in order to establish a long time existence result for a class of nonlinear parabolic problems.

Mots clés: Logarithmic Sobolev inequalities; parabolic $BMO$ spaces; anisotropic Lizorkin-Triebel spaces; harmonic analysis.; harmonic analysis

Domaines: Equations aux dérivées partielles [math.AP]

Hassan Ibrahim, Régis Monneau. On a parabolic logarithmic Sobolev inequality. 2008. ⟨hal-00348682⟩