On a parabolic logarithmic Sobolev inequality

2009/03/08 by H. Ibrahim, R. Monneau, Ibrahim, H. +1
Mathematics · #39B05 #42B25 #42B35 #54C35 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:39B05 #msc:42B25 #msc:42B35 #msc:54C35

paper · pdf · doi:10.48550/arxiv.0903.1436

arxiv created 2009/03/08 · arxiv updated 2009/12/01

Abstract

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^∞ 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.

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