On a parabolic logarithmic Sobolev inequality
Ibrahim, Hassan; Monneau, Régis (2009), On a parabolic logarithmic Sobolev inequality, Journal of Functional Analysis, 257, 3, p. 903-930. http://dx.doi.org/10.1016/j.jfa.2009.01.008
TypeArticle accepté pour publication ou publié
External document linkhttp://arxiv.org/abs/0903.1436v1
Journal nameJournal of Functional Analysis
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Abstract (EN)In order to extend the blow-up criterion of solutions to the Euler equations, Kozono and Taniuchi [H. Kozono, Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Comm. Math. Phys. 214 (2000) 191–200] 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∞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.
Subjects / KeywordsLogarithmic Sobolev inequalities; Parabolic BMO spaces; Anisotropic Lizorkin–Triebel spaces; Harmonic analysis
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