On a parabolic logarithmic Sobolev inequality |
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Authors: | H Ibrahim R Monneau |
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Institution: | a Université Paris-Est, CERMICS, Ecole des Ponts, 6 et 8 avenue Blaise Pascal, Cité Descartes Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2, France b LaMA-Liban, Lebanese University, P.O. Box 826 Tripoli, Lebanon c CEREMADE, Université Paris-Dauphine, Place De Lattre de Tassigny, 75775 Paris Cedex 16, France |
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Abstract: | 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∞ 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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Keywords: | Logarithmic Sobolev inequalities Parabolic BMO spaces Anisotropic Lizorkin-Triebel spaces Harmonic analysis |
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