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On a parabolic logarithmic Sobolev inequality. (English) Zbl 1180.46024

Authors’ abstract: In order to extend the blow-up criterion of solutions to the Euler equations, H. Kozono and Y. Taniuchi [Commun. Math. Phys. 214, No. 1, 191–200 (2000; Zbl 0985.46015)] proved a logarithmic Sobolev inequality by means of the isotropic (elliptic) BMO norm. In this paper, we show a parabolic version of the Kozono-Taniuchi inequality by means of the anisotropic (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.
Reviewer: Liu Zheng (Anshan)

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26D10 Inequalities involving derivatives and differential and integral operators
26D15 Inequalities for sums, series and integrals
35K55 Nonlinear parabolic equations

Citations:

Zbl 0985.46015
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References:

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