On a parabolic logarithmic Sobolev inequality

TL;DR

This paper extends a logarithmic Sobolev inequality to a parabolic setting using anisotropic BMO norms, providing new tools for analyzing long-time solutions of nonlinear parabolic equations.

Contribution

It introduces a parabolic version of the Kozono-Taniuchi inequality using anisotropic BMO norms, advancing the mathematical framework for parabolic PDE analysis.

Findings

Established a parabolic logarithmic Sobolev inequality.

Provided an upper bound for the $L^{ abla}$ norm via anisotropic BMO and Sobolev norms.

Applied the inequality to prove long-time existence for certain nonlinear parabolic problems.

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 $L^{\infty}$ 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.