Preservation of log-Sobolev inequalities under some Hamiltonian flows

Bo Xia

Vol. 305 (2020), No. 1, 339–352

DOI: 10.2140/pjm.2020.305.339

Abstract

We prove that the probability measure induced by the BBM flow satisfies a logarithmic Sobolev type inequality. Precisely, we suppose the initial data $u_0$ induces a Gaussian measure on $H^s$ with $s\in \left[1-\frac{\gamma }{2},\frac{\gamma }{2}\right]$ for $\gamma \in \left(\frac{3}{2},2\right]$. Then the induced measure $\nu$ under BBM flow satisfies, for any $𝜀$ small enough,

$${\mathbb{𝔼}}_{\nu }\left[f^{2}log\frac{f^2}{{\mathbb{𝔼}}_{\nu }\left[f^2\right]}\right]\le C_{𝜀}{\left({\mathbb{𝔼}}_{\nu }\left[|\nabla f{|}_{H^{s+\gamma ∕2}}^{2+𝜀}\right]\right)}^{\frac{2}{2+𝜀}},$$

where $C_{𝜀}$ is an $𝜀$-dependent constant.

Keywords

Hamiltonian flows, BBM equation, log-Sobolev inequalities

Mathematical Subject Classification 2010

Primary: 35B99

Milestones

Received: 20 August 2018

Revised: 8 March 2019

Accepted: 30 October 2019

Published: 17 March 2020