Preservation of log-Sobolev inequalities under some Hamiltonian flows

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 $s \in [1 - \frac{γ}{2}, \frac{γ}{2}]$ for $γ \in (\frac{3}{2}, 2]$ . Then the induced measure $ν$ under BBM flow satisfies, for any $𝜀$ small enough,

𝔼_{ν} [f^{2} log \frac{f^{2}}{𝔼_{ν} [f^{2}]}] \leq C_{𝜀} {(𝔼_{ν} [| \nabla f |_{H^{s + γ ∕ 2}}^{2 + 𝜀}])}^{\frac{2}{2 + 𝜀}},

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

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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

Authors

Bo Xia
School of Mathematical Sciences University of Science and Technology of China Hefei China

Vol. 305, No. 1, 2020

Bo Xia

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors