#### Vol. 305, No. 1, 2020

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Preservation of log-Sobolev inequalities under some Hamiltonian flows

### Bo Xia

Vol. 305 (2020), No. 1, 339–352
##### 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.

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