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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
$\mathit{\epsilon}$
small enough,
$${\mathbb{\mathbb{E}}}_{\nu}\left[{f}^{2}log\frac{{f}^{2}}{{\mathbb{\mathbb{E}}}_{\nu}\left[{f}^{2}\right]}\right]\le {C}_{\mathit{\epsilon}}{\left({\mathbb{\mathbb{E}}}_{\nu}\left[\nabla f{}_{{H}^{s+\gamma \u22152}}^{2+\mathit{\epsilon}}\right]\right)}^{\frac{2}{2+\mathit{\epsilon}}},$$ 
where
${C}_{\mathit{\epsilon}}$ is
an
$\mathit{\epsilon}$dependent
constant.

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Keywords
Hamiltonian flows, BBM equation, logSobolev inequalities

Mathematical Subject Classification 2010
Primary: 35B99

Milestones
Received: 20 August 2018
Revised: 8 March 2019
Accepted: 30 October 2019
Published: 17 March 2020

