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

We prove that the probability measure induced by the BBM flow satisfies a logarithmic Sobolev type inequality. Precisely, we suppose the initial data u0 induces a Gaussian measure on Hs with s [1 γ 2 , γ 2 ] for γ (3 2,2]. Then the induced measure ν under BBM flow satisfies, for any 𝜀 small enough,

𝔼ν[f2 log f2 𝔼ν[f2]] C𝜀(𝔼ν[|f|Hs+γ22+𝜀]) 2 2+𝜀 ,

where C𝜀 is an 𝜀-dependent constant.

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Hamiltonian flows, BBM equation, log-Sobolev inequalities
Mathematical Subject Classification 2010
Primary: 35B99
Received: 20 August 2018
Revised: 8 March 2019
Accepted: 30 October 2019
Published: 17 March 2020
Bo Xia
School of Mathematical Sciences
University of Science and Technology of China