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arXiv · 2606.03182

Logarithmic Sobolev inequality and hypercontractivity for the Navier-Stokes Fokker-Planck operator

Zhi-Wei Wang·Samuel L. Braunstein
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Abstract

The stochastic incompressible Navier-Stokes equations on $\TT^3$, completed by the fluctuation-dissipation noise, have a Fokker-Planck generator that decomposes into a self-adjoint Ornstein-Uhlenbeck (dissipative) part and an antisymmetric (convective) part. We prove two results about this generator. First, the logarithmic Sobolev inequality holds with the same optimal constant as the pure Ornstein-Uhlenbeck operator, $c_\mathrm{LSI} = \nu\lambda_1$ (where $\nu$ is the viscosity and $\lambda_1$ is the smallest nonzero eigenvalue of the Laplacian on $\TT^3$), independent of the number of retained Fourier modes. Second, the full semigroup is hypercontractive with the same rate as the Ornstein-Uhlenbeck semigroup. Both results follow from a single structural property: the convective generator is antisymmetric in $L^2(P_\mathrm{eq})$ (where $P_\mathrm{eq}$ is the Gibbs measure), and therefore contributes nothing to the Dirichlet form or the $L^q$ norm evolution. The antisymmetry is a consequence of two properties of the incompressible Navier-Stokes nonlinearity: energy conservation and phase-space volume preservation (the Liouville property). These are the same properties that underpin the fluctuation-dissipation theorem for the nonlinear Navier-Stokes equations.

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Zhi-Wei Wang, Samuel L. Braunstein. 2026-06-02. Logarithmic Sobolev inequality and hypercontractivity for the Navier-Stokes Fokker-Planck operator. https://arxiv.org/abs/2606.03182

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