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

The stochastic incompressible Navier-Stokes equations on ${\text{TT}}^3$, 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, $c_{\mathrm{L}\mathrm{S}\mathrm{I}}=\nu {\lambda }_1$ (where $\nu$ is the viscosity and ${\lambda }_1$ is the smallest nonzero eigenvalue of the Laplacian on ${\text{TT}}^3$), 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 $L^2(P_{\mathrm{e}\mathrm{q}})$ (where $P_{\mathrm{e}\mathrm{q}}$ 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.