Search

This article addresses the local boundedness and Hölder continuity of weak solutions to kinetic Fokker--Planck equations with general transport operators and rough coefficients. These results are due to the mixing effect of diffusion and transport. Although the equation is parabolic only in the velocity variable, it has a hypoelliptic structure provided that the transport part ∂ t + b(v). ∇x is nondegenerate in some sense. We achieve the results by revisiting the method, proposed by Golse, Imbert, Mouhot, and Vasseur in the case b(v) = v, that combines the elliptic De Giorgi--Nash--Moser theory with velocity averaging lemmas. [ABSTRACT FROM AUTHOR]

We consider the three-dimensional magnetohydrodynamics (MHD) equations in the presence of a spatially degenerate stochastic forcing as a model for magnetostrophic turbulence in the Earth's uid core. We examine the multiparameter singular limit of vanishing Rossby number ε and magnetic Reynold's number δ, and establish that (i) the formal limit (with ε = δ = 0), a stochastically driven active scalar equation, possesses a unique ergodic invariant measure, and (ii) any suitable sequence of statistically invariant states of the full MHD system converges weakly, as ε δ → 0, to the unique invariant measure of the limit equation. This latter convergence result does not require any conditions on the relative rates at which ε, δ decay. Our analysis of the limit equation relies on a recently developed theory of hypoellipticity for infinite-dimensional stochastic dynamical systems. We carry out a detailed study of the interactions between the nonlinear and stochastic terms to demonstrate that a Hörmander bracket condition is satisfied for the limit equation, which yields a contraction property in a suitable Wasserstein metric. This contraction property reduces the convergence of invariant states in the multiparameter limit to the convergence of solutions at finite times. However, in view of the phase space mismatch between the small parameter system and the limit equation, and due to the multiparameter nature of the problem, further analysis is required to establish the singular limit. In particular, we develop methods to lift the contraction for the limit equation to the extended phase space, including the velocity and magnetic fields. Moreover, for the convergence of solutions at finite times we make use of a probabilistic modification of the Grönwall inequality, relying on a delicate stopping time argument. [ABSTRACT FROM AUTHOR]

We are dealing with possibly degenerate second-order parabolic operators whose coefficients are infinitely differentiable with respect to the space variables and only measurable with respect to the time variable. We impose the Hörmander condition on the diffusion coefficients and prove that the solutions of the corresponding equations with right-hand sides which are infinitely differentiable in the space variables in a space-time domain also have this property. [ABSTRACT FROM AUTHOR]