Lipschitz Continuity of Solutions to Drift-Diffusion Equations in the Presence of Nonlocal Terms

Abstract

We analyze the propagation of Lipschitz continuity of solutions to various linear and nonlinear drift-diffusion systems, with and without incompressibility constraints. Diffusion is assumed to be either fractional or classical. We derive conditions that guarantee the propagation of Lipschitz regularity by the incompressible NSE in the form of a non-local, one dimensional viscous Burgers-type inequality. We show the analogous inequality is always satisfied for the generalized viscous Burgers–Hilbert equation, in any spatial dimension, leading to global regularity. We also obtain a regularity criterion for the Navier–Stokes equation with fractional dissipation $$(-\Delta )^{\alpha }$$ , regardless of the power of the Laplacian $$\alpha \in (0,1]$$ , in terms of Hölder-type assumptions on the solution. If we drop the pressure term and analyze a linear drift-diffusion problem instead, we obtain regularity under a super-critical assumption on the drift when the power of diffusion is in (1/2, 1]. We also generalize previous known regularity results under subcritical assumptions on the drift when the power of diffusion is between (0, 1/2]. The technique we use builds upon the evolution of moduli of continuity as introduced by Kiselev, Nazarov, Volberg and Shterenberg.