Drift-diffusion equations with saturation on $\mathbb{R}^d$

Abstract

We study a class of nonlinear drift--diffusion equations on $\mathbb{R}^d$ with saturation-type mobilities, not necessarily concave. Under structural assumptions on the mobility and potentials, bounded-domain approximations and local Caccioppoli estimates yield weak solutions for all positive times and an $L^1$-contraction estimate. The same estimates provide local space--time compactness of time translates, whose subsequential limits are weak solutions. Under additional confinement assumptions, uniform mass-tail estimates yield a mass-preserving, free-energy dissipating $C_0$-semigroup of $L^1(\mathbb{R}^d)$-contractions. In this confining setting, every trajectory converges strongly in $L^1(\mathbb{R}^d)$ to a stationary state with the same mass, and the resulting time-limit operator is an $L^1$-contraction. Under the same assumptions, we also identify the unique global and $L^1$-local minimiser of the free energy at each fixed mass and prove that it is stationary under the semigroup.