Drift-diffusion equations with saturation on Rᵈ

We study a class of nonlinear drift--diffusion equations on Rᵈ 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 L¹-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 C₀-semigroup of L¹(Rᵈ)-contractions. In this confining setting, every trajectory converges strongly in L¹(Rᵈ) to a stationary state with the same mass, and the resulting time-limit operator is an L¹-contraction. Under the same assumptions, we also identify the unique global and L¹-local minimiser of the free energy at each fixed mass and prove that it is stationary under the semigroup.