Aggregation-Confinement-Diffusion Evolutions with Saturation: Regularity and Long-Time Asymptotics

TL;DR

This paper proves regularity and long-term convergence of solutions to a degenerate diffusion equation with saturation, involving local and nonlocal terms, in bounded domains with no-flux boundaries.

Contribution

It establishes H"older regularity for solutions with saturation-induced degeneracy and demonstrates their uniform convergence to stationary states over time.

Findings

Proved H"older regularity of solutions.

Showed solutions converge uniformly to stationary states.

Analyzed the effects of saturation on diffusion dynamics.

Abstract

We establish H\"older regularity for the weak solution to a degenerate diffusion equation in the presence of a local (drift) potential and nonlocal (interaction) term, posed in a bounded domain with no-flux boundary conditions. The degeneracy is due to saturation, i.e., it occurs when the solution reaches its maximal value. As a byproduct of the established regularity and the underlying dissipative structure of the evolution, we prove the uniform convergence of contractive solutions to a stationary state as $t \to \infty$.