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

Abstract

We establish Hölder 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 $t\to \infty$.

Reformulation of the convergence proof in terms of contractive solutions and added illustrations & references

References in corpus (7)

• Phase Segregation Dynamics in Particle Systems with Long Range Interactions I: Macroscopic Limits
• A new class of transport distances
• Long-time behaviour and phase transitions for the McKean--Vlasov equation on the torus
• The McKean-Vlasov Equation in Finite Volume
• On gradient flow and entropy solutions for nonlocal transport equations with nonlinear mobility
• Structure preserving primal dual methods for gradient flows with nonlinear mobility transport distances
• Bound-Preserving Finite-Volume Schemes for Systems of Continuity Equations with Saturation