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

Yousef Alamri

TL;DR

This work analyzes a saturated aggregation-diffusion equation with local drift and nonlocal interactions, focusing on regularity and long-time behavior in a bounded domain with no-flux boundaries. Using an intrinsic scaling framework and energy-dissipation structure, it establishes interior Hölder regularity for weak solutions and derives oscillation-decay estimates via a two-pronged De Giorgi-type argument. The dissipation analysis shows that, when the nonlocal term vanishes ($W=0$), solutions converge uniformly in space to a stationary state, with convergence guaranteed by energy dissipation and $L^1$-contraction. The results provide a rigorous foundation for global regularity and stabilization in models with saturation and Boltzmann-type diffusion, with potential extensions to porous-medium-type diffusion.

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 $t \to \infty$.

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

TL;DR

This work analyzes a saturated aggregation-diffusion equation with local drift and nonlocal interactions, focusing on regularity and long-time behavior in a bounded domain with no-flux boundaries. Using an intrinsic scaling framework and energy-dissipation structure, it establishes interior Hölder regularity for weak solutions and derives oscillation-decay estimates via a two-pronged De Giorgi-type argument. The dissipation analysis shows that, when the nonlocal term vanishes (), solutions converge uniformly in space to a stationary state, with convergence guaranteed by energy dissipation and -contraction. The results provide a rigorous foundation for global regularity and stabilization in models with saturation and Boltzmann-type diffusion, with potential extensions to porous-medium-type diffusion.

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 .

Paper Structure

This paper contains 14 sections, 18 theorems, 189 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Assumptions
  3. Statement of results
  4. Hölder Regularity
  5. Convergence to steady-state
  6. Hölder regularity
  7. Notion of weak solution
  8. Intrinsic geometry
  9. Notations and auxiliary results
  10. Energy estimates
  11. Behavior near the saturation value
  12. Behavior away from the saturation value
  13. Oscillation decay
  14. Energy dissipation and asymptotics

Key Result

Theorem 1.1

Let $\rho$ be a bounded weak solution for PDE in $\Omega_T$, then $\rho$ is locally Hölder continuous. More precisely, there exist constants $\Gamma>1$ and $\alpha \in (0,1)$ such that for every compact set $K \subset \Omega_T$, for every pair $(x_1,t_1),(x_2,t_2) \in K$. The constants $\alpha$ and $\Gamma$ are independent of $(x_1,t_1),(x_2,t_2)$ and can be determined a priori only in terms of

Figures (1)

  • Figure 1: The standard parabolic cylinder $Q(R^2,R)$ contained in the rescaled cylinder $Q(\theta_1 R^2,R)$.

Theorems & Definitions (38)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.1: Fast geometric convergence
  • ...and 28 more