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Global Schauder Regularity and Convergence for Uniformly Degenerate Parabolic Equations

Qing Han, Jiongduo Xie

TL;DR

The paper develops a global Schauder framework for linear uniformly degenerate parabolic equations on bounded domains, using weighted Hölder spaces to account for degeneracy along the lateral boundary. It proves existence and higher regularity of solutions up to the boundary, and establishes several layers of long-time convergence: $L^{\infty}$, $C^{2+\alpha}_{\ast}$, and $C^{k,2+\alpha}_{\ast}$-convergence to the Dirichlet problem for the limit elliptic operator $L_{0}$ as $t\to\infty$, with convergence rates governed by the characteristic polynomial $P(\mu)$. The approach leverages maximum principles, intermediate Schauder theory, and an induction scheme to propagate regularity from tangential to normal directions and time. The results provide a rigorous global description of both regularity and asymptotic behavior for uniformly degenerate parabolic equations, with explicit boundary data constraints and precise convergence norms that are relevant for applications where degeneracy and long-time dynamics arise.

Abstract

In this paper, we study the global Hölder regularity of solutions to uniformly degenerate parabolic equations. We also study the convergence of solutions as time goes to infinity under extra assumptions on the characteristic exponents of the limit uniformly degenerate elliptic equations.

Global Schauder Regularity and Convergence for Uniformly Degenerate Parabolic Equations

TL;DR

The paper develops a global Schauder framework for linear uniformly degenerate parabolic equations on bounded domains, using weighted Hölder spaces to account for degeneracy along the lateral boundary. It proves existence and higher regularity of solutions up to the boundary, and establishes several layers of long-time convergence: , , and -convergence to the Dirichlet problem for the limit elliptic operator as , with convergence rates governed by the characteristic polynomial . The approach leverages maximum principles, intermediate Schauder theory, and an induction scheme to propagate regularity from tangential to normal directions and time. The results provide a rigorous global description of both regularity and asymptotic behavior for uniformly degenerate parabolic equations, with explicit boundary data constraints and precise convergence norms that are relevant for applications where degeneracy and long-time dynamics arise.

Abstract

In this paper, we study the global Hölder regularity of solutions to uniformly degenerate parabolic equations. We also study the convergence of solutions as time goes to infinity under extra assumptions on the characteristic exponents of the limit uniformly degenerate elliptic equations.
Paper Structure (13 sections, 46 theorems, 459 equations)

This paper contains 13 sections, 46 theorems, 459 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Main Results
  4. The Existence of Solutions
  5. The Higher Regularity of Solutions
  6. The Regularity of the Derivative in $x'$
  7. The Regularity of the Derivative in $x_{n}$
  8. The Regularity of the Derivative in $t$
  9. The Convergence of Solutions
  10. $L^{\infty}$-Convergence
  11. $C^{2+\alpha}_{\ast}$-Convergence
  12. $C^{k,2+\alpha}_{\ast}$-Convergence
  13. Appendix: Intermediate Schauder Theory

Key Result

Theorem 1.2

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ with a $C^{1}$-boundary $\partial\Omega$ and $\rho$ be a $C^1(\bar{\Omega})\cap C^{2}(\Omega)$-defining function with $\rho\nabla^2\rho\in C(\bar{\Omega})$ and $\rho\nabla^2\rho=0$ on $\partial\Omega$. For some constant $T>0$, let $Q_{T}=\Omega\ti

Theorems & Definitions (81)

  • Example 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5: Existence and uniqueness
  • Theorem 1.6: Optimal regularity
  • Definition 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Corollary 1.10
  • ...and 71 more