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Well-posedness for a class of parabolic equations with singular-degenerate coefficients

Junyuan Fang, Tuoc Phan

TL;DR

The paper proves well-posedness and regularity for linear parabolic equations in divergence form with a singular-degenerate volumetric heat capacity β in a Muckenhoupt class and measurable A, under small mean oscillation. A weighted Sobolev framework W^{1,p}_*(Ω_T,β) is developed, and a weighted Aubin–Lions compactness theorem facilitates limit-passage in a perturbation/freezing coefficient scheme. The authors introduce non-homogeneous weighted parabolic cylinders and a quasi-distance ρ_β, enabling uniform Lipschitz and energy estimates and a level-set density approach to obtain interior and boundary regularity, culminating in global W^{1,p} estimates. This work extends classical parabolic regularity theory to media with singular-degenerate coefficients and provides a foundation for further weighted mixed-norm or Lorentz-space analyses. Overall, the results establish existence, uniqueness, and detailed regularity for solutions in the weighted setting, advancing the mathematical understanding of parabolic problems with degenerate and singular coefficients.

Abstract

This paper studies a class of linear parabolic equations with measurable coefficients in divergence form whose volumetric heat capacity coefficients are assumed to be in some Muckenhoupt class of weights. As such, the coefficients can be degenerate, singular, or both degenerate and singular. A class of weighted parabolic cylinders with a non-homogeneous quasi-distance function, and a class of weighted parabolic Sobolev spaces intrinsically suitable for the class of equations are introduced. Under some smallness assumptions on the mean oscillations of the coefficients, regularity estimates, existence, and uniqueness of weak solutions in the weighted Sobolev spaces are proved. To achieve the results, we apply the level-set method introduced by Caffarelli and Peral. Several weighted inequalities and a weighted Aubin-Lions compactness theorem for sequences in weighted parabolic Sobolev spaces are established.

Well-posedness for a class of parabolic equations with singular-degenerate coefficients

TL;DR

The paper proves well-posedness and regularity for linear parabolic equations in divergence form with a singular-degenerate volumetric heat capacity β in a Muckenhoupt class and measurable A, under small mean oscillation. A weighted Sobolev framework W^{1,p}_*(Ω_T,β) is developed, and a weighted Aubin–Lions compactness theorem facilitates limit-passage in a perturbation/freezing coefficient scheme. The authors introduce non-homogeneous weighted parabolic cylinders and a quasi-distance ρ_β, enabling uniform Lipschitz and energy estimates and a level-set density approach to obtain interior and boundary regularity, culminating in global W^{1,p} estimates. This work extends classical parabolic regularity theory to media with singular-degenerate coefficients and provides a foundation for further weighted mixed-norm or Lorentz-space analyses. Overall, the results establish existence, uniqueness, and detailed regularity for solutions in the weighted setting, advancing the mathematical understanding of parabolic problems with degenerate and singular coefficients.

Abstract

This paper studies a class of linear parabolic equations with measurable coefficients in divergence form whose volumetric heat capacity coefficients are assumed to be in some Muckenhoupt class of weights. As such, the coefficients can be degenerate, singular, or both degenerate and singular. A class of weighted parabolic cylinders with a non-homogeneous quasi-distance function, and a class of weighted parabolic Sobolev spaces intrinsically suitable for the class of equations are introduced. Under some smallness assumptions on the mean oscillations of the coefficients, regularity estimates, existence, and uniqueness of weak solutions in the weighted Sobolev spaces are proved. To achieve the results, we apply the level-set method introduced by Caffarelli and Peral. Several weighted inequalities and a weighted Aubin-Lions compactness theorem for sequences in weighted parabolic Sobolev spaces are established.
Paper Structure (26 sections, 38 theorems, 454 equations, 1 figure)

This paper contains 26 sections, 38 theorems, 454 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem settings and motivations
  3. Main results
  4. Main difficulties and approaches
  5. Relevant literature
  6. Organization of the paper
  7. Weighted inequalities and weighted parabolic cylinders
  8. Weights and weighted inequalities
  9. Weighted parabolic cylinders and a quasi-distance function
  10. Function spaces, a compactness lemma, and weak solutions
  11. Function spaces and a compactness lemma
  12. Weak solutions and special dilation properties
  13. Lipschitz estimates for simple equations
  14. Interior regularity estimates
  15. Interior energy estimates
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $p \in (1, \infty)$, $\nu \in (0,1)$, and $M_0 \geq 1$. Then there exists a constant $\delta = \delta (n, \nu, p, M_0) \in (0,1)$ sufficiently small such that the following assertions hold. Assume that ellip-cond and beta-cond hold, $\partial \Omega \in C^1$, and for some $R_0 \in (0,1)$. Then, for every $F \in L^p(\Omega_T)^n$ with $T>0$, there exists a unique weak solution $u \in \mathcal{W

Figures (1)

  • Figure 1: Weighted cylinders $Q_{r,\, \beta}(z_0)$ centered at $z_0=(x_0, t_0)$.

Theorems & Definitions (74)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 64 more