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Continuity of the spatial gradient of weak solutions to very singular parabolic equations involving the one-Laplacian

Shuntaro Tsubouchi

TL;DR

The paper resolves the longstanding question of spatial gradient continuity for weak solutions to a very singular parabolic equation combining a one-Laplacian and a $p$-Laplace operator. By introducing a truncation of the gradient and a parabolic approximation $E_{\varepsilon}$, the authors obtain local Hölder continuity for the truncated gradient ${\mathcal G}_{2\delta,\varepsilon}(\nabla u_{\varepsilon})$ and derive uniform estimates via De Giorgi truncation, heat-flow comparisons, and Campanato-type growth. The convergence analysis then transfers these regularity properties to the original solution, yielding $\nabla u\in C^{0}(\Omega_T)$ under the stated $p$-range and forcing conditions, with the approach avoiding intrinsic parabolic scaling. The methods extend prior stationary results to the time-dependent setting and have implications for models such as Bingham flows and crystal-surface evaporation, where facets induce nonuniform parabolicity. Overall, the work provides a robust gradient-regularity framework for highly singular parabolic equations, combining truncation techniques with classical parabolic theory to achieve sharp continuity results.

Abstract

We consider weak solutions to very singular parabolic equations involving a one-Laplace-type operator, which is singular and degenerate, and a $p$-Laplace-type operator with $\frac{2n}{n+2}<p<\infty$, where $n\ge 2$ denotes the space dimension. This type of equation is used to describe the motion of a Bingham flow. It has been a long-standing open problem of whether the spatial gradients of weak solutions are continuous in space and time. This paper aims to give an affirmative answer for a wide class of such equations. This equation becomes no longer uniformly parabolic near the facet, the place where the spatial gradient vanishes. To achieve our goal, we show local a priori Hölder continuity of gradients suitably truncated near facets. For this purpose, we consider a parabolic approximate problem and appeal to standard methods, including De Giorgi's truncation and comparisons with Dirichlet heat flows. Our method is a parabolic adjustment of our method developed to prove the corresponding statements for stationary problems.

Continuity of the spatial gradient of weak solutions to very singular parabolic equations involving the one-Laplacian

TL;DR

The paper resolves the longstanding question of spatial gradient continuity for weak solutions to a very singular parabolic equation combining a one-Laplacian and a -Laplace operator. By introducing a truncation of the gradient and a parabolic approximation , the authors obtain local Hölder continuity for the truncated gradient and derive uniform estimates via De Giorgi truncation, heat-flow comparisons, and Campanato-type growth. The convergence analysis then transfers these regularity properties to the original solution, yielding under the stated -range and forcing conditions, with the approach avoiding intrinsic parabolic scaling. The methods extend prior stationary results to the time-dependent setting and have implications for models such as Bingham flows and crystal-surface evaporation, where facets induce nonuniform parabolicity. Overall, the work provides a robust gradient-regularity framework for highly singular parabolic equations, combining truncation techniques with classical parabolic theory to achieve sharp continuity results.

Abstract

We consider weak solutions to very singular parabolic equations involving a one-Laplace-type operator, which is singular and degenerate, and a -Laplace-type operator with , where denotes the space dimension. This type of equation is used to describe the motion of a Bingham flow. It has been a long-standing open problem of whether the spatial gradients of weak solutions are continuous in space and time. This paper aims to give an affirmative answer for a wide class of such equations. This equation becomes no longer uniformly parabolic near the facet, the place where the spatial gradient vanishes. To achieve our goal, we show local a priori Hölder continuity of gradients suitably truncated near facets. For this purpose, we consider a parabolic approximate problem and appeal to standard methods, including De Giorgi's truncation and comparisons with Dirichlet heat flows. Our method is a parabolic adjustment of our method developed to prove the corresponding statements for stationary problems.
Paper Structure (24 sections, 25 theorems, 171 equations)

This paper contains 24 sections, 25 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Our strategy
  3. Literature overview and comparisons with the paper
  4. Mathematical models
  5. Notations, main results and outline of the paper
  6. Approximation problems
  7. Relaxation of divergence operators and some basic estimates
  8. Convergence of approximate solutions
  9. Uniform a priori estimates and proof of Theorem \ref{['Thm: Main theorem parabolic']}
  10. Basic estimates of approximate solutions
  11. A basic weak formulation
  12. De Giorgi's truncation
  13. $L^{2}$-energy estimates
  14. Local gradient bounds
  15. Preliminary
  16. ...and 9 more sections

Key Result

Theorem 1.2

Let $p$, $q$, $E_{1}$ and $E_{p}$ satisfy (Eq (Section 1): p-q condition) and (Eq (Section 1): Positively one-homogeneous)--(Eq (Section 1): modulus of continuity of Hess Ep). Assume that $u$ is a weak solution to (Eq (Section 1): General Parabolic Eq) with $f\in L^{q}(\Omega_{T})\cap L^{p^{\prime}}

Theorems & Definitions (50)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Remark 2.6
  • ...and 40 more