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$\mathcal{C}^α$-regularity for nonlinear non-diagonal parabolic systems

Miroslav Bulíček, Jens Frehse

TL;DR

This work develops $C^\alpha$-regularity theory for nonlinear, non-diagonal parabolic systems with $p$-growth under a splitting structure. By combining energy estimates, convexity-based bounds, and structure-driven weighted gradient controls, the authors implement a Caccioppoli-type framework and hole-filling arguments to obtain interior Hölder continuity in space and time for $p>\tfrac{d}{2}$; ellipticity further yields full space-time Hölder continuity via interpolation. The analysis covers both the elliptic-like and non-elliptic regimes, and, crucially, extends Hölder regularity results beyond the radial (Uhlenbeck) case to a broader class of nonlinear parabolic systems. The results provide new weighted Morrey-type estimates and a robust regularity theory applicable to systems far from radial symmetry, with implications for qualitative behavior of solutions in high dimensions.

Abstract

In the elliptic theory for $p$-Laplacian-like problems, the Hölder continuity of solutions has been proven for problems arising as Euler--Lagrange equations of a convex potential with $p$-growth that additionally satisfies the splitting condition. In this article, we extend these results to the parabolic setting. We investigate nonlinear parabolic systems whose structure parallels the elliptic case but incorporates time dependence. Assuming suitable space-time regularity of $F$ and natural structural conditions analogous to the stationary theory, we establish $\mathcal{C}^α$-regularity of weak solutions in space and time whenever the growth parameter $p>d/2$. This extends the classical result for parabolic systems, which is valid only for $p>d-2$. This is the only regularity result for systems that are far from the radial (Uhlenbeck) structure.

$\mathcal{C}^α$-regularity for nonlinear non-diagonal parabolic systems

TL;DR

This work develops -regularity theory for nonlinear, non-diagonal parabolic systems with -growth under a splitting structure. By combining energy estimates, convexity-based bounds, and structure-driven weighted gradient controls, the authors implement a Caccioppoli-type framework and hole-filling arguments to obtain interior Hölder continuity in space and time for ; ellipticity further yields full space-time Hölder continuity via interpolation. The analysis covers both the elliptic-like and non-elliptic regimes, and, crucially, extends Hölder regularity results beyond the radial (Uhlenbeck) case to a broader class of nonlinear parabolic systems. The results provide new weighted Morrey-type estimates and a robust regularity theory applicable to systems far from radial symmetry, with implications for qualitative behavior of solutions in high dimensions.

Abstract

In the elliptic theory for -Laplacian-like problems, the Hölder continuity of solutions has been proven for problems arising as Euler--Lagrange equations of a convex potential with -growth that additionally satisfies the splitting condition. In this article, we extend these results to the parabolic setting. We investigate nonlinear parabolic systems whose structure parallels the elliptic case but incorporates time dependence. Assuming suitable space-time regularity of and natural structural conditions analogous to the stationary theory, we establish -regularity of weak solutions in space and time whenever the growth parameter . This extends the classical result for parabolic systems, which is valid only for . This is the only regularity result for systems that are far from the radial (Uhlenbeck) structure.

Paper Structure

This paper contains 12 sections, 1 theorem, 78 equations.

Table of Contents

  1. Introduction and statement of the result
  2. Proof of Theorem \ref{['main']}
  3. First a priori estimates
  4. Estimates based on the convexity of $F$
  5. Estimates based on the structure of $F$
  6. No ellipticity condition
  7. Estimate with ellipticity condition
  8. Estimates for critical $\gamma=d-p$
  9. Caccioppoli inequality, case $p\ge 2$
  10. Beyond the critical Morrey exponent
  11. No ellipticity condition
  12. Ellipticity condition

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{R}^d$ be open, and let $F$ satisfy the assumptions 0.0, 5r2, and 5.55--6.1 with $p>\max\{2,d/2\}$. Assume that $f\in W^{1,1}(0,T; L^2(\Omega; \mathbb{R}^{N}))$ and $u_0\in L^2(\Omega; \mathbb{R}^N)$. Then there exists $\alpha_{\min}>0$ such that for any weak solution $u\i for any compact $K \subset \Omega$. Moreover, if $u_0\in W^{1,p}(\Omega)$ then we can set $\delta=0

Theorems & Definitions (1)

  • Theorem 1.1