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Regularity criteria for the surface growth model with a forcing term

Yuqian Cheng, Zhisu Li, Xuening Wei

Abstract

Based on a compactness method, we establish regularity criteria for suitable weak solutions to the surface growth model with a forcing term. These criteria imply that the Hölder regularity of solutions follows from smallness conditions on several scale-invariant quantities. As a consequence, we obtain a partial regularity result stating that the one-dimensional biparabolic Hausdorff measure of the singular set is zero.

Regularity criteria for the surface growth model with a forcing term

Abstract

Based on a compactness method, we establish regularity criteria for suitable weak solutions to the surface growth model with a forcing term. These criteria imply that the Hölder regularity of solutions follows from smallness conditions on several scale-invariant quantities. As a consequence, we obtain a partial regularity result stating that the one-dimensional biparabolic Hausdorff measure of the singular set is zero.
Paper Structure (14 sections, 23 theorems, 110 equations)

This paper contains 14 sections, 23 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and definitions
  4. Local energy estimate
  5. Parabolic Poincaré inequality
  6. Compactness of suitable weak solutions
  7. Hölder continuity via the compactness method
  8. Key approximation lemma
  9. Contraction lemma and decay lemma
  10. Proof of Theorem \ref{['thm.main-G+F']}
  11. Proofs of Theorem \ref{['thm.main-U']} and Theorem \ref{['thm.main-L']}
  12. Interpolation inequalities
  13. Regularity criterion in terms of $U_r$
  14. Regularity criterion in terms of $L_r$

Key Result

Theorem 1.1

There exist universal constants $\delta_0=\delta_0(p)>0$ and $0<\alpha=\alpha(p)<1$ such that for any $0<r\leq1$, if then $u\in C^{\alpha,\alpha/4}(Q_{r/4})$.

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Corollary 1.1: Partial regularity
  • Definition 2.1: Suitable weak solution
  • Definition 2.2: Biparabolic Hausdorff measure
  • Lemma 2.1
  • ...and 35 more