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Besov regularity of non-linear parabolic PDEs on non-convex polyhedral domains

Stephan Dahlke, Markus Hansen, Cornelia Schneider

TL;DR

This work addresses the regularity of parabolic evolution equations on non-smooth polyhedral domains, with a focus on Besov regularity in the adaptivity scale to justify adaptive schemes. It develops a framework based on Kondratiev spaces and Besov spaces, and uses operator pencils to manage boundary singularities, extending linear parabolic regularity to semilinear problems via Schauder's fixed point theorem. The main contributions are (i) Besov regularity results for linear and nonlinear parabolic problems on polyhedral domains, (ii) a nonlinear fixed-point approach that preserves Besov-regularity under small nonlinear perturbations, and (iii) explicit embeddings that show Besov smoothness can exceed Sobolev smoothness, thus enabling superior adaptive approximation orders. The findings have practical impact on the design and analysis of adaptive/time-m-space schemes for parabolic PDEs in non-convex geometries, with implications for numerical efficiency and convergence rates.

Abstract

This paper is concerned with the regularity of solutions to parabolic evolution equations. We consider semilinear problems on non-convex domains. Special attention is paid to the smoothness in the specific scale $B^r_{τ,τ}$, $\frac{1}τ=\frac rd+ \frac 1p$ of Besov spaces. The regularity in these spaces determines the approximation order that can be achieved by adaptive and other nonlinear approximation schemes. We show that for all cases under consideration the Besov regularity is high enough to justify the use of adaptive algorithms. Our proofs are based on Schauder's fixed point theorem.

Besov regularity of non-linear parabolic PDEs on non-convex polyhedral domains

TL;DR

This work addresses the regularity of parabolic evolution equations on non-smooth polyhedral domains, with a focus on Besov regularity in the adaptivity scale to justify adaptive schemes. It develops a framework based on Kondratiev spaces and Besov spaces, and uses operator pencils to manage boundary singularities, extending linear parabolic regularity to semilinear problems via Schauder's fixed point theorem. The main contributions are (i) Besov regularity results for linear and nonlinear parabolic problems on polyhedral domains, (ii) a nonlinear fixed-point approach that preserves Besov-regularity under small nonlinear perturbations, and (iii) explicit embeddings that show Besov smoothness can exceed Sobolev smoothness, thus enabling superior adaptive approximation orders. The findings have practical impact on the design and analysis of adaptive/time-m-space schemes for parabolic PDEs in non-convex geometries, with implications for numerical efficiency and convergence rates.

Abstract

This paper is concerned with the regularity of solutions to parabolic evolution equations. We consider semilinear problems on non-convex domains. Special attention is paid to the smoothness in the specific scale , of Besov spaces. The regularity in these spaces determines the approximation order that can be achieved by adaptive and other nonlinear approximation schemes. We show that for all cases under consideration the Besov regularity is high enough to justify the use of adaptive algorithms. Our proofs are based on Schauder's fixed point theorem.

Paper Structure

This paper contains 17 sections, 10 theorems, 115 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sobolev, Kondratiev, and Besov spaces
  4. Kondratiev spaces
  5. Kondratiev spaces on domains of polyhedral type
  6. Besov spaces
  7. Parabolic PDEs and operator pencils
  8. The fundamental parabolic problems
  9. Operator pencils
  10. Regularity results in Sobolev and Kondratiev spaces
  11. Regularity results in Sobolev and Kondratiev spaces for Problem I
  12. Schauder's fixed point theorem
  13. Regularity results in Sobolev and Kondratiev spaces for Problem II
  14. Regularity results in Besov spaces
  15. Locally convex spaces
  16. ...and 2 more sections

Key Result

Theorem 3.1

Let $1<p<\infty$, $m\in \mathbb{N}$, $I\subset \mathbb{R}$ be some bounded interval, and $X$ a Banach space. Then

Figures (5)

  • Figure 1: Polyhedron
  • Figure 2: Corner domain $D_c$ ($l=0$) and edge domain $D_e$ ($l'=0$)
  • Figure 3: Dihedron $\mathcal{D}_{\xi}$
  • Figure 4: Eigenvalues of operator pencil $\mathfrak{A}_i(\lambda)$
  • Figure 5: Nonlinear solution in $B_0$

Theorems & Definitions (22)

  • Theorem 3.1: Generalized Sobolev's embedding theorem
  • Theorem 3.2
  • Definition 3.3
  • Remark 3.4
  • Corollary 3.5
  • Remark 3.6
  • Theorem 3.7: Embeddings between Kondratiev and Besov spaces
  • Remark 3.8
  • Remark 4.1: Assumptions on the time-dependent bilinear form
  • Remark 4.2: Operator pencils for parabolic problems
  • ...and 12 more