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Higher-order derivative estimates for the heat equation on a smooth domain

Yoshinori Furuto, Tsukasa Iwabuchi

TL;DR

The paper develops higher-order derivative estimates for the heat equation on smooth bounded and exterior domains, establishing local-in-time $L^p$ bounds for derivatives of arbitrary order, including the endpoint cases $p=1$ and $p=\infty$. It extends the classical semigroup bounds by introducing endpoint-appropriate techniques, such as spectral multiplier methods, resolvent and elliptic estimates, and a contradiction framework for $L^\infty$ control. A key contribution is the extension to fractional Dirichlet Laplacians, proving $L^p$-derivative bounds for the semigroup $e^{-tA^{\alpha/2}}$ with optimal scaling $t^{-k/\alpha}$. The work provides a robust, kernel-free approach to higher-order derivative control that applies to both standard and fractional Dirichlet problems, with potential impact on regularity theory and numerical analysis for parabolic problems on complex domains.

Abstract

We consider the linear heat equation on a bounded domain and on an exterior domain. We study estimates of any order derivatives of the solution locally in time in the Lebesgue spaces. We give a proof of the estimates in the end-point cases $p = 1, \infty$. We also obtain derivative estimates for the equation with the fractional Dirichlet Laplacian.

Higher-order derivative estimates for the heat equation on a smooth domain

TL;DR

The paper develops higher-order derivative estimates for the heat equation on smooth bounded and exterior domains, establishing local-in-time bounds for derivatives of arbitrary order, including the endpoint cases and . It extends the classical semigroup bounds by introducing endpoint-appropriate techniques, such as spectral multiplier methods, resolvent and elliptic estimates, and a contradiction framework for control. A key contribution is the extension to fractional Dirichlet Laplacians, proving -derivative bounds for the semigroup with optimal scaling . The work provides a robust, kernel-free approach to higher-order derivative control that applies to both standard and fractional Dirichlet problems, with potential impact on regularity theory and numerical analysis for parabolic problems on complex domains.

Abstract

We consider the linear heat equation on a bounded domain and on an exterior domain. We study estimates of any order derivatives of the solution locally in time in the Lebesgue spaces. We give a proof of the estimates in the end-point cases . We also obtain derivative estimates for the equation with the fractional Dirichlet Laplacian.

Paper Structure

This paper contains 6 sections, 9 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $L^1$ estimate in Theorem \ref{['thm:p']}
  4. $L^\infty$ estimate in Theorem \ref{['thm:p']}
  5. Proof of Theorem \ref{['thm:20250127-1']}
  6. Estimate of commutator for resolvent

Key Result

Theorem 1.1

Let $d \geq 2, \gamma \in (\mathbb{Z}_{\geq 0})^d$. Suppose that $\Omega$ is a bounded domain or an exterior domain of ${\mathbb R}^d$ with $C^{|\gamma| + 2}$ boundary and $1 \leq p \leq \infty$. Then, a positive constant $C$ exists such that for every $u_0 \in C_c^\infty (\Omega)$, $u(t) := S(t) u_ where $\gamma = (\gamma_1, \gamma_2, \ldots, \gamma_d) \in (\mathbb{Z}_{\geq 0})^d$ and $|\gamma| =

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Proposition 2.2
  • proof : Proof of Lemma \ref{['lemma:C2']}.
  • Proposition 2.3: paper:FuIwKo-2024
  • Proposition 3.1
  • Theorem 4.1
  • Lemma A.1
  • Lemma A.2