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Higher-order derivative estimates for the parabolic Lamé system on a smooth bounded domain

Yoshinori Furuto, Tsukasa Iwabuchi

Abstract

We consider the parabolic Lamé system on a bounded domain. We focus on two types of inequalities for higher-order derivatives of solutions. The first is related to an $L^p$-$L^p$ estimate locally in time in the Lebesgue space setting, which includes the endpoint cases $p=1$ and $p=\infty$. The second concerns an equivalent norm of Besov spaces by means of the solution of the parabolic Lamé system.

Higher-order derivative estimates for the parabolic Lamé system on a smooth bounded domain

Abstract

We consider the parabolic Lamé system on a bounded domain. We focus on two types of inequalities for higher-order derivatives of solutions. The first is related to an - estimate locally in time in the Lebesgue space setting, which includes the endpoint cases and . The second concerns an equivalent norm of Besov spaces by means of the solution of the parabolic Lamé system.
Paper Structure (7 sections, 12 theorems, 86 equations)

This paper contains 7 sections, 12 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proposition for the proof of Theorem \ref{['thm:1']} when $p = 1$
  4. Proposition for the proof of Theorem \ref{['thm:1']} when $p = \infty$
  5. $L^1$ estimate in Theorem \ref{['thm:1']}
  6. $L^\infty$ estimate in Theorem \ref{['thm:1']}
  7. Proof of Theorem \ref{['thm:2']}

Key Result

Theorem 1.3

Let $d \geq 2$, $\gamma \in (\mathbb{Z}_{\geq 0})^d$ and $1 \leq p \leq \infty$. Suppose that $\Omega$ is a bounded domain of ${\mathbb R}^d$ with $C^{|\gamma| + 2}$ boundary. Then, a positive constant $C$ exists such that for every $u_0 \in C_c^\infty (\Omega)$, $u(t) := e^{t \mathcal{L}} u_0$ in $ For $t \geq 1$, the norm decays exponentially.

Theorems & Definitions (23)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark
  • Remark
  • Proposition 2.1: paper:DaTo-2022
  • Definition 2.2
  • Lemma 2.3
  • ...and 13 more