Higher-order derivative estimates for the parabolic Lam\'{e} system on a smooth bounded domain

TL;DR

This paper establishes higher-order derivative estimates for the parabolic Lamé system on smooth bounded domains, including $L^p$-$L^p$ estimates and Besov space norm equivalences, advancing understanding of regularity in elasticity models.

Contribution

It provides new $L^p$-$L^p$ estimates and Besov space characterizations for solutions to the parabolic Lamé system, including endpoint cases.

Findings

Established $L^p$-$L^p$ estimates for higher derivatives.

Derived equivalence of Besov space norms via solutions.

Included endpoint cases $p=1$ and $p=ty$.

Abstract

We consider the parabolic Lam\'{e} 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 Lam\'{e} system.