Regularity of Euler-Bernoulli and Kirchhoff-Love Thermoelastic Plates with Fractional Coupling

TL;DR

This paper investigates the regularity and analyticity of solutions to a class of thermoelastic plate systems with fractional coupling, providing sharp Gevrey class estimates and conditions for semigroup analyticity.

Contribution

It determines precise Gevrey regularity classes and analyticity conditions for solutions of thermoelastic plates with fractional coupling, extending prior work.

Findings

Gevrey class sharpness depends on fractional power and system type.

Semigroup analyticity occurs at specific fractional coupling parameters.

Explicit regularity thresholds are established for both Euler-Bernoulli and Kirchhoff-Love plates.

Abstract

I In this work, we present the study of the regularity of the solutions of the abstract system\eqref{Eq1.10} that includes the Euler-Bernoulli($\omega=0$) and Kirchoff-Love($\omega>0$) thermoelastic plates, we consider for both fractional couplings given by $A^\sigma\theta$ and $A^\sigma u_t$, where $A$ is a strictly positive and self-adjoint linear operator and the parameter $\sigma\in[0,\frac{3}{2}]$. Our research stems from the work of \cite{MSJR}, \cite{OroJRPata2013}, and \cite{KLiuH2021}. Our contribution was to directly determine the Gevrey sharp classes: for $\omega=0$, $s_{01}>\frac{1}{2\sigma-1}$ and $s_{02}> \sigma$ when $\sigma\in (\frac{1}{2},1)$ and $\sigma\in (1,\frac{3}{2})$ respectively. And $s_\omega>\frac{1}{4(\sigma-1)}$ for case $\omega>0$ when $\sigma\in (1,\frac{5}{4})$. This work also contains direct proofs of the analyticity of the corresponding semigroups $e^{t\mathbb{A}_\omega}$: In the case $\omega=0$ the analyticity of the semigroup $e^{t\mathbb{A}_0}$ occurs when $\sigma=1$ and for the case $\omega>0$ the semigroup $e^{t\mathbb{A}_\omega}$ is analytic for the parameter $\sigma\in[5/4, 3/2]$. The abstract system is given by: \begin{equation}\label{Eq1.10} \left\{\begin{array}{c} u_{tt}+\omega Au_{tt}+A^2u-A^\sigma\theta=0,\\ \theta_t+A\theta+A^\sigma u_t=0. \end{array}\right. \end{equation} where $\omega\geq 0$.