A Note on the Regularity of Thermoelastic Plates with Fractional Rotational Inertial Force

TL;DR

This paper investigates the regularity properties of solutions to thermoelastic plates with fractional rotational inertial forces, revealing new Gevrey class regularity results for a range of fractional parameters and boundary conditions.

Contribution

It establishes new Gevrey class regularity results for the semigroup associated with thermoelastic plates with fractional rotational forces, extending previous findings to a broader parameter range.

Findings

Semigroup is of Gevrey class s > (3 - τ)/(2 - 2τ) for τ in [1/2, 1)

Semigroup is not analytic for τ in (0, 1]

Results apply to hinged plates with Dirichlet boundary conditions

Abstract

The present work intends to complement the study of the regularity of the solutions of the thermoelastic plate with rotacional forces. The rotational forces involve the spectral fractional Laplacian, with power parameter $\tau\in [0,1]$ ( $\gamma(-\Delta)^\tau u_{tt}$). Previous research regarding regularity showed that, as for the analyticity of the semigroup $S(t)=e^{\mathbb{B}t}$ for the Euler-Bernoulli Plate($\tau=0$) model, the first result was established by Liu and Renardy, \cite{LiuR95} in the case of hinged and clamped boundary conditions, for the case $\tau=1$ (Plate Kirchoff-Love) Lasiecka and Triggiani showed, that the semigroup is not differentiable \cite{LT1998, LT2000} and more recently in 2020 Tebou et al.\cite{Tebou2020} showed that for $\tau\in (0,\frac{1}{2})$, $S(t)$ is of class Gevrey $s>\frac{2-\tau}{2-4\tau}$. Our main contribution here is to show that $S(t)$ is of Gevrey class $s>\frac{3-\tau}{2-2\tau}$ when the parameter $\tau$ lies in the interval $[\frac{1}{2},1)$ and also show that $S(t)$ is not analytic for $\tau\in (0,1]$ both results for Hinged plate/ Dirichlet temperature boundary conditions.