A note on local higher regularity in the dynamic linear relaxed micromorphic model

TL;DR

This paper investigates the local higher regularity of solutions in the dynamic linear relaxed micromorphic model, showing improved regularity results for displacement and micro-distortion fields using energy estimates.

Contribution

It demonstrates that solutions have higher local regularity than previously known, specifically in ${\rm H}^1_{\rm loc}$ for displacement and micro-distortion, and ${\rm H}^1$-regularity for curl of micro-distortion.

Findings

Displacement field is ${\rm H}^1_{\rm loc}$-regular.

Micro-distortion tensor $P$ is ${\rm H}^1_{\rm loc}$-regular.

Curl of $P$ is ${\rm H}^1$-regular with smooth data.

Abstract

We consider the regularity question of solutions for the dynamic initial-boundary value problem for the linear relaxed micromorphic model. This generalized continuum model couples a wave-type equation for the displacement with a generalized Maxwell-type wave equation for the micro-distortion. Naturally solutions are found in ${\rm H}^1$ for the displacement $u$ and ${\rm H}({\rm Curl})$ for the microdistortion $P$. Using energy estimates for difference quotients, we improve this regularity. We show ${\rm H}^1_{\rm loc}$-regularity for the displacement field, ${\rm H}^1_{\rm loc}$-regularity for the micro-distortion tensor $P$ and that ${\rm Curl}\,P$ is ${\rm H}^1$-regular if the data is sufficiently smooth.