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

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 ${\mathrm{H}}^1$ for the displacement $u$ and $\mathrm{H}(\mathrm{Curl})$ for the microdistortion $P$. Using energy estimates for difference quotients, we improve this regularity. We show ${\mathrm{H}}_{\mathrm{loc}}^1$-regularity for the displacement field, ${\mathrm{H}}_{\mathrm{loc}}^1$-regularity for the micro-distortion tensor $P$ and that $\mathrm{Curl}P$ is ${\mathrm{H}}^1$-regular if the data is sufficiently smooth.

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Cited by in corpus (2)

• Analytical solutions of the cylindrical bending problem for the relaxed micromorphic continuum and other generalized continua (including full derivations)
• A hybrid H1xH(curl) finite element formulation for a relaxed micromorphic continuum model of antiplane shear