Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity

Martin Kalousek; Joshua Kortum; Anja Schlömerkemper

doi:10.3934/dcdss.2020331

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2021, Volume 14, Issue 1: 17-39. Doi: 10.3934/dcdss.2020331

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Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity

University of Würzburg, Institute of Mathematics, Emil-Fischer-Straße 40, 97074 Würzburg, Germany

^* Corresponding author: Anja Schlömerkemper
^* Corresponding author: Anja Schlömerkemper

Dedicated to Alexander Mielke on the occasion of his 60th birthday

Received: June 2019

Revised: September 2019

Early access: April 2020

Published: January 2021

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Abstract

The paper is concerned with the analysis of an evolutionary model for magnetoviscoelastic materials in two dimensions. The model consists of a Navier-Stokes system featuring a dependence of the stress tensor on elastic and magnetic terms, a regularized system for the evolution of the deformation gradient and the Landau-Lifshitz-Gilbert system for the dynamics of the magnetization.

First, we show that our model possesses global in time weak solutions, thus extending work by Benešová et al. 2018. Compared to that work, we include the stray field energy and relax the assumptions on the elastic energy density. Second, we prove the local-in-time existence of strong solutions. Both existence results are based on the Galerkin method. Finally, we show a weak-strong uniqueness property.

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Mathematics Subject Classification: Primary: 35Q35, 35Q74; Secondary: 74F15.

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References

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