A damped elastodynamics system under the global injectivity condition: Local wellposedness in $L^p$-spaces

TL;DR

This paper models cardiac tissue mechanics using elastodynamics equations with a pressure variable enforcing global injectivity, proving local well-posedness in $L^p$-spaces with damping.

Contribution

It introduces a coupled elastodynamics-pressure system with global injectivity constraints and establishes local existence of solutions in $L^p$-spaces.

Findings

Existence of local-in-time solutions in $L^p$-spaces.

Inclusion of damping term facilitates analysis.

Mathematical modeling of cardiac tissue deformation.

Abstract

The purpose of this paper is to model mathematically mechanical aspects of cardiac tissues. The latter constitute an elastic domain whose total volume remains constant. The time deformation of the heart tissue is modeled with the elastodynamics equations dealing with the displacement field as main unknown. These equations are coupled with a pressure whose variations characterize the heart beat. This pressure variable corresponds to a Lagrange multiplier associated with the so-called global injectivity condition. We derive the corresponding coupled system with nonhomogeneous boundary conditions where the pressure variable appears. For mathematical convenience a damping term is added, and for a given class of strain energies we prove the existence of local-in-time solutions in the context of the $L^p$-parabolic maximal regularity.