The Vanishing-Diffusion Limit of an Incompressible Visco-Morphoelastic System: Weak Solutions and a Jaumann Defect

Abstract

We study an incompressible visco-morphoelastic system in three dimensions in which the Eulerian effective strain is transported by the Zaremba-Jaumann rate and coupled to a Navier-Stokes momentum balance through a Kelvin-Voigt stress. The strain remains symmetric and splits into a scalar transport-relaxation equation for its trace and a deviatoric equation containing the nonlinear stretching and Jaumann terms. We regularize both equations by diffusion of order $\epsilon$, which admits a mechanical interpretation as weakly nonlocal remodeling with a characteristic length scale. For fixed $\epsilon >0$ we construct Leray-Hopf type weak solutions by a truncated Galerkin scheme and remove the truncation a posteriori using a maximum principle for the scalar variable. As $\epsilon \to 0$, the scalar variable converges strongly in $L^p((0,T)\times \Omega )$ for every $1\le p<\infty$, by a DiPerna-Lions commutator argument on bounded Lipschitz domains, allowing the stretching term to be identified in the limit. In contrast, the Jaumann commutator is a product of two weakly convergent sequences, and its limit cannot be identified from the available estimates. We therefore obtain a defect \[\boldsymbol{\mathcal{D}}\in L^2(0,T;(H^2(Ω;\mathbb{S}_0))^\ast)\] in the limiting deviatoric equation. We give two sufficient conditions for $\mathcal{D}=0$. The energy-level cancellation used in related corotational viscoelastic models does not close here because the coefficient of the stretching term is itself an unknown.

arXiv comment30 pages