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 $\varepsilon$, which admits a mechanical interpretation as weakly nonlocal remodeling with a characteristic length scale. For fixed $\varepsilon&gt;0$ 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 $\varepsilon\to0$, the scalar variable converges strongly in $L<sup>p((0,T)\timesΩ)$ for every $1\le p&lt;\infty$, 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 $\boldsymbol{\mathcal{D}}=0$. 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.