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 ε, which admits a mechanical interpretation as weakly nonlocal remodeling with a characteristic length scale. For fixed ε>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 ε→0, the scalar variable converges strongly in Lp((0,T)×Ω) for every 1≤ p<∞, 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 𝒟∈ L2(0,T;(H2(Ω;𝕊0))∗) in the limiting deviatoric equation. We give two sufficient conditions for 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.
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BibTeX
@misc{banerjee2026vanishingdiffusion,
title = {{The Vanishing-Diffusion Limit of an Incompressible Visco-Morphoelastic System: Weak Solutions and a Jaumann Defect}},
author = {Swarupananda Banerjee and Amartya Chakrabortty and Hari Shankar Mahato and Raja Sekhar G P},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01487},
eprint = {2610.01487},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01487},
}
RIS
TY - UNPB
TI - The Vanishing-Diffusion Limit of an Incompressible Visco-Morphoelastic System: Weak Solutions and a Jaumann Defect
AU - Swarupananda Banerjee
AU - Amartya Chakrabortty
AU - Hari Shankar Mahato
AU - Raja Sekhar G P
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01487
UR - https://arxiv.org/abs/2610.01487
AB - 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 ε, which admits a mechanical interpretation as weakly nonlocal remodeling with a characteristic length scale. For fixed ε>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 ε→0, the scalar variable converges strongly in L^p((0,T)×Ω) for every 1≤ p<∞, 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 𝒟∈ L^2(0,T;(H^2(Ω;𝕊_0))^∗) in the limiting deviatoric equation. We give two sufficient conditions for 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.
ER -