Existence of a nonzero Jaumann defect

Determine whether there exists a sequence of weak solutions to the vanishing-diffusion regularized incompressible visco-morphoelastic system for which the limiting Jaumann defect is nonzero, or prove that the defect necessarily vanishes under the available structural constraints.

Background

The limiting formulation contains a Jaumann defect because the products $\Phi^\varepsilon W(v^\varepsilon)$ and $W(v^\varepsilon)\Phi^\varepsilon$ involve weakly convergent factors. The paper gives sufficient additional strong-convergence conditions under which the defect vanishes, but neither condition follows from the uniform energy estimates.

The authors point out that no sequence exhibiting a nonzero defect is known in this setting. Any such construction would need oscillations in the deviatoric strain and velocity gradient while remaining compatible with the strong convergence and transport equation for the trace variable.