Uniform Beale–Kato–Majda-type gradient control

Establish whether the uniform condition \(\nabla v^\varepsilon\in L^1(0,T;L^\infty(\Omega))\) holds for Leray–Hopf solutions of the vanishing-diffusion regularized incompressible visco-morphoelastic system, and develop the corresponding uniform strong-solution theory for the coupled system.

Background

The vanishing-diffusion analysis produces a defect in the limiting deviatoric equation because the Jaumann commutator contains two factors that converge only weakly. One sufficient route to eliminate the defect would be to obtain strong convergence through additional spatial regularity of the deviatoric strain or the velocity gradient.

The paper proposes a Beale–Kato–Majda-type condition, ∇vε∈L1(0,T;L∞(Ω))\nabla v^\varepsilon\in L^1(0,T;L^\infty(\Omega)) uniformly in ε\varepsilon, together with suitable initial-data regularity. However, the available estimates do not imply this condition, and proving it would require a strong-solution theory with bounds uniform in the diffusion parameter for the fully coupled system.

References

In three dimensions it is not known to hold for Leray--Hopf solutions of the momentum equation, and establishing it for Rmodel+ would mean developing a strong-solution theory, with bounds uniform in $$, for the coupled system rather than for the momentum equation alone, since the latter is forced by $\lambda\nabla!\cdot!$.

Rmodel+:

$\left\{ \begin{aligned} &\rho \partial_t v^ + \rho (v^ \cdot \nabla) v^ + \nabla \pi^ = (\mu_1 D(v^) + \lambda ^) + f && \quad\text{in } S ,\\[0.5em] &\partial_t q^ - \Delta q^ + v^ \cdot \nabla q^ + \alpha q^ = 3\beta && \quad\text{in } S ,\\[0.5em] &\partial_t ^ - \Delta ^ + (v^ \cdot \nabla) ^ + ^ W(v^) - W(v^) ^ + (q^ - 1) D(v^) + \alpha ^ = 0 && \quad\text{in } S ,\\[0.5em] & v^ = 0 && \quad\text{in } S ,\\[0.5em] &v^(0) = v_0,\,q^(0)=q_0 = tr(E_0),\, ^(0) =_0=dev\,E_0 && \quad\text{in },\\[0.5em] &v^=0,\, \nabla q^\cdot\bm{n}=0,\, \nabla^\cdot\bm{n}=0 && \quad\text{on } S\times\partial,\\ \end{aligned} \right. \tag{\text{{\bf RM}}} $

— The Vanishing-Diffusion Limit of an Incompressible Visco-Morphoelastic System: Weak Solutions and a Jaumann Defect  (2610.01487 - Banerjee et al., 1 Oct 2026) in Section 5, Conclusion and outlook, subsection “Removing the defect by additional regularity”

Whether the small-data and short-time theories available for Oldroyd-type systems can be adapted in this way is not clear to us, and we do not pursue the two-dimensional case here either.

— The Vanishing-Diffusion Limit of an Incompressible Visco-Morphoelastic System: Weak Solutions and a Jaumann Defect  (2610.01487 - Banerjee et al., 1 Oct 2026) in Section 5, Conclusion and outlook, subsection “Removing the defect by additional regularity”