Relative-energy formulation for the variable-coefficient coupling

Determine whether the relative-energy method used to construct energy-variational solutions for related viscoelastic systems can be adapted to the incompressible visco-morphoelastic system with coupling \((q-1)D(v)\), and whether the pointwise bounds on the trace variable \(q\) enable that adaptation.

Background

A relative-energy formulation could provide a closed limiting solution concept without retaining an undetermined Jaumann defect. The paper notes, however, that the present system differs from the viscoelastic models treated in the cited literature because the deviatoric equation contains the variable coefficient q−1q-1.

The scalar trace variable satisfies pointwise upper and lower bounds established by a maximum-principle argument. The authors explicitly leave unresolved whether these bounds are useful for adapting the relative-energy approach.

References

Applying the relative energy method to Dmodel would require adapting it to the coupling $(q-1)D$, whose variable coefficient is absent from the models treated in . Whether the pointwise bounds on $q$ from Lemma~\ref{lem:qbdd} are of use here we leave open.

Dmodel:

{ρ∂tv+ρ(v⋅∇)v+∇π=(μ1D(v)+λ)+fin S,∂tq+v⋅∇q+αq=3βin S,∂t+(v⋅∇)+W(v)−W(v)+(q−1)D(v)+α=0in S,v=0in S,v(0)=v0, q(0)=q0=tr(E0), (0)=0=dev E0in ,v=0on S×∂,(DM)\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 + v \cdot \nabla q + \alpha q = 3\beta && \quad\text{in } S ,\\[0.5em] &\partial_t + (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 && \quad\text{on } S\times\partial,\\ \end{aligned} \right. \tag{\text{{\bf DM}}}

— 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 “Widening the solution concept”