Chain rule for the energy-dissipation balance at weak-solution regularity

Establish a chain rule yielding the energy-dissipation balance for the degenerate viscoelastic phase-separation system at the regularity of the weak solutions considered in the paper.

Background

The paper constructs weak solutions for a fourth-order viscoelastic phase-separation system with degenerate mobility by combining a dynamic transport metric for the phase field with a weighted L2 metric for a transformed stress variable. In contrast to the constant-mobility setting, the relevant transport structure does not provide a comparable geodesic-convexity property for the Dirichlet energy. Consequently, the authors do not obtain the energy-dissipation balance through a chain-rule argument; instead, they derive an energy-dissipation inequality from the discrete variational scheme and De Giorgi interpolants. Establishing such a chain rule would provide a stronger variational characterization of the limiting weak solutions and could potentially yield an energy-dissipation balance rather than only an inequality.