Treatment of the fifth-order coupling term

Determine how to treat the term \(\nabla\operatorname{div}j\), which contains fifth derivatives of the concentration \(w\), in the analysis of temperature-dependent Cahn–Hilliard equations when \(\eta_{\nabla w}\neq0\).

Background

For the general Helmholtz energy ψ(w,θ,∇w)\psi(w,\theta,\nabla w), the temperature evolution equation contains η∇w⋅∇div⁡j\eta_{\nabla w}\cdot\nabla\operatorname{div}j. Under the usual constitutive relation for the flux jj, this term introduces fifth-order derivatives of the concentration. The subsequent analysis therefore restricts attention to Type I systems with η∇w=0\eta_{\nabla w}=0, which corresponds to temperature-independent ε\varepsilon in the specified energy form.

The authors explicitly leave unresolved the analytical treatment required when this higher-order term is present.

References

In Sections \ref{SL}--\ref{SS}, we only consider Type I with \eta_{\nabla w}=0 since we do not know in what way we have to treat the term \nabla\operatorname{div}j which contains 5th derivative of w.

— Remarks on temperature-dependent Cahn--Hilliard equations  (2610.13140 - Giga et al., 9 Oct 2026) in Section 2, subsection “Temperature evolution”