Dissipative membrane coupling

The paper defines a free energy and dual dissipation potential by $$\mathcal{E}(v)=\int_{\mathscr{D}}E(v(x))\,\mathrm{d}x,\qquad \mathcal{R}^{*}(v,\xi)=\frac12\int_{\mathscr{D}}A(x)\mathsf{m}(v)|\nabla\xi|^{2}\,\mathrm{d}x,$$ where \(v\) is the state, \(E\) is the scalar energy density, \(A(x)\) is mobility, \(\mathsf{m}(v)\) is state-dependent mobility, and \(\xi\) is a dual potential. The corresponding chemical potential is \(\mu=E'(v)\), and the Onsager operator is \(\mathbb{K}(v)\xi=-\operatorname{div}(A(x)\mathsf{m}(v)\nabla\xi)\). The thin-layer scaling is $$A_{\varepsilon}(x)=\begin{cases}A_{-},&x\in(-1,0),\\b\varepsilon,&x\in(0,\varepsilon),\\A_{+},&x\in(\varepsilon,1),\end{cases}$$ so a region of width \(\varepsilon\) and mobility \(b\varepsilon\) has finite resistance in the limit. The limiting interface has equal flux on both sides, $$A_{-}\partial_xv^{\alpha}(0^-)=A_{+}\partial_xv^{\alpha}(0^+)=q.$$ The adapted neural kinetic relation is $$q=\kappa\left[\exp\!\left(\frac{\mu_{-}-\mu_{+}}{2\tau}\right)-\exp\!\left(-\frac{\mu_{-}-\mu_{+}}{2\tau}\right)\right]=2\kappa\sinh\!\left(\frac{\mu_{-}-\mu_{+}}{2\tau}\right),$$ where \(\mu_{-}=E'(u_{-})\), \(\mu_{+}=E'(u_{+})\), \(\kappa>0\) is a learned interface mobility, and \(\tau>0\) is a temperature or scale parameter. Its essential property is $$q(\mu_{-}-\mu_{+})=2\kappa(\mu_{-}-\mu_{+})\sinh\!\left(\frac{\mu_{-}-\mu_{+}}{2\tau}\right)\ge0,$$ because \(z\sinh(z)\ge0\). Thus exchange moves mass from high chemical potential to low chemical potential and cannot increase the energy under the corresponding continuous-time dynamics. For the paper's power-law family, $$E''_{p}(u)=u^{p-2},\qquad \mathsf{m}(u)=u^{\beta},$$ with \(\beta\ge0\) and \(p\ge\max\{1,(3-\beta)/2\}\); this gives \(\mu(u)=(u^{p-1}-1)/(p-1)\) for \(p\ne1\), and \(\mu(u)=\log u\) for \(p=1\).