Allen–Cahn and Cahn–Hilliard double-well gradient flows

Let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be a bounded connected Lipschitz domain, let $d\ge 1$, and fix constants $\epsilon >0$, $M>0$, and $\kappa >0$. Put $F_{\epsilon }(s)=\frac{(s^2-1{)}^2}{4{\epsilon }^2},f_{\epsilon }(s)=F_{\epsilon }^{\mathrm{\prime }}(s)=\frac{s^3-s}{{\epsilon }^2},$ and define the free energy on $H^1(\mathrm{\Omega };\mathbb{R})\cap L^4(\mathrm{\Omega };\mathbb{R})$ by ${\mathcal{E}}_{\epsilon }(u)={\int }_{\mathrm{\Omega }}(\frac{\kappa }{2}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2+F_{\epsilon }(u))dx\mathrm{.}$ Here $H^1=W^{1,2}$ and weak derivatives have the meaning in Integer-order Sobolev spaces and their norms. The finite-energy initial class consists of $u_0\in H^1(\mathrm{\Omega })\cap L^4(\mathrm{\Omega })$.

The Allen–Cahn gradient flow is $u_t=-M\mu ,\mu =-\kappa \mathrm{\Delta }u+f_{\epsilon }(u),{\mathrm{\partial }}_{n}u=0,u(0)=u_0\mathrm{.}$ The Cahn–Hilliard gradient flow is $u_t=M\mathrm{\Delta }\mu ,\mu =-\kappa \mathrm{\Delta }u+f_{\epsilon }(u),{\mathrm{\partial }}_{n}u={\mathrm{\partial }}_n\mu =0,u(0)=u_0\mathrm{.}$ For weak solutions, the no-flux conditions are encoded by the natural weak formulations with all $H^1(\mathrm{\Omega })$ test functions. In dimensions $d\ge 5$, the finite-energy class $H^1\cap L^4$ alone does not in general make $f_{\epsilon }(u)$ a continuous functional on $H^1(\mathrm{\Omega })$: it gives only $f_{\epsilon }(u)\in L^{4\mathrm{/}3}$, while $H^1(\mathrm{\Omega })$ need not embed into $L^4(\mathrm{\Omega })$. Thus any weak solution tested against all $H^1$ functions must have the additional integrability needed to make the nonlinear pairing continuous. The chemical potential $\mu$ is the variational derivative of ${\mathcal{E}}_{\epsilon }$, and the mobilities select the $L^2$ metric for Allen–Cahn and the mean-zero $H^{-1}$ metric for Cahn–Hilliard. In the Cahn–Hilliard flow, the no-flux equation conserves ${\int }_{\mathrm{\Omega }}u$; the Allen–Cahn flow has no such conservation law in general. No maximum-principle property is part of these definitions.

The convergence theorem on this page is restricted to one space dimension. Assuming the Axiom of Choice for the following standard embedding consequences, if $d=2$ and $\mathrm{\Omega }$ is a bounded $W^{1,p}$ extension domain for every $1<p<2$ as defined in Sobolev extension domains and extension operators, then The critical Sobolev embedding into every finite $L^q$ gives $H^1(\mathrm{\Omega })\hookrightarrow L^4(\mathrm{\Omega })$. If $d=3$ and $\mathrm{\Omega }$ is a bounded $W^{1,2}$ extension domain, then Sobolev embedding on bounded extension domains for $p<n$ gives $H^1(\mathrm{\Omega })\hookrightarrow L^4(\mathrm{\Omega })$. These are the exact extra extension hypotheses for the quartic-energy claim. In dimensions $d\ge 4$ no assertion that every $H^1$ state has finite quartic energy is made; the explicit initial class above remains $H^1\cap L^4$. No sharp-interface limit as $\epsilon \downarrow 0$ is part of either definition.