Periodic Cahn–Hilliard incompressible flow system

Fix an integer $d\ge 1$, a time interval $[0,T]$ with $T>0$, constants $\epsilon ,M,\nu >0$, and a potential $W\in C^2(\mathbb{R})$ that is bounded below. Use the torus, integral, periodic-lift, and time-regularity conventions of periodic torus calculus, with all spatial derivatives taken on the lifts. For vectors $a,b\in {\mathbb{R}}^d$, write $a\cdot b={\sum }_{j=1}^d{a}_j{b}_j$ and $\mathrm{\mid }a{\mathrm{\mid }}^2=a\cdot a$; set $(\mathrm{\Delta }u{)}_i={\sum }_{j=1}^d{\mathrm{\partial }}_j^2{u}_i$, $(\mathrm{\nabla }u{)}_{ij}={\mathrm{\partial }}_j{u}_i$, and $(u\cdot \mathrm{\nabla })f={\sum }_{j=1}^d{u}_j{\mathrm{\partial }}_{j}f$, componentwise when $f$ is vector-valued. Write $\mathrm{\nabla }\cdot u=\mathrm{div}u$. A classical tuple consists of periodic fields $u:[0,T]\times {\mathbb{T}}^d\to {\mathbb{R}}^d$, $\varphi :[0,T]\times {\mathbb{T}}^d\to \mathbb{R}$, and $\pi :[0,T]\times {\mathbb{T}}^d\to \mathbb{R}$, with $u\in C^1([0,T];C_{\mathrm{p}\mathrm{e}\mathrm{r}}^2)$, $\varphi \in C^1([0,T];C_{\mathrm{p}\mathrm{e}\mathrm{r}}^1)\cap C([0,T];C_{\mathrm{p}\mathrm{e}\mathrm{r}}^2)$, and $\pi \in C([0,T];C_{\mathrm{p}\mathrm{e}\mathrm{r}}^1)$. Define the chemical-potential field, which is not an independent unknown, by

${\mu }_c:={\epsilon }^{-1}{W}^{\mathrm{\prime }}(\varphi )-\epsilon \mathrm{\Delta }\varphi ,$

and require ${\mu }_c\in C([0,T];C_{\mathrm{p}\mathrm{e}\mathrm{r}}^2)$ so that its Laplacian is pointwise defined. The dimensionless periodic incompressible phase-flow system is

${\mathrm{\partial }}_t\varphi +u\cdot \mathrm{\nabla }\varphi =M\mathrm{\Delta }{\mu }_c,{\mathrm{\partial }}_{t}u+(u\cdot \mathrm{\nabla })u+\mathrm{\nabla }\pi =\nu \mathrm{\Delta }u-\varphi \mathrm{\nabla }{\mu }_c,\mathrm{\nabla }\cdot u=0.$

Here $\varphi$ is a scalar order parameter; “diffuse interface” refers only to representing interfacial structure through this field and its spatial-gradient energy. The scalar $\pi$ is the multiplier enforcing the divergence constraint. The associated total energy is

$E(t):={\int }_{{\mathbb{T}}^d}(\frac{1}{2}\mathrm{\mid }u(t,x){\mathrm{\mid }}^2+\frac{\epsilon }{2}\mathrm{\mid }\mathrm{\nabla }\varphi (t,x){\mathrm{\mid }}^2+{\epsilon }^{-1}W(\varphi (t,x)))dx\mathrm{.}$

All coefficients and fields in this definition are dimensionless mathematical quantities; no physical units, empirical calibration, sharp-interface limit, or claim that $\varphi$ is a probability or positive density is asserted. A physical identification requires additional density, phase-scale, surface-energy, length/time-scale, and chemical-potential units.

Torus, integral, and periodic-field conventions