Finite phase-field free energy does not impose an interval bound

Let $L>0$ metres, $U>0$ metres per second, ${\rho }_{\ast }>0$ kilograms per cubic metre, ${\mathrm{\Phi }}_{\ast }>0$ in a chosen phase unit $[\mathrm{\Phi }]$, and $S>0$ dimensionless. Let $\epsilon ,M,\nu >0$ be dimensionless, set $\beta ={\rho }_{\ast }{U}^2\mathrm{/}\epsilon$, ${\lambda }_{\varphi }={\rho }_{\ast }{U}^2{L}^2\epsilon \mathrm{/}{\mathrm{\Phi }}_{\ast }^2$, $\eta ={\rho }_{\ast }UL\nu$, and $\mathcal{M}=M{\mathrm{\Phi }}_{\ast }^{2}L\mathrm{/}({\rho }_{\ast }U)$, and choose the dimensionless potential $W(q)=(q^2-1{)}^2\mathrm{/}4$. Put $T_{\ast }=L\mathrm{/}U$, ${\mathrm{\Omega }}_L={\mathbb{R}}^3\mathrm{/}(2\pi L\mathbb{Z}{)}^3$, $f_b(\phi )=\beta W(\phi \mathrm{/}{\mathrm{\Phi }}_{\ast })$, $\mu =f_b^{\mathrm{\prime }}(\phi )-{\lambda }_{\varphi }{\mathrm{\Delta }}_x\phi$, and $D_{ij}=({\mathrm{\partial }}_{x_j}{u}_i+{\mathrm{\partial }}_{x_i}{u}_j)\mathrm{/}2$.

$\sigma =-p{I}_3+2\eta D+(f_b(\phi )-\phi \mu +\frac{{\lambda }_{\varphi }}{2}\mathrm{\mid }{\mathrm{\nabla }}_x\phi {\mathrm{\mid }}^2)I_3-{\lambda }_{\varphi }{\mathrm{\nabla }}_x\phi \otimes {\mathrm{\nabla }}_x\phi \mathrm{.}$

On this periodic domain, let $u$ be a velocity field in metres per second, $\phi$ a scalar phase field in unit $[\mathrm{\Phi }]$, and $p$ a scalar pressure in pascals. For a scalar field $g$, ${\mathrm{\nabla }}_{x}g=({\mathrm{\partial }}_{x_1}g,{\mathrm{\partial }}_{x_2}g,{\mathrm{\partial }}_{x_3}g)$ and ${\mathrm{\Delta }}_{x}g={\sum }_j{\mathrm{\partial }}_{x_j}^{2}g$; for a vector field $u$, ${\mathrm{\nabla }}_x\cdot u={\sum }_j{\mathrm{\partial }}_{x_j}{u}_j$; $I_3$ is the $3\times 3$ identity. Consider the adopted equations ${\mathrm{\partial }}_t\phi +u\cdot {\mathrm{\nabla }}_x\phi =\mathcal{M}{\mathrm{\Delta }}_x\mu$, ${\mathrm{\nabla }}_x\cdot u=0$, and ${\rho }_{\ast }({\mathrm{\partial }}_{t}u+(u\cdot {\mathrm{\nabla }}_x)u)={\mathrm{div}}_x\sigma$, where $\sigma =-p{I}_3+2\eta D+(f_b-\phi \mu +{\lambda }_{\varphi }\mathrm{\mid }{\mathrm{\nabla }}_x\phi {\mathrm{\mid }}^2\mathrm{/}2)I_3-{\lambda }_{\varphi }{\mathrm{\nabla }}_x\phi \otimes {\mathrm{\nabla }}_x\phi$ and $({\mathrm{div}}_x\sigma {)}_i={\sum }_j{\mathrm{\partial }}_{x_j}{\sigma }_{ij}$. Define the kinetic-plus-phase free energy by $\mathcal{G}(t)={\int }_{{\mathrm{\Omega }}_L}({\rho }_{\ast }\mathrm{\mid }u{\mathrm{\mid }}^2\mathrm{/}2+f_b(\phi )+{\lambda }_{\varphi }\mathrm{\mid }{\mathrm{\nabla }}_x\phi {\mathrm{\mid }}^2\mathrm{/}2)dx$. The constant fields $u=0$, $\phi =2{\mathrm{\Phi }}_{\ast }$, and $p=0$ satisfy these equations for all $t\in [0,T_{\ast }S]$. Their chemical potential is the constant $\mu =6\beta \mathrm{/}{\mathrm{\Phi }}_{\ast }$, and their kinetic-plus-phase free energy is $\mathcal{G}(t)=(2\pi L{)}^3\beta W(2)=\frac{9}{4}(2\pi L{)}^3\beta <\mathrm{\infty }$. Thus this smooth model solution has normalized order parameter $\phi \mathrm{/}{\mathrm{\Phi }}_{\ast }=2$ outside $[-1,1]$ while its free energy is finite and constant. The model's energy and equations do not impose that interval as a pointwise constraint; this example is not a maximum-principle theorem, stability claim, or reported observation.

Adopted dimensional phase-field equations and quartic-potential model

Conditional smooth periodic free-energy identity