论文标题
在耦合的cahn-hilliard方程系统上
On a system of coupled Cahn-Hilliard equations
论文作者
论文摘要
我们考虑一个由Cahn-Hilliard方程组成的系统，并在$ \ Mathbb {r}^d $，$ d = 2，3 $的有限域中与cahn-hilliard-oono方程组成。该系统通过两个订单参数$ u $和$ v $来解释聚合物混合物中的巨量和微相分离。该系统的自由能是双变量相互作用电位，其中包含两个阶参数的混合熵和合适的耦合项。该方程均具有初始条件和均匀的Neumann边界条件，包括$ U，V $以及相应的化学电位。我们首先证明所产生的问题在薄弱的意义上是很好的。然后，在保守的情况下，我们确定弱解决方案会立即正规化。此外，在两个空间维度中，我们显示了$ u $和$ v $的严格分离属性，即它们在有限的时间内均远离纯阶段$ \ pm 1 $。最后，我们研究了有限能量解决方案的长期行为，尤其是它会收敛到单个固定态。
We consider a system which consists of a Cahn-Hilliard equation coupled with a Cahn-Hilliard-Oono equation in a bounded domain of $\mathbb{R}^d$, $d = 2, 3$. This system accounts for macrophase and microphase separation in a polymer mixture through two order parameters $u$ and $v$. The free energy of this system is a bivariate interaction potential which contains the mixing entropy of the two order parameters and suitable coupling terms. The equations are endowed with initial conditions and homogeneous Neumann boundary conditions both for $u,v$ and for the corresponding chemical potentials. We first prove that the resulting problem is well posed in a weak sense. Then, in the conserved case, we establish that the weak solution regularizes instantaneously. Furthermore, in two spatial dimensions, we show the strict separation property for $u$ and $v$, namely, they both stay uniformly away from the pure phases $\pm 1$ in finite time. Finally, we investigate the long-time behavior of a finite energy solution showing, in particular, that it converges to a single stationary state.
