论文标题
具有广义Cahn-Hilliard流的单径向界面的溶液
Solutions with single radial interface of the generalized Cahn-Hilliard flow
论文作者
论文摘要
我们考虑通用的抛物线抛物线cahn -hilliard方程$$ u_t =-Δ\ weft [ΔU-W'（u）\右]+w''（u）\ left [ΔU-W'（u） $ n $ n = 2 $或$ n \ geq 4 $，$ w（\ cdot）$是典型的双孔电势函数，$ \ widetilde {\ mathbb r} $由$ \ wideTilde {\ mathbb r} = \ weft \ weft \ left \ weft \ ews { \ begin {array} {rl} （0，\ infty），＆\ quad \ mbox {if} n = 2， （ - \ infty，0），＆\ quad \ mbox {if} n \ geq 4。 \ end {array} \正确的。 $$我们构建具有接口的径向解决方案$ u（t，x）$。在主要顺序上，该解决方案由稳态$ω（| x |）$的旅行副本组成，该副本满足$ω''（y）-w'（ω（y））= 0 $。它的界面与以下形式的范围的主要顺序副本相似 几何学。当$ n = 1 $或$ 3 $时，结果是微不足道的解决方案。
We consider the generalized parabolic Cahn-Hilliard equation $$ u_t=-Δ\left[Δu -W'(u)\right]+W''(u)\left[Δu -W'(u)\right] \qquad \forall\, (t, x)\in \widetilde{\mathbb R}\times{\mathbb R}^n, $$ where $n=2$ or $n\geq 4$, $W(\cdot)$ is the typical double-well potential function and $\widetilde{\mathbb R}$ is given by $$ \widetilde{\mathbb R}=\left\{ \begin{array}{rl} (0, \infty), &\quad \mbox{if } n=2, (-\infty, 0), & \quad\mbox{if } n\geq 4. \end{array} \right. $$ We construct a radial solution $u(t, x)$ possessing an interface. At main order this solution consists of a traveling copy of the steady state $ω(|x|)$, which satisfies $ω''(y)-W'(ω(y))=0$. Its interface is resemble at main order copy of the sphere of the following form $$ |x|=\sqrt[4]{-2(n-3)(n-1)^2t}, \qquad \forall\, (t, x)\in \widetilde{\mathbb R}\times{\mathbb R}^n, $$ which is a solution to the Willmore flow in Differential Geometry. When $n=1$ or $3$, the result consists trivial solutions.
