论文标题
通过混合抑制外延薄膜生长
Suppression of epitaxial thin film growth by mixing
论文作者
论文摘要
我们考虑以下在二维圆环上具有梯度非线性的四阶抛物线方程，在情况下有和不向后进行不可压缩的矢量字段，$ 2 <p <3 $：\ begin {equination {equination {equination {equination*} \ partial_t u +（-Δ）^2 u = - \ nabla \ cdot（| \ nabla u |^{p-2} {p-2} \ nabla u）。 \ end {等式*}对这种方程式的研究来自模拟薄膜的外延生长的数学模型。在两种情况下，我们都证明，对于任何初始数据的局部存在，均为$ l^2 $。我们的主要结果是：在对流案例中，如果施加的对流足够混合，则可以证明解决方案的全球存在，并且该解决方案将指数融合到均匀的混合状态。虽然在没有优势的情况下，在$ h^2 \ cap w^{1，\ infty} $中存在初始数据，以便在有限的时间内爆炸。
We consider following fourth-order parabolic equation with gradient nonlinearity on the two-dimensional torus with and without advection of an incompressible vector field in the case $2<p<3$: \begin{equation*} \partial_t u + (-Δ)^2 u = -\nabla\cdot(|\nabla u|^{p-2}\nabla u). \end{equation*} The study of this form of equations arises from mathematical models that simulate the epitaxial growth of the thin film. We prove the local existence of mild solutions for any initial data lies in $L^2$ in both cases. Our main result is: in the advective case, if the imposed advection is sufficiently mixing, then the global existence of solution can be proved, and the solution will converge exponentially to a homogeneous mixed state. While in the absence of advection, there exist initial data in $H^2\cap W^{1,\infty}$ such that the solution will blow up in finite time.
