论文标题
Cahn-Hilliard-Cook方程的完全离散方案的强收敛率
Strong convergence rates of a fully discrete scheme for the Cahn-Hilliard-Cook equation
论文作者
论文摘要
本文的第一个目的是检查空间维度中的Cahn-Hilliard-Cook（CHC）方程的存在，独特性和规律性$ d \ leq 3 $。通过在无限的尺寸方程式中应用光谱盖金方法，我们阐述了有限维度近似问题的适当性和规律性。关键思想在于将随机问题{\ color {black {black} {带有添加噪声}}}变成等效的随机方程。借助能量参数，在gagliardo-nirenberg的不等式的帮助下，在一个维度上获得了对等效随机方程的规律性，并在两个和三个维度上完成。此外，近似溶液被证明是与原始CHC方程的独特温和解决方案的强烈收敛，其时空的规律性可以通过类似的参数来实现。此外，研究了该问题的完全离散的近似，在空间中通过光谱Galerkin方法和及时的向后欧拉方法进行。以前获得的问题的规律性结果有助于我们确定完全离散方案的强收敛率。
The first aim of this paper is to examine existence, uniqueness and regularity for the Cahn-Hilliard-Cook (CHC) equation in space dimension $d\leq 3$. By applying a spectral Galerkin method to the infinite dimensional equation, we elaborate the well-posedness and regularity of the finite dimensional approximate problem. The key idea lies in transforming the stochastic problem {\color{black}{with additive noise}} into an equivalent random equation. The regularity of the solution to the equivalent random equation is obtained, in one dimension, with the aid of the Gagliardo-Nirenberg inequality and done in two and three dimensions, by the energy argument. Further, the approximate solution is shown to be strongly convergent to the unique mild solution of the original CHC equation, whose spatio-temporal regularity can be attained by similar arguments. In addition, a fully discrete approximation of such problem is investigated, performed by the spectral Galerkin method in space and the backward Euler method in time. The previously obtained regularity results of the problem help us to identify strong convergence rates of the fully discrete scheme.
