关于由时空白噪声散度驱动的二维Cahn-Hilliard方程的注记

中文摘要 AI 辅助

我们研究由时空白噪声散度驱动的二维Cahn-Hilliard方程，该方程代表保守形式的Kawasaki动力学。标准启发式论证表明其解是分布，导致非线性项内的乘积无定义。我们证明了全局时间唯一解理论，除了处理无定义乘积所需的最少标准重整化程序外，证明还运用了确定性分析工具并利用了方程的独特结构，这一点至关重要。

英文摘要

We consider the two-dimensional Cahn-Hilliard equation forced by divergence of space-time white noise that represents Kawasaki dynamics in conservative form. The standard heuristic argument shows that the solution is a distribution and thus the product within the nonlinear term is ill-defined, making this stochastic partial differential equation challenging. We prove the global-in-time unique solution theory after multiple Da Prato-Debussche tricks where the last equation is solved in a weak formulation. Besides a minimum amount of the standard renormalization procedure to deal with ill-defined products, our proof consists of applications of deterministic analysis tools and taking advantage of the unique structure of the equation, which is crucial. We also prove the local-in-time unique solution theory of a mild solution, demonstrating that our improved estimate gets us arbitrarily close, although not quite, to being able to prove the global-in-time solution theory with the last equation in not only a weak formulation but also a mild formulation. Additionally, our approach can be readily extended to prove the global-in-time existence, although not path-wise uniqueness, of solution to the singular surface quasi-geostrophic equations with the resulting equation after multiple Da Prato-Debussche tricks in weak formulation.

发表机构

• University of Nebraska（内布拉斯加大学）

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