Remarks on the two-dimensional Cahn-Hilliard equation forced by divergence of space-time white noise

Abstract

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.