The stochastic Cahn-Hilliard equation in critical spaces

Abstract

We study stochastic Cahn-Hilliard equations in bounded smooth domains with a double-well potential, transport-type noise, and natural Neumann boundary conditions in dimensions $d\le 4$. By employing stochastic maximal regularity techniques and deriving suitable energy estimates, we prove local and global well-posedness. The initial data considered here are allowed to belong to the critical trace space $B_{q,p}^{d/q-1}$, which is locally invariant under the natural scaling of the Cahn-Hilliard equation. In particular, for arbitrary $\epsilon >1/3$ one can find $q$ sufficiently large such that uniqueness and global existence of a probabilistically strong solution hold for every initial datum $u_0\in H^{\epsilon ,q}(\mathcal{O})$. If $u_0\in H^1(\mathcal{O})$, then these global solutions have $L_t^2{H}_x^3\cap C_t{H}_x^1$-regularity on finite time intervals.

arXiv comment46 pages. Contribution to the Oberwolfach Report for the Seminar 'Stochastic Partial Differential Equations in Critical Spaces' organized by Antonio Agresti and Mark Veraar. Comments are welcome!