The stochastic Cahn-Hilliard equation in critical spaces

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 $B_{q,p}^{d\mathrm{/}q-1}$, which is locally invariant under the natural scaling of the Cahn-Hilliard equation. In particular, for arbitrary $\epsilon >1\mathrm{/}3$ 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.