The stochastic Cahn-Hilliard equation in critical spaces

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  • Authors: Simon Bau, Gideon Chiusole, Sarah Geiss, Katharina Klioba, Tobias Werner
  • Preprint year: 2026
  • First public date: 2026-08-21
  • arXiv: 2608.21014

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 $B^{d/q-1}_{q,p}$, which is locally invariant under the natural scaling of the Cahn-Hilliard equation. In particular, for arbitrary $\varepsilon>1/3$ one can find $q$ sufficiently large such that uniqueness and global existence of a probabilistically strong solution hold for every initial datum $u_0\in H^{\varepsilon, q}(\mathscr{O})$. If $u_0\in H^1(\mathscr{O})$, then these global solutions have $L_t^2H_x^3 \cap C_tH_x^1$-regularity on finite time intervals.

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BibTeX

@article{arxiv260821014,
  title = {The stochastic Cahn-Hilliard equation in critical spaces},
  author = {Simon Bau and Gideon Chiusole and Sarah Geiss and Katharina Klioba and Tobias Werner},
  year = {2026},
  eprint = {2608.21014},
  archivePrefix = {arXiv},
  primaryClass = {math.PR}
}

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