AI 中文总结
该研究针对$d\le4$的有界光滑区域中带双井势、输运型噪声和自然Neumann边界条件的随机Cahn-Hilliard方程，运用随机极大正则性技术与能量估计，证明了其局部和整体适定性，明确了不同初始数据下解的正则性等关键结果。
AI 中文摘要
我们研究了维数$d\le4$、具有双井势、输运型噪声和自然Neumann边界条件的有界光滑区域中的随机Cahn-Hilliard方程。通过运用随机极大正则性技术并推导合适的能量估计，我们证明了其局部和整体适定性。本文考虑的初始数据允许属于临界迹空间$B^{d/q-1}_{q,p}$，该空间在Cahn-Hilliard方程的自然缩放下是局部不变的。特别地，对于任意$\varepsilon>1/3$，可找到足够大的$q$，使得对每个初始数据$u_0\in H^{\varepsilon, q}(\mathscr{O})$，概率强解的唯一性和整体存在性成立；若$u_0\in H^1(\mathscr{O})$，则这些整体解在有限时间区间上具有$L_t^2H_x^3 \cap C_tH_x^1$正则性。
英文摘要
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.
Comments46 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!
