On October 23, "Resolvent Estimates in L∞ for the Stokes Operator in Nonsmooth Domains" by Zhongwei Shen, Professor at the Institute for Theoretical Sciences, Westlake University and Jun Geng, Professor at the Lanzhou University, was published online in Inventiones Mathematicae, a leading international journal of mathematics.
This paper achieves a major breakthrough in the resolvent estimates and semigroup estimates of the Stokes operator in bounded domains. Professors Zhongwei Shen and Jun Geng introduced and proved a novel class of pressure estimates, which they combined with the localized Lq resolvent estimates established in their prior work. As a result, they demonstrated that the Stokes operator generates a uniformly bounded analytic semigroup on the spaces L∞σ(Ω) and C0,σ(Ω) under the optimal assumptions on the boundary of the domain.
The study successfully addresses two long-standing challenges: (1) the L∞ estimate, and (2) the non-smooth domains. Regarding the point (1), Abe and Giga, in 2013, used a blow-up method to prove the analyticity of the Stokes semigroup in the spaces of bounded solenoidal functions in smooth domains. As for (2), Zhongwei Shen tackled the problem on Lipschitz domains in 2012, establishing the resolvent estimates in Lq spaces for q close to 2 in dimensions d≥3. The challenge with rough domains, such as those with Lipschitz or C1 boundaries, lies in the difficulty of obtaining quantitative improvements when locally magnifying the boundary, and many gradient regularity estimates fail in such domains. The main technical contribution of this paper is the introduction of new pressure estimates that connect the pressure to the gradient of the velocity in the Lq average, paving a new path for research in related fields.
Link: https://link.springer.com/article/10.1007/s00222-025-01383-4
