Year: 2025
Type: article
Abstract: We establish resolvent estimates in spaces of bounded solenoidal functions for the Stokes operator in a bounded domain $\Omega $ Ω in $\mathbb{R}^{d}$ R d under the assumptions that $\Omega $ Ω is $C^{1}$ C 1 for $d\ge 3$ d ≥ 3 and Lipschitz for $d=2$ d = 2 . As a corollary, it follows that the Stokes operator generates a uniformly bounded analytic semigroup in the spaces of bounded solenoidal functions in $\Omega $ Ω . The smoothness conditions on $\Omega $ Ω are sharp. The case of exterior domains with nonsmooth boundaries is also studied. The key step in our proof involves new estimates that connect the pressure to the gradient of the velocity in the $L^{q}$ L q average, but only on scales above certain level.
Source: Inventiones mathematicae
Authors: Jun Geng, Zhongwei Shen
Institutions: Lanzhou University, Westlake University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4415466836