The Navier-Stokes equations in exterior Lipschitz domains: Lp-theory

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:269

Article

Tolksdorf, Patrick¹  Watanabe, Keiichi² 

[1] Johannes Gutenberg Univ Mainz, Inst Math, Staudingerweg 9, D-55099 Mainz, Germany

[2] Waseda Univ, Grad Sch Fundamental Sci & Engn, Dept Pure & Appl Math, Shinjuku Ku, 3-4-1 Ookubo, Tokyo 1698555, Japan

关键词: Navier-Stokes equations;    Stokes semigroup;    Lipschitz domains;    Exterior domains;    R-bounded;   

DOI  :  10.1016/j.jde.2020.04.015

来源: Elsevier

【 摘 要 】

We show that the Stokes operator defined on L-sigma(p)(Omega) for an exterior Lipschitz domain Omega subset of R-n (n >= 3) admits maximal regularity provided that p satisfies vertical bar 1/p - 1/2 vertical bar < 1/(2n) + epsilon for some epsilon > 0. In particular, we prove that the negative of the Stokes operator generates a bounded analytic semigroup on L-sigma(p)(Omega) for such p. In addition, L-p-L-q-mapping properties of the Stokes semigroup and its gradient with optimal decay estimates are obtained. This enables us to prove the existence of mild solutions to the Navier-Stokes equations in the critical space L-infinity (0, T; L-sigma(3)(Omega)) (locally in time and globally in time for small initial data). (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free