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

Patrick Tolksdorf*, Keiichi Watanabe

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


We show that the Stokes operator defined on Lσp(Ω) for an exterior Lipschitz domain Ω⊂Rn (n≥3) admits maximal regularity provided that p satisfies |1/p−1/2|<1/(2n)+ε for some ε>0. In particular, we prove that the negative of the Stokes operator generates a bounded analytic semigroup on Lσp(Ω) for such p. In addition, Lp-Lq-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(0,T;Lσ3(Ω)) (locally in time and globally in time for small initial data).

Original languageEnglish
Pages (from-to)5765-5801
Number of pages37
JournalJournal of Differential Equations
Issue number7
Publication statusPublished - 2020 Sept 15


  • Exterior domains
  • Lipschitz domains
  • Navier–Stokes equations
  • R-bounded
  • Stokes semigroup

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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