Decay estimates of gradient of the Stokes semigroup in exterior Lipschitz domains

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Abstract

This paper develops Lp-Lq decay estimates of the gradient of the Stokes semigroup (T(t))t≥0 generated by the negative of the Stokes operator in exterior Lipschitz domains Ω⊂Rn, n≥3. More precisely, the Lp-Lq estimates of ∇T(t) with optimal rates are proved if p and q satisfy |1/p−1/2|<1/(2n)+ε, |1/q−1/2|<1/(2n)+ε, and p≤q≤n with some ε>0, which should be useful for the study of stability of the several physically relevant flows. As an application, we obtain the existence theorem of global-in-time strong solutions to the three-dimensional Navier–Stokes equations in the critical space L(0,∞;Lσ3(Ω)) provided that the initial velocity is small in the L3-norm.

Original languageEnglish
Pages (from-to)277-312
Number of pages36
JournalJournal of Differential Equations
Volume346
DOIs
Publication statusPublished - 2023 Feb 15

Keywords

  • Exterior domains
  • Lipschitz domains
  • Navier–Stokes equations
  • Stokes semigroup

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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