TY - JOUR

T1 - Decay estimates of gradient of the Stokes semigroup in exterior Lipschitz domains

AU - Watanabe, Keiichi

N1 - Funding Information:
This research was partly supported by JSPS Japan Society for the Promotion of Science Grant Number 20K22311 and 21K13826 and the Waseda University Grant for Special Research Projects (Project number: 2021C-583 ).
Publisher Copyright:
© 2022 Elsevier Inc.

PY - 2023/2/15

Y1 - 2023/2/15

N2 - 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.

AB - 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.

KW - Exterior domains

KW - Lipschitz domains

KW - Navier–Stokes equations

KW - Stokes semigroup

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U2 - 10.1016/j.jde.2022.11.045

DO - 10.1016/j.jde.2022.11.045

M3 - Article

AN - SCOPUS:85143132825

VL - 346

SP - 277

EP - 312

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

ER -