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

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.