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This paper develops L p - L q 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 ⊂ R n , n ≥ 3. More precisely, the L p - L q estimates of ∇ T (t) with optimal rates are proved if p and q satisfy | 1 / p − 1 / 2 | < 1 / (2 n) + ε , | 1 / q − 1 / 2 | < 1 / (2 n) + 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 , ∞ ; L σ 3 (Ω)) provided that the initial velocity is small in the L 3 -norm. [ABSTRACT FROM AUTHOR]

We show that the Stokes operator defined on L σ p (Ω) for an exterior Lipschitz domain Ω ⊂ R n (n ≥ 3) admits maximal regularity provided that p satisfies | 1 / p − 1 / 2 | < 1 / (2 n) + ε for some ε > 0. In particular, we prove that the negative of the Stokes operator generates a bounded analytic semigroup on L σ p (Ω) 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 ∞ (0 , T ; L σ 3 (Ω)) (locally in time and globally in time for small initial data). [ABSTRACT FROM AUTHOR]