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

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 Q subset of Rn, n >= 3. More precisely, the Lp-Lq estimates of backward difference T (t) with optimal rates are proved if p and q satisfy |1/p - 1/2| < 1/(2n) + 6, |1/ q - 1/2| < 1/(2n) + c, and p <= q <= n with some c > 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 infinity(0, infinity; L3 sigma (Q)) provided that the initial velocity is small in the L3-norm. (c) 2022 Elsevier Inc. All rights reserved.

Automated summary

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 Q subset of Rn, n >= 3. The Lp-Lq estimates of backward difference T (t) with optimal rates are proved if p and q satisfy |1/p - 1/2| < 1/(2n) + 6, |1/ q - 1/2| < 1/(2n) + c, and p <= q <= n with some c > 0. 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 infinity(0, infinity; L3 sigma (Q)) provided that the initial velocity is small in the L3-norm.