Global well-posedness of the Navier–Stokes equations in homogeneous Besov spaces on the half-space

Abstract

Consider the Stokes equations in the half-space R+n, n≧2. It is shown that the negative of the Stokes operator defined on the homogeneous Besov space B˙p,q,σs(R+n) generates a bounded strongly continuous semigroup in B˙p,qs(R+n) provided that 1<p<∞, 1≦q<∞, and −1+1/p<s<1/p. As a by-product, the maximal Lq-regularity of the Stokes operator is obtained, admitting the limiting case q=1. This will be applied to develop the global well-posedness result for the incompressible Navier–Stokes equations in the maximal L1-regularity class provided that the initial data are small in B˙p,1−1+n/p(R+n) with n−1<p<∞. The dependence of the solution on initial data and the large time behavior of the solution are investigated.