Characterization of initial data in the homogeneous Besov space for solutions in the Serrin class of the Navier-Stokes equations

Consider the Cauchy problem of the Navier-Stokes equations in Rⁿ with initial data a in the homogeneous Besov space [Formula presented] for n<p<∞ and 1≦q≦∞. We show that the Stokes flow e^tΔa can be controlled in L^α,q(0,∞;B˙_r,1 ⁰(Rⁿ)) for [Formula presented] with p≦r<∞, where L^α,q denotes the Lorentz space. As an application, the global existence theorem of mild solutions for the small initial data is established in the above class which is slightly stronger than Serrin's. Conversely, if the global solution belongs to the usual Serrin class L^α,q(0,∞;L^r(Rⁿ)) for r and α as above with 1<q≦∞, then the initial data a necessarily belongs to B˙_r,q ^−1+nr(Rⁿ). Moreover, we prove that such solutions are analytic in the space variables. Our method for the proof of analyticity is based on a priori estimates of higher derivatives of solutions in L^p(Rⁿ) with Hölder continuity in time.