Necessary and sufficient condition on initial data in the Besov space for solutions in the Serrin class of the Navier–Stokes equations

The Cauchy problem of the Navier–Stokes equations in Rⁿ with the initial data a in the Besov space Bp,q-1+np(Rn) for n< p< ∞ and 1 ≤ q≤ ∞ is considered. We construct the local solution in Lα,q(0,T;Br,10(Rn)) for p≤ r< ∞ satisfying 2α+nr=1 with the initial data a∈Bp,q-1+np(Rn), where L^α,q denotes the Lorentz space. Conversely, if the solution belongs to L^α,q(0 , T; L^r(Rⁿ)) with 2α+nr=1, then the initial data a necessarily belong to Br,q-1+nr(Rn). It implies that the initial data in the Besov space Bp,q-1+np(Rn) are a necessary and sufficient condition for the existence of solutions in the Serrin class.