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 Lα,q denotes the Lorentz space. Conversely, if the solution belongs to Lα,q(0 , T; Lr(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.
ASJC Scopus subject areas
- Mathematics (miscellaneous)