Necessary and sufficient conditions on local strong solvability of the Navier-Stokes system

Consider in a smooth bounded domain 3 and a time interval [0, T), 0 T , the Navier-Stokes system with the initial value [image omitted] and the external force f = div F, F L2(0, T; L2()). As is well-known, there exists at least one weak solution on [0, T) x in the sense of Leray-Hopf; then it is an important problem to develop conditions on the data u0, f as weak as possible to guarantee the existence of a unique strong solution u Ls(0, T; Lq()) satisfying Serrin's condition [image omitted] with 2 s , 3 q at least if T 0 is sufficiently small. Up to now there are known several sufficient conditions, yielding a larger class of corresponding local strong solutions, step by step, during the past years. Our following result is optimal in a certain sense yielding the largest possible class of such local strong solutions: let E be the weak solution of the linearized system (Stokes system) in [0, T) x with the same data u0, f. Then we show that the smallness condition ELs(0,T;Lq()) epsilon* with some constant epsilon* = epsilon*(, q) 0 is sufficient for the existence of such a strong solution u in [0, T). This leads to the following sufficient and necessary condition: Given F L2(0, ; L2()), there exists a strong solution u Ls(0, T; Lq()) in some interval [0, T), 0 T , if and only if E Ls(0, T'; Lq()) with some 0 T' .