On stability of weak Navier–Stokes solutions with largeL3,∞initial data

We consider the Cauchy problem for the Navier–Stokes equation in ℝ3×]0,∞[ with the initial datum u0∈Lweak3, a critical space containing nontrivial (−1)−homogeneous fields. For small ||u0||Lweak3 one can get global well-posedness by perturbation theory. When ||u0||Lweak3 is not small, the perturbation theory no longer applies and, very likely, the local-in-time well-posedness and uniqueness fails. One can still develop a good theory of weak solutions with the following stability property: If u(n) are weak solutions corresponding the the initial datum u0(n), and u0(n) converge weakly* in Lweak3 to u0, then a suitable subsequence of u(n) converges to a weak solution u corresponding to the initial condition u0. This is of interest even in the special case u0≡0.