On stability of weak Navier–Stokes solutions with large L^3,∞ initial data

We consider the Cauchy problem for the Navier–Stokes equation in ℝ³×]0,∞[ with the initial datum u₀ ∈ L³ _weak, a critical space containing nontrivial (−1)−homogeneous fields. For small ||u₀||L³ _weak one can get global well-posedness by perturbation theory. When ||u₀||L³ _weak 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 u⁽ⁿ⁾ ₀, and u⁽ⁿ⁾ ₀ converge weakly* in L³ _weak 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.