Abstract. The main subject of this paper concerns the establishment of certain classes of initial data, which grant short time uniqueness of the associated weak Leray-Hopf solutions of the three dimensional Navier-Stokes equations. In particular, our main theorem that this holds for any solenodial initial data, with finite L 2 (R 3 ) norm, that also belongs to certain subsets of VMO −1 (R 3 ). As a corollary of this, we obtain the same conclusion for any solenodial u 0 belonging todenotes the closure of test functions in the critical Besov spaceḂOur results rely on the establishment of certain continuity properties near the initial time, for weak Leray-Hopf solutions of the Navier-Stokes equations, with these classes of initial data. Such properties seem to be of independent interest. Consequently, we are also able to show if a weak Leray-Hopf solution u satisfies certain extensions of the Prodi-Serrin condition on R 3 ×]0, T [, then it is unique on R 3 ×]0, T [ amongst all other weak Leray-Hopf solutions with the same initial value. In particular, we show this is the case if u ∈ L q,s (0,

We introduce a notion of global weak solution to the Navier-Stokes equations in three dimensions with initial values in the critical homogeneous Besov spacesḂ −1+ 3 p p,∞ , p > 3. These solutions satisfy a certain stability property with respect to the weak- * convergence of initial conditions. To illustrate this property, we provide applications to blow-up criteria, minimal blow-up initial data, and forward self-similar solutions. Our proof relies on a new splitting result in homogeneous Besov spaces that may be of independent interest.

Assuming that T is a potential blow up time for the Navier-Stokes system in R 3 + , we show that the norm of the velocity field in the Lorenz space L 3,q with q < ∞ goes to ∞ as time t approaches T .

This paper addresses a question concerning the behaviour of a sequence of global solutions to the Navier-Stokes equations, with the corresponding sequence of smooth initial data being bounded in the (non-energy class) weak Lebesgue space L 3,∞ . It is closely related to the question of what would be a reasonable definition of global weak solutions with a non-energy class of initial data, including the aforementioned Lorentz space. This paper can be regarded as an extension of a similar problem regarding the Lebesgue space L 3 to the weak Lebesgue space L 3,∞ , whose norms are both scale invariant with the respect to the Navier-Stokes scaling.

This paper is concerned with two dual aspects of the regularity question of the Navier-Stokes equations. First, we prove a local in time localized smoothing effect for local energy solutions. More precisely, if the initial data restricted to the unit ball belongs to the scale-critical space L 3 , then the solution is locally smooth in space for some short time, which is quantified. This builds upon the work of Jia andŠverák, who considered the subcritical case. Second, we apply these localized smoothing estimates to prove a concentration phenomenon near a possible Type I blow-up. Namely, we show if (0, T * ) is a singular point thenThis result is inspired by and improves concentration results established by Li, Ozawa, and Wang and Maekawa, Miura, and Prange. We also extend our results to other critical spaces, namely L 3,∞ and the Besov spaceḂ −1+ 3 p p,∞ , p ∈ (3, ∞).