Uniqueness Results for Weak Leray-Hopf Solutions of the Navier-Stokes System with Initial Values in Critical Spaces

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 $L_2(\mathbb{R}^3)$ norm, that also belongs to to certain subsets of $VMO^{-1}(\mathbb{R}^3)$. As a corollary of this, we obtain the same conclusion for any solenodial $u_{0}$ belonging to $L_{2}(\mathbb{R}^3)\cap \mathbb{\dot{B}}^{-1+\frac{3}{p}}_{p,\infty}(\mathbb{R}^3)$, for any $3<p<\infty$. Here, $\mathbb{\dot{B}}^{-1+\frac{3}{p}}_{p,\infty}(\mathbb{R}^3)$ denotes the closure of test functions in the critical Besov space ${\dot{B}}^{-1+\frac{3}{p}}_{p,\infty}(\mathbb{R}^3)$. Our 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 $\mathbb{R}^3 \times ]0,T[$, then it is unique on $\mathbb{R}^3 \times ]0,T[$ amongst all other weak Leray-Hopf solutions with the same initial value. In particular, we show this is the case if $u\in L^{q,s}(0,T; L^{p,s}(\mathbb{R}^3))$ or if it's $L^{q,\infty}(0,T; L^{p,\infty}(\mathbb{R}^3))$ norm is sufficiently small, where $3<p< \infty$, $1\leq s<\infty$ and $3/p+2/q=1$.