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

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 $L_2(\mathbb{R}^3)$ 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 $3/p+2/q=1$.