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

TL;DR

This paper establishes short-time uniqueness of weak Leray-Hopf solutions to the 3D Navier-Stokes equations for initial data in certain critical spaces, extending known results and providing new continuity properties near initial time.

Contribution

It introduces new classes of initial data ensuring uniqueness and proves continuity properties of solutions near initial time, advancing understanding of Navier-Stokes solution behavior.

Findings

Uniqueness for initial data in $VMO^{-1}$ and certain Besov spaces.

Continuity properties of solutions near initial time.

Extension of Prodi-Serrin conditions ensuring solution uniqueness.

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 $3/p+2/q=1$.