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

• Published in: arXiv.org
• Main Author: Barker, T
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 08.12.2016
• Subjects: Fluid dynamics; Fluid flow; Function space; Mathematical analysis; Mathematics - Analysis of PDEs; Navier-Stokes equations; Uniqueness
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1610.08348

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 \(3/p+2/q=1\).