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

Uniqueness Results for Weak Leray–Hopf Solutions of the Navier–Stokes System with Initial... 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 $$L_2(\mathbb {R}^3)$$ L 2 ( R 3 ) norm, that also belongs to certain subsets of $${\textit{VMO}}^{-1}(\mathbb {R}^3)$$ VMO - 1 ( R 3 ) . As a corollary of this, we obtain the same conclusion for any solenodial $$u_{0}$$ u 0 belonging to $$L_{2}(\mathbb {R}^3)\cap \mathbb {\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }(\mathbb {R}^3)$$ L 2 ( R 3 ) ∩ B ˙ p , ∞ - 1 + 3 p ( R 3 ) , for any $$3<p<\infty$$ 3 < p < ∞ . Here, $$\mathbb {\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }(\mathbb {R}^3)$$ B ˙ p , ∞ - 1 + 3 p ( R 3 ) denotes the closure of test functions in the critical Besov space $${\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }(\mathbb {R}^3)$$ B ˙ p , ∞ - 1 + 3 p ( 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 $$\mathbb {R}^3 \times ]0,T[$$ R 3 × ] 0 , T [ , then it is unique on $$\mathbb {R}^3 \times ]0,T[$$ 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 $$u\in L^{q,s}(0,T; L^{p,s}(\mathbb {R}^3))$$ u ∈ L q , s ( 0 , T ; L p , s ( R 3 ) ) or if it’s $$L^{q,\infty }(0,T; L^{p,\infty }(\mathbb {R}^3))$$ L q , ∞ ( 0 , T ; L p , ∞ ( R 3 ) ) norm is sufficiently small, where $$3<p< \infty$$ 3 < p < ∞ , $$1\le s<\infty$$ 1 ≤ s < ∞ and $$3/p+2/q=1$$ 3 / p + 2 / q = 1 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Mathematical Fluid Mechanics Springer Journals

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

, Volume 20 (1) – Jan 30, 2017
28 pages

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Publisher
Springer International Publishing
Copyright
Copyright © 2017 by The Author(s)
Subject
Physics; Fluid- and Aerodynamics; Mathematical Methods in Physics; Classical and Continuum Physics
ISSN
1422-6928
eISSN
1422-6952
D.O.I.
10.1007/s00021-017-0315-8
Publisher site
See Article on Publisher Site

### 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 $$L_2(\mathbb {R}^3)$$ L 2 ( R 3 ) norm, that also belongs to certain subsets of $${\textit{VMO}}^{-1}(\mathbb {R}^3)$$ VMO - 1 ( R 3 ) . As a corollary of this, we obtain the same conclusion for any solenodial $$u_{0}$$ u 0 belonging to $$L_{2}(\mathbb {R}^3)\cap \mathbb {\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }(\mathbb {R}^3)$$ L 2 ( R 3 ) ∩ B ˙ p , ∞ - 1 + 3 p ( R 3 ) , for any $$3<p<\infty$$ 3 < p < ∞ . Here, $$\mathbb {\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }(\mathbb {R}^3)$$ B ˙ p , ∞ - 1 + 3 p ( R 3 ) denotes the closure of test functions in the critical Besov space $${\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }(\mathbb {R}^3)$$ B ˙ p , ∞ - 1 + 3 p ( 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 $$\mathbb {R}^3 \times ]0,T[$$ R 3 × ] 0 , T [ , then it is unique on $$\mathbb {R}^3 \times ]0,T[$$ 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 $$u\in L^{q,s}(0,T; L^{p,s}(\mathbb {R}^3))$$ u ∈ L q , s ( 0 , T ; L p , s ( R 3 ) ) or if it’s $$L^{q,\infty }(0,T; L^{p,\infty }(\mathbb {R}^3))$$ L q , ∞ ( 0 , T ; L p , ∞ ( R 3 ) ) norm is sufficiently small, where $$3<p< \infty$$ 3 < p < ∞ , $$1\le s<\infty$$ 1 ≤ s < ∞ and $$3/p+2/q=1$$ 3 / p + 2 / q = 1 .

### Journal

Journal of Mathematical Fluid MechanicsSpringer Journals

Published: Jan 30, 2017

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