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

Abstract

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 (R 3 ) norm, that also belongs to certain subsets of VMO −1 (R 3 ). As a corollary of this, we obtain the same conclusion for any solenodial u 0 belonging todenotes the closure of test functions in the critical Besov spaceḂOur 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 R 3 ×]0, T [, then it is unique on 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 ∈ L q,s (0,

Localized Smoothing for the Navier–Stokes Equations and Concentration of Critical Norms Near Singularities

“…Notice that the result we prove in Section 4 is stronger. Indeed, considering the mild solution a associated to an L 3 continuous divergence-free extension of the critical data u 0 | B 1 (0) , we prove that 1 (4) u − a ∈ C 0,ν par (B 1…”

“…In particular, in the Caffarelli, Kohn and Nirenberg-type iteration, we need to exploit the local decay of the kinetic energy near the initial time, because the critical drift is more singular in the Besov case than in the L 3 case. Such an insight was used before for global estimates by Barker [5] to prove weak-strong uniqueness, in Barker, Seregin andŠverák's paper [7] on global L 3,∞ solutions and by Albritton and Barker [2] for global Besov solutions. However, to the best of our knowledge, it is the first time that the decay of the kinetic energy near initial time is used in local estimates, such as a Caffarelli, Kohn and Nirenberg-type iteration.…”

“…Related arguments were previously used in the context of the Navier-Stokes equations by Lin in [37] and by Ladyzhenskaya and Seregin in [32]. Such arguments applied to the system (5) crucially use that for a in subcritical spaces, we have parabolic Hölder continuity in B 1 2 (0) × (− 1 4 , 0) for the following linear system: (6) ∂ t w−∆w+a·∇w+w·∇a+∇q = ∇·G, ∇·a = 0, ∇·w = 0 in B 1 (0)×(−1, 0), for a subcritical forcing term G ∈ L 5m 3 (B 1 (0)×(−1, 0)). Unfortunately, when u 0 is critical and hence a, G belong to scale invariant spaces with respect to the Navier-Stokes scaling 2 such as L 5 (B 1 (0) × (−1, 0)), we do not expect solutions of (6) to be Hölder continuous.…”

“…Let t 0 ∈ [−1, 0] and η ∈ (0, 1) be fixed. For all δ ∈ (0, 3), there existsC * (δ) ∈ (0, ∞), for all E ∈ (0, ∞), there exists ε * (δ, η, E) ∈ (0, ∞), for all a such that sup s∈(−1,0) |s − t 0 | a(·, s) L ∞ (B 1 (0)) < ∞ and all local suitable solution v to (19) in Q 1 (0, 0) such that (0) |v(x, s)| 2 dx + ŝ −1B 1 (0) |∇v| 2 dxds ≤ E(s − t 0 ) η + , ∀s ∈ (−1, 0), a(·, s) L ∞ (B 1 (0)) ≤ ε * and dxds ≤ ε *imply that for all (x, t) ∈Q 1/2 (0, 0), for all r ∈ (0r −δ 5. By definition (·)+ := max(0, ·) 6.…”

Blowup criterion for Navier–Stokes equation in critical Besov space with spatial dimensions d ≥ 4

“…On the other hand, there are lots of interests in relaxing the condition (1.2), for instance, Phuc showed the same conclusion for the 3D Leray-Hopf solution u by assuming u ∈ L ∞ (0, T ; L 3,m ) with 3 ≤ m < ∞, see [38]. Besides, according to [22,4], the same result also applies for u in 3D provided u ∈ L ∞ (0, T ;Ḃ s p p,q ) with 3 < p, q < ∞, a natural extension to the higer dimension in such setting is one of aims of our current paper. Finally, we mention a very interesting work, Buckmaster and Vicol [8] recently demonstrates a nonuniqueness result for the periodic weak solution in T 3 with finite kinetic energy, unfortunately, this weak solution is still not known as a Leray-Hopf solution.…”