Global existence, regularity, and uniqueness of infinite energy solutions to the Navier-Stokes equations

Abstract

This paper addresses several problems associated to local energy solutions (in the sense of Lemarié-Rieusset) to the Navier-Stokes equations with initial data which is sufficiently small at large or small scales as measured using truncated Morrey-type quantities, namely: (1) global existence for a class of data including the critical L 2based Morrey space; (2) initial and eventual regularity of local energy solutions to the Navier-Stokes equations with initial data sufficiently small at small or large scales; (3) small-large uniqueness of local energy solutions for data in the critical L 2 -based Morrey space. A number of interesting corollaries are included, including eventual regularity in familiar Lebesgue, Lorentz, and Morrey spaces, a new local generalized Von Wahl uniqueness criteria, as well as regularity and uniqueness for local energy solutions with small discretely self-similar data.1. If u 0 ∈ M 2,r where 0 ≤ r < 1, then u has eventual regularity and t 1/2 u(•, t) L ∞ is bounded for sufficiently large t. If u 0 ∈ M 2,r and 1 < r ≤ 3, then u has initial regularity and t 1/2 u(•, t) L ∞ is bounded for sufficiently small t. If u, then u has initial and eventual regularity and t 1/2 u(•, t) L ∞ is bounded for sufficiently small and large t. In particular, this is true if u 0 ∈ L 3,q for 1 ≤ q < ∞.

Mild criticality breaking for the Navier-Stokes equations

“…Finally, combining the short-time estimate (5) with the large-time estimate (7) and the intermediate-time estimate (10) yields the desired conclusion.…”

Characterisation of the pressure term in the incompressible Navier–Stokes equations on the whole space

“…uloc have been provided, such as Kikuchi and Seregin in 2007 [14] or Bradshaw and Tsai in 2019 [4]. The formulas proposed for the pressure, however, are actually equivalent, as they all imply that u is solution to the (MNS) problem.…”

Quantitative regularity for the Navier-Stokes equations via spatial concentration

“…Our first Theorem involves applying the backward propagation of concentration stated in Proposition 2 below to give a new necessary condition for solutions to the Navier-Stokes equations to possess a Type I blow-up. Note that if u is a finite-energy solution that first blows-up at T * > 0 we say that T * is a Type I blow-up if (9) u L ∞ (0,T * ;L 3,∞ (R 3 )) ≤ M.…”

“…All these arguments are qualitative and achieved by contradiction and compactness arguments. It is interesting to note that in contrast to (7) it is not known 9 , even abstractly, if there exists a G : (0, ∞) → (0, ∞) such that if u is a finite-energy solution of the Navier-Stokes equations belonging to C ∞ (R 3 × (0, 1]) then ( 16)…”

“…. 8 We say u : 9 Assume for contradiction that (16) does not hold. Then we have a sequence of solutions…”