Quantitative regularity for the Navier-Stokes equations via spatial concentration

Abstract

This paper is concerned with quantitative estimates for the Navier-Stokes equations.First we investigate the relation of quantitative bounds to the behaviour of critical norms near a potential singularity with Type I bound uWe demonstrate that this potential blow-up rate is optimal for a certain class of potential non-zero backward discretely self-similar solutions. Second, we quantify the result of Seregin ( 2012), which says that if u is a smooth finiteenergy solution to the Navier-Stokes equations on R 3 × (0, 1) withTo prove our results we develop a new strategy for proving quantitative bounds for the Navier-Stokes equations. This hinges on local-in-space smoothing results (near the initial time) established by Jia and Šverák ( 2014), together with quantitative arguments using Carleman inequalities given by Tao (2019).

The localized characterization for the singularity formation in the Navier-Stokes equations

“…In conclusion, if one characterizes the local behaviors of solutions to the Navier-Stokes equations near a potential singularity by critical norm L 3 , the optimal blow-up rate was obtained by Barker and Prange [3]. But, if we consider the local characterization of singularity by L 3 norm, there is an unpleasant problem.…”

“…Recently, Barker and Prange [3] show under the assumption ||u|| L ∞ t L 3,∞ ≤ M, the optimal blow-up rate at the potential singular point (T * , 0) is…”

Improved quantitative regularity for the Navier-Stokes equations in a scale of critical spaces

“…A On(1) N 1− 3(n−1) q −β using Holder's inequality, (35), (2), and summing the geometric series, assuming additionally that max(− 2 q , 1 − 3(n−1) q ) < β ≤ min(α q , 2 − 2(n−1) q −). The "low-high" term is analogous.…”

“…A notable example is the criterion of Prodi-Serrin-Ladyzhenskaya [15,20,11], which implies that a necessary condition for blowup is that u L p t L q x becomes unbounded, where 2 p + 3 q = 1 and q ∈ (3, ∞]. Extension of this result to the endpoint space…”

“…One of the main challenges in reducing the number of exponentials in the bounds (and consequently, the number of logarithms in the blowup rate) is finding suitable spacetime regions where the solution is regular enough to apply Carleman inequalities, particularly the Carleman inequality related to backward uniqueness of the heat equation. This has been done by using ǫ-regularity criteria, either based on enstrophy localization (see [22], following [21]) or Caffarelli-Kohn-Nirenberg partial regularity (see [2], following [4]). In order to find a region for which there is a known backward uniqueness Carleman inequality and where a suitable norm is sufficiently small, it has been necessary to use the pigeonhole principle which is the source of the one of exponentials in the final result.…”