Abstract
In this short survey paper, we focus on some new developments in the study of the regularity or potential singularity formation for solutions of the 3D Navier–Stokes equations. Some of the motivating questions are the following. Are certain norms accumulating/concentrating on small scales near potential blow-up times? At what speed do certain scale-invariant norms blow-up? Can one prove explicit quantitative regularity estimates? Can one break the criticality barrier, even slightly? We emphasize that these questions are closely linked together. Many recent advances for the Navier–Stokes equations are directly inspired by results and methods from the field of nonlinear dispersive equations.
| Original language | English |
|---|---|
| Number of pages | 28 |
| Journal | Vietnam Journal of Mathematics |
| Early online date | 29 Dec 2023 |
| DOIs | |
| Publication status | Published - 29 Dec 2023 |
Funding
Both authors thank the Institute of Advanced Studies of Cergy Paris University for their hospitality. CP is partially supported by the Agence Nationale de la Recherche, project BORDS, grant ANR-16-CE40-0027-01, project SINGFLOWS, grant ANR-18-CE40-0027-01, project CRISIS, grant ANR-20-CE40-0020-01, by the CY Initiative of Excellence, project CYNA (CY Nonlinear Analysis) and project CYFI (CYngular Fluids and Interfaces).
| Funders | Funder number |
|---|---|
| French National Research Agency | ANR-20-CE40-0020-01, ANR-18-CE40-0027-01, ANR-16-CE40-0027-01 |
Keywords
- Kolmogorov scales
- Navier–Stokes equations
- Norm concentration
- Quantitative estimates
- Regularity criteria
- Slight criticality breaking
- Supercritical norms
ASJC Scopus subject areas
- General Mathematics