Non-uniqueness of weak solutions to hyperviscous Navier–Stokes equations: on sharpness of J.-L. Lions exponent

By Tianwen Luo, Edriss S. Titi. Calculus of Variations and Partial Differential Equations, 2018

83 citations recorded by the source corpus.

Using the convex integration technique for the three-dimensional Navier–Stokes equations introduced by Buckmaster and Vicol, it is shown the existence of non-unique weak solutions for the 3D Navier–Stokes equations with fractional hyperviscosity (-Δ)θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )^{\theta }$$\end{document}, whenever the exponent θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} is less than Lions’ exponent 5/4, i.e., when θ<5/4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta < 5/4$$\end{document}.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c8f57122b26af00e83a6704ca9c4e4d8728a538f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Maria Colombo
2. Vlad Vicol
3. T. Buckmaster
4. Martina Hofmanov'a
5. Patrick Winkert
6. Xiangchan Zhu
7. Rongchan Zhu
8. Elia Brué

In Control and Systems Engineering

1. T. Buckmaster
2. Maria Colombo
3. Vlad Vicol
4. Jacob Bedrossian
5. Nader Masmoudi

In Mathematical Physics

1. Maria Colombo
2. T. Buckmaster
3. Vlad Vicol
4. Patrick Winkert
5. Leszek Gasinski
6. Giuseppe Mingione
7. Elia Brué
8. Cristiana De Filippis

Citations in the corpus

• Wild solutions of the Navier–Stokes equations whose singular sets in time have Hausdorff dimension strictly less than 1 (2018, background): “When compared to [35] the weak solutions constructed in this paper have the additional property that their set of singular times has Hausdorff dimension strictly less than 1.”
• Convex integration and phenomenologies in turbulence (2019, methodology): “ such that ˜(z) = ( 1 if 0 z1 z if z2 and with z2˜(z) 4zfor z2(1;2). For such a function ˜we then define ˆ(x;t) = 4cRq+1˜ (cRq+1) 1 R‘(x;t) : This simplified definition of ˆwas introduced in [138]. The main properties of ˆare that pointwise in (x;t) we have R ‘(x;t) ˆ(x;t) = 1 4 (cRq+1) 1jR‘(x;t)j ˜(”