An alternative approach to regularity for the Navier–Stokes equations in critical spaces

By Carlos E. Kenig, G. Koch. 2009

69 citations recorded by the source corpus.

Abstract In this paper we present an alternative viewpoint on recent studies of regularity of solutions to the Navier–Stokes equations in critical spaces. In particular, we prove that mild solutions which remain bounded in the space H ˙ 1 2 do not become singular in finite time, a result which was proved in a more general setting by L. Escauriaza, G. Seregin and V. Sverak using a different approach. We use the method of “concentration-compactness” + “rigidity theorem” using “critical elements” which was recently developed by C. Kenig and F. Merle to treat critical dispersive equations. To the authorsʼ knowledge, this is the first instance in which this method has been applied to a parabolic equation.

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

View Paper: https://www.semanticscholar.org/paper/2254f75d204d569e76898ff71699062131e3ff0d

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. Aynur Bulut
2. Terence Tao
3. Carlos E. Kenig
4. Sanghyuk Lee
5. D. Tataru
6. B. Dodson
7. Zihua Guo
8. Jonathan Bennett

In Control and Systems Engineering

1. Aynur Bulut
2. Terence Tao
3. E.J. Candes
4. Suhas N. Diggavi
5. David L. Donoho
6. Martin J. Wainwright
7. Tong Zhang
8. Cun-Hui Zhang

In Mathematical Physics

1. Aynur Bulut
2. Terence Tao
3. Carlos E. Kenig
4. B. Dodson
5. Van H. Vu
6. Zihua Guo
7. D. Tataru
8. L. Erdős

Citations in the corpus

• Quantitative bounds for critically bounded solutions to the Navier-Stokes equations (2019, background): “There are now many proofs, variants and generalisations [7], [11], [8], [17], [15], [9] [5], [1], [2], [18], [21] of this theorem, including extensions to higher dimensions or other domains than Euclidean spaces, replacing L3 with another critical Besov or Lorenz space, or replacing the limit superi”
• Finite time blowup for an averaged three-dimensional Navier-Stokes equation (2014, background): “t specic properties of the Navier-Stokes equation that are not shared by the averaged versions. For instance, the arguments in [12] (see also BLOWUP FOR AVERAGED NAVIER-STOKES 9 the subsequent paper [22] for an alternate treatment), which establish global regularity for Navier-Stokes subject to a hy”
• Global Well-Posedness and Scattering for the Defocusing Energy-Supercritical Cubic Nonlinear Wave Equation (2010, result): “We also remark that similar results showing that the boundedness of a critical norm implies global well-posedness are known for Navier-Stokes, which is also a supercritical problem with respect to the control given by the known conservation laws and monotonicity formulae; see the work of Escauriaza,”