On the global wellposedness of the 3-D Navier–Stokes equations with large initial data

By Jean-Yves Chemin, Isabelle Gallagher. 2005

97 citations recorded by the source corpus.

Abstract We give a condition for the periodic, three-dimensional, incompressible Navier–Stokes equations to be globally wellposed. This condition is not a smallness condition on the initial data, as the data is allowed to be arbitrarily large in the scale invariant space B ∞ , ∞ −1 , which contains all the known spaces in which there is a global solution for small data. The smallness condition is rather a nonlinear type condition on the initial data; an explicit example of such initial data is constructed, which is arbitrarily large and yet gives rise to a global, smooth solution.

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

View Paper: https://www.semanticscholar.org/paper/cc13225bb498b0f173c42c8ca68601c14c516057

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In Applied Mathematics

1. Changxing Miao
2. Zhifei Zhang
3. Qionglei Chen
4. Isabelle Gallagher
5. Jean-Yves Chemin
6. Nader Masmoudi
7. Ping Zhang
8. Zhen Lei

In Control and Systems Engineering

1. Changxing Miao
2. Isabelle Gallagher
3. Jean-Yves Chemin
4. Zhifei Zhang
5. Qionglei Chen
6. Jiahong Wu
7. Nader Masmoudi
8. Sadek Gala

In Mathematical Physics

1. Changxing Miao
2. Isabelle Gallagher
3. Jean-Yves Chemin
4. Qionglei Chen
5. Zhifei Zhang
6. Nader Masmoudi
7. Ping Zhang
8. Jiahong Wu

Citations in the corpus

• Global well‐posedness for compressible Navier‐Stokes equations with highly oscillating initial velocity (2009, background): “Let us emphasize that this result allows us to construct global solutions for highly oscillating initial data which may have a large norm in Ḣ n 2 −1 or Ln.”
• PARACONTROLLED DISTRIBUTIONS AND SINGULAR PDES (2012, unknown): “The result on the torus can be shown using the same arguments, see for example [CG06].”
• Global regularity for some classes of large solutions to the Navier-Stokes equations (2008, unknown): “There are two positive numbers ε 0 and η such that for any divergence free vector field v 0 satisfying then, for any positive ε smaller than ε 0 , the initial data generates a global smooth solution of ( NS ) on T 2 × R .”
• Large, global solutions to the Navier-Stokes equations, slowly varying in one direction (2007, background): “In particular, the following theorem is proved in [4].”