Large, global solutions to the Navier-Stokes equations, slowly varying in one direction

In to previous papers by the authors, classes of initial data to the three dimensional, incompressible Navier-Stokes equations were presented, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The aim of this article is to provide new examples of arbitrarily large initial data giving rise to global solutions, in the whole space. Contrary to the previous examples, the initial data has no particular oscillatory properties, but varies slowly in one direction. The proof uses the special structure of the nonlinear term of the equation.

[1]  Jean Leray,et al.  Sur le mouvement d'un liquide visqueux emplissant l'espace , 1934 .

[2]  D. Iftimie,et al.  D Navier-Stokes equations seen as a perturbation of the 2 D Navier-Stokes equations Bulletin , 2000 .

[3]  I. Gallagher,et al.  Asymptotics and stability for global solutions to the Navier-Stokes equations , 2003 .

[4]  Y. Meyer,et al.  Solutions auto-similaires des équations de Navier-Stokes , 1994 .

[5]  George R. Sell,et al.  Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions , 1993 .

[6]  Hiroshi Fujita,et al.  On the Navier-Stokes initial value problem. I , 1964 .

[7]  On the global wellposedness of the 3-D Navier–Stokes equations with large initial data , 2005, math/0508374.

[8]  G. Sell,et al.  Navier-Stokes equations in thin 3D domains with Navier boundary conditions , 2007 .

[9]  I. Gallagher.  The tridimensional Navier-Stokes equations with almost bidimensional data: stability, uniqueness, and life span , 1997 .

[10]  Wellposedness and stability results for the Navier-Stokes equations in ${\mathbf R}^{3}$ , 2006, math/0611044.

[11]  Yoshikazu Giga,et al.  Solutions in Lr of the Navier-Stokes initial value problem , 1985 .

[12]  J. Chemin.  Remarks on global existence for the incompressible Navier-Stokes system , 1992 .

[13]  Alfred P. Sloanfellowship.  Well-posedness for the Navier-stokes Equations , 1999 .

[14]  T. Sideris,et al.  Global stability of large solutions to the 3D Navier-Stokes equations , 1994 .

[15]  D. Iftimie,et al.  The 3D Navier-Stokes equations seen as a perturbation of the 2D Navier-Stokes equations , 1999 .

[16]  J. Chemin,et al.  Wellposedness and stability results for the Navier-Stokes equations in R 3 , 2006 .

[17]  E. Titi,et al.  Invariant helical subspaces for the Navier-Stokes equations , 1990 .

[18]  R. A. Silverman,et al.  The Mathematical Theory of Viscous Incompressible Flow , 1972 .

[19]  Global solubility of the three-dimensional Navier-Stokes equations with uniformly large initial vorticity , 2003 .

[20]  M. R. Ukhovskii,et al.  Axially symmetric flows of ideal and viscous fluids filling the whole space , 1968 .