The role of oscillations in the global wellposedness of the 3-D incompressible anisotropic Navier-Stokes equations

Jean-Yves Chemin[1]; Ping Zhang[2]

  • [1] Laboratoire J.-L. Lions, Case 187 Université Pierre et Marie Curie, 75230 Paris Cedex 05, FRANCE
  • [2] Academy of Mathematics & Systems Science, CAS Beijing 100080, CHINA.

Séminaire Équations aux dérivées partielles (2005-2006)

  • Volume: 2005-2006, page 1-18

Abstract

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Corresponding to the wellposedness result [2] for the classical 3-D Navier-Stokes equations ( N S ν ) with initial data in the scaling invariant Besov space, ℬ p , ∞ - 1 + 3 p , here we consider a similar problem for the 3-D anisotropic Navier-Stokes equations ( A N S ν ) , where the vertical viscosity is zero. In order to do so, we first introduce the Besov-Sobolev type spaces, ℬ 4 - 1 2 , 1 2 and ℬ 4 - 1 2 , 1 2 ( T ) . Then with initial data in the scaling invariant space ℬ 4 - 1 2 , 1 2 , we prove the global wellposedness for ( A N S ν ) provided the norm of initial data is small enough compared to the horizontal viscosity. In particular, this result implies the global wellposedness of ( A N S ν ) with high oscillatory initial data.

How to cite

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Chemin, Jean-Yves, and Zhang, Ping. "The role of oscillations in the global wellposedness of the 3-D incompressible anisotropic Navier-Stokes equations." Séminaire Équations aux dérivées partielles 2005-2006 (2005-2006): 1-18. <http://eudml.org/doc/11143>.

@article{Chemin2005-2006,
abstract = {Corresponding to the wellposedness result [2] for the classical 3-D Navier-Stokes equations $(NS_\nu )$ with initial data in the scaling invariant Besov space, $\{\cal B\}^\{-1+\frac\{3\}\{p\}\}_\{p,\infty \},$ here we consider a similar problem for the 3-D anisotropic Navier-Stokes equations $(ANS_\nu ),$ where the vertical viscosity is zero. In order to do so, we first introduce the Besov-Sobolev type spaces, $\{\cal B\}^\{-\frac\{1\}\{2\},\frac\{1\}\{2\}\}_4$ and $\{\cal B\}^\{-\frac\{1\}\{2\},\frac\{1\}\{2\}\}_4(T).$ Then with initial data in the scaling invariant space $\{\cal B\}^\{-\frac\{1\}\{2\},\frac\{1\}\{2\}\}_4,$ we prove the global wellposedness for $(ANS_\nu )$ provided the norm of initial data is small enough compared to the horizontal viscosity. In particular, this result implies the global wellposedness of $(ANS_\nu )$ with high oscillatory initial data.},
affiliation = {Laboratoire J.-L. Lions, Case 187 Université Pierre et Marie Curie, 75230 Paris Cedex 05, FRANCE; Academy of Mathematics & Systems Science, CAS Beijing 100080, CHINA.},
author = {Chemin, Jean-Yves, Zhang, Ping},
journal = {Séminaire Équations aux dérivées partielles},
keywords = {anisotropic Navier-Stokes equations; global wellposedness; high oscillatory initial data},
language = {eng},
pages = {1-18},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {The role of oscillations in the global wellposedness of the 3-D incompressible anisotropic Navier-Stokes equations},
url = {http://eudml.org/doc/11143},
volume = {2005-2006},
year = {2005-2006},
}

TY - JOUR
AU - Chemin, Jean-Yves
AU - Zhang, Ping
TI - The role of oscillations in the global wellposedness of the 3-D incompressible anisotropic Navier-Stokes equations
JO - Séminaire Équations aux dérivées partielles
PY - 2005-2006
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2005-2006
SP - 1
EP - 18
AB - Corresponding to the wellposedness result [2] for the classical 3-D Navier-Stokes equations $(NS_\nu )$ with initial data in the scaling invariant Besov space, ${\cal B}^{-1+\frac{3}{p}}_{p,\infty },$ here we consider a similar problem for the 3-D anisotropic Navier-Stokes equations $(ANS_\nu ),$ where the vertical viscosity is zero. In order to do so, we first introduce the Besov-Sobolev type spaces, ${\cal B}^{-\frac{1}{2},\frac{1}{2}}_4$ and ${\cal B}^{-\frac{1}{2},\frac{1}{2}}_4(T).$ Then with initial data in the scaling invariant space ${\cal B}^{-\frac{1}{2},\frac{1}{2}}_4,$ we prove the global wellposedness for $(ANS_\nu )$ provided the norm of initial data is small enough compared to the horizontal viscosity. In particular, this result implies the global wellposedness of $(ANS_\nu )$ with high oscillatory initial data.
LA - eng
KW - anisotropic Navier-Stokes equations; global wellposedness; high oscillatory initial data
UR - http://eudml.org/doc/11143
ER -

References

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