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Some examples of global solutions associated with large initial data for the incompressible Navier Stokes system

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Some examples of global solutions associated with large initial data for the incompressible Navier-Stokes system Isabelle Gallagher Institut de Mathematiques de Jussieu, Universite Paris 7 Luminy, March 25, 2008 I. Gallagher (IMJ, Paris 7) Examples of large solutions to Navier-Stokes Luminy, March 25, 2008 1 / 22

  • largest adapted space

  • solution associated

  • weak solution

  • navier-stokes luminy

  • tataru space

  • divergence-free vector


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Isabelle Gallagher
Luminy, March 25, 2008
InstitutdeMathe´matiquesdeJussieu,Universit´eParis7
Some examples of global solutions associated with large initial data for the incompressible Navier-Stokes system
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The Cauchy problem Presentation of the equations Fundamental properties Weak solutions Strong solutions Towards the largest adapted space The Koch and Tataru space
1
Outline of the talk
Situations when large data can generate a global solution Previous results Two examples
2
nstoNavier-StokeLsmuni,yaMcr2h,5
tu+u∙ ruΔu=−rpdivu= 0
(NS)
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3 withdivu=Xjujandu∙ ru=Xujju=Xj(uju). j=1j=1j=1 Remark :The pressure can be eliminated byprojection onto divergence-free vector fields:
Presentation of the equations
2
P= Id− rΔ1div.
Viscous, incompressible, homogeneous fluid, in two or three space dimensions
Velocityu= (u1u2u3)(tx), pressurep(tx)
Cauchy data :u|t=0=u0.
002,2/3833s7)ExampIMJ,Parillgaeh(r.IaG
33raP,)7simaxEselplaofesrgutolnsioI.Gallagher(IMJ
P= Id− rΔ1div.
tu+u∙ ruΔu=−rpdivu= 0
(NS)
3 withdivu=Xjujandu∙ ru=Xujju=Xj(uju). j=1j=1j=1 Remark :The pressure can be eliminated byprojection onto divergence-free vector fields:
Cauchy data :u|t=0=u0.
Velocityu= (u1u2u3)(tx), pressurep(tx)
Viscous, incompressible, homogeneous fluid, in two or three space dimensions
22
Presentation of the equations
083/02,52hcraM,ynimusLketo-ServiNato
I.aP,JMI(rehgallaGflsolempxa)Es7ritsNoitnoosulraegLumiokesr-Stavie33
3 withdivu=Xjujandu∙ ru=Xujju=Xj(uju). j=1j=1j=1 Remark :The pressure can be eliminated byprojection onto divergence-free vector fields:
Presentation of the equations
Velocityu= (u1u2u3)(tx), pressurep(tx)
Cauchy data :u|t=0=u0.
2/2
P= Id− rΔ1div.
Viscous, incompressible, homogeneous fluid, in two or three space dimensions
(NS)
tu+u∙ ruΔu=−rpdivu= 0
archny,M008325,2
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