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

Isabelle Gallagher

Description

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

Sujets

• Gallagher
• Institut de Mathématiques de Jussieu
• Weak solution

Informations

• Publié par: pefav
• Nombre de lectures: 12
• Langue: English

Extrait

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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

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∂ t u+u∙ ru−Δu=−rp divu= 0

(NS)

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3 withdivu= X ∂ j u j andu∙ ru= X u j ∂ j u= X ∂ j (u j u). j=1j=1j=1 Remark :The pressure can be eliminated byprojection onto divergence-free vector fields:

Presentation of the equations

2

P= Id− rΔ −1 div.

• Viscous, incompressible, homogeneous fluid, in two or three space dimensions

• Velocityu= (u 1 u 2 u 3 )(tx), pressurep(tx)

Cauchy data :u |t=0 =u 0 .

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P= Id− rΔ −1 div.

∂ t u+u∙ ru−Δu=−rp divu= 0

(NS)

3 withdivu= X ∂ j u j andu∙ ru= X u j ∂ j u= X ∂ j (u j u). j=1j=1j=1 Remark :The pressure can be eliminated byprojection onto divergence-free vector fields:

Cauchy data :u |t=0 =u 0 .

• Velocityu= (u 1 u 2 u 3 )(tx), pressurep(tx)

• Viscous, incompressible, homogeneous fluid, in two or three space dimensions

22

Presentation of the equations

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3 withdivu= X ∂ j u j andu∙ ru= X u j ∂ j u= X ∂ j (u j u). j=1j=1j=1 Remark :The pressure can be eliminated byprojection onto divergence-free vector fields:

Presentation of the equations

• Velocityu= (u 1 u 2 u 3 )(tx), pressurep(tx)

Cauchy data :u |t=0 =u 0 .

2/2

P= Id− rΔ −1 div.

• Viscous, incompressible, homogeneous fluid, in two or three space dimensions

(NS)

∂ t u+u∙ ru−Δu=−rp divu= 0

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