A Caccioppoli inequality is of the form

where is a constant independent of and , , and .

Caccioppoli inequality can be considered as the reverse Poincare inequality

holds for .

Of course does not hold for any , but if satisfies some second order elliptic equation or certain variational problems, can satisfies .

For example, if is the weak solution of the divergence form

or in

here are just bounded measurable and strictly elliptic. .

Choose a cut off function and in and . Let , here we use for abbreviation. Then

Noticing that on , we can get

If satisfies , we can also get the same conclusion.

MA6000A: Theory of Partial Differential Equations. Roger Moser

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## Comments

Dear Professor ;

I am interested in your Website and in the Caccioppoli inequality and in the Navier-Stokes equations.

I write to you for asked if we can apply the Caccioppoli inequality for the unsteady Stokes system on the weak solutions of the unsteady 3D Navier-Stokes system.

Or if there existes anothre type Caccioppoli inequality or reverse Poincare Inequality for the unsteady 3D Navier-Stokes equations.

Best regards,

Hafid

Dear Hafid,

I am not any professor. I am just a graduate student in Rutgers. I can not say anything about Caccioppoli inequality for unsteady 3D Navier-stokes equations. I just began to learn Navier-Stokes equations, mostly on 2D case. But i am interested in your research. May be there is something I can help you. If you possible, can you give me some references to read?

Thank you very much for your response.

1. On the boundary regularity of suitable weak solutions to the Navier-Stokes equations

Jörg Wolf

http://link.springer.com/article/10.1007/s11565-010-0091-3

or

http://www.cirm.univ-mrs.fr/videos/2008/exposes/288/Wolf.pdf

2. ON THE CACCIOPPOLI INEQUALITY OF THE UNSTEADY

STOKES SYSTEM

BUM JA JIN

http://www.math.ualberta.ca/ijnamb/Volume-4-2013/No-3-13/2013-03-02.pdf

All the best.

Hafid.

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