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May  2009, 8(3): 785-802. doi: 10.3934/cpaa.2009.8.785

Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations

P.E. Kloeden 1, , Pedro Marín-Rubio 2, and  José Real 3,

1. 

FB Mathematik, Johann Wolfgang Goethe Universität, Postfach 11 19 32, D-60054 Frankfurt a.M.
2. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla
3. 

Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla

Received  May 2008 Revised  October 2008 Published  February 2009

A new proof of existence of solutions for the three dimensional system of globally modified Navier-Stokes equations introduced in [3] by Caraballo, Kloeden and Real is obtained using a smoother Galerkin scheme. This is then used to investigate the relationship between invariant measures and statistical solutions of this system in the case of temporally independent forcing term. Indeed, we are able to prove that a stationary statistical solution is also an invariant probability measure under suitable assumptions.
Keywords: Modified Navier-Stokes model, statistical solutions, invariant measures..
Mathematics Subject Classification: 35Q30, 35K90, 37L3.
Citation: P.E. Kloeden, Pedro Marín-Rubio, José Real. Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 785-802. doi: 10.3934/cpaa.2009.8.785
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