Averaging of trajectory attractors of evolution equations with rapidly oscillating terms

Marko Vishik and Vladimir Chepyzhov

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Submission date: 10. Aug. 2000
Pages: 40
published in: Matematiceskij sbornik, 129 (2000) 1, p. 13-50 
Bibtex

Abstract:
We consider evolution equations with terms which oscillate rapidly with respect to spatial or time variables. We prove that the trajectory attractors of these equations tend to the trajectory attractors of equations whose terms are the averages of the corresponding terms of the initial equations. We do not assume that the corresponding Cauchy problems are uniquely solvable. At the same time if the Cauchy problems for the equations under consideration have unique solution, then they generate semigroups having global attractors. These global attractors also converge to the global attractors of the averaged equations in the corresponding space.

We apply these results to the following equations and systems of mathematical physics: the 3D and 2D Navier-Stokes systems with rapidly oscillationg external forces, the reaction-diffusion systems, the complex Ginzburg-Landau equation, the generalized Chafee-Infante equation, the dissipative hyperbolic equations with rapidly oscillating terms and coefficients.