Global dissipativity and inertial manifolds for diffusive burgers equations with low-wavenumber instability

Jesenko Vukadinovic

doi:10.3934/dcds.2011.29.327

Article Contents

2011, Volume 29, Issue 1: 327-341. Doi: 10.3934/dcds.2011.29.327

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Global dissipativity and inertial manifolds for diffusive burgers equations with low-wavenumber instability

Jesenko Vukadinovic^1,

1.
Department of Mathematics, 1S-208, The City University of New York – CSI, 2800 Victory Boulevard, Staten Island, NY 10314

Received: July 31, 2009

Revised: March 31, 2010

Published: December 2010

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Abstract

The global well-posedness, the existence of globally absorbing sets and the existence of inertial manifolds are investigated in a class of diffusive (viscous) Burgers equations. The class includes diffusive Burgers equation with nontrivial forcing, Burgers-Sivashinsky equation and Quasi-Steady equation of cellular flames. Global dissipativity is proven in two space dimensions for periodic boundary conditions. For the proof of the existence of inertial manifolds, the spectral-gap condition, which Burgers-type equations do not satisfy in their original form, is circumvented by employing the Cole-Hopf transform. The procedure is valid in both one and two space dimensions.

Keywords:

Viscous Burgers equations; Kuramoto-Sivashinsky equation; Quasi-Steady equation; global attractor; inertial manifolds.

Mathematics Subject Classification: 35K55, 35B41, 35B42, 37L25.

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