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2009, 2009(Special): 729-738. doi: 10.3934/proc.2009.2009.729

Effective estimates of the higher Sobolev norms for the Kuramoto-Sivashinsky equation

Milena Stanislavova 1, and  Atanas Stefanov 1,

1. 

Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd, Lawrence, KS 66045-7523

Received  June 2008 Revised  May 2009 Published  September 2009

We consider the Kuramoto-Sivashinsky (KS) equation in finite domains of the form $[-L,L]$. Our main result provides effective new estimates for higher Sobolev norms of the solutions in terms of powers of $L$ for the one-dimentional differentiated KS. We illustrate our method on a simpler model, namely the regularized Burger's equation. The underlying idea in this result is that a priori control of the $L^2$ norm is enough in order to conclude higher order regularity and in fact, it allows one to get good estimates on the high-frequency tails of the solution.
Keywords: Gevrey regularity, regularized Burger's equation, Kuramoto-Sivashinsky equation.
Mathematics Subject Classification: 35B35, 35G30, 35B40, 35K30, 37K40.
Citation: Milena Stanislavova, Atanas Stefanov. Effective estimates of the higher Sobolev norms for the Kuramoto-Sivashinsky equation. Conference Publications, 2009, 2009 (Special) : 729-738. doi: 10.3934/proc.2009.2009.729
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