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

Pages: 729 - 738, Issue Special, September 2009

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Milena Stanislavova - Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd, Lawrence, KS 66045-7523, United States (email)
Atanas Stefanov - Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd, Lawrence, KS 66045-7523, United States (email)

Abstract: 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:  Kuramoto-Sivashinsky equation, regularized Burger's equation, Gevrey regularity
Mathematics Subject Classification:  35B35, 35G30, 35B40, 35K30, 37K40

Received: June 2008;      Revised: May 2009;      Published: September 2009.