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We consider the Kuramoto-Sivashinsky (KS) equation in finite domains of theform $[-L,L]^d$. Our main result provides refined Gevrey estimates for thesolutions of the one dimensional differentiated KS, which in turn implyeffective new estimates for higher Sobolev norms of the solutions in terms ofpowers of $L$. We illustrate our method on a simpler model, namely theregularized Burger's equation. We also show local well-posedness for the twodimensional KS equation and provide an explicit criteria for (eventual) blow-upin terms of its $L^2$ norm. The common underlying idea in both results is that{\it a priori} control of the $L^2$ norm is enough in order to conclude higherorder regularity and allows one to get good estimates on the high-frequencytails of the solutions.

The Kuramoto-Sivashinsky equation in R^1 and R^2: effective estimates of the high-frequency tails and higher Sobolev norms