The Kuramoto-Sivashinsky equation in R¹ and R²: effective estimates of the high-frequency tails and higher Sobolev norms

We consider the Kuramoto-Sivashinsky (KS) equation in finite domains of the form $[-L,L]^d$. Our main result provides refined Gevrey estimates for the solutions of the one dimensional differentiated KS, which in turn imply effective new estimates for higher Sobolev norms of the solutions in terms of powers of $L$. We illustrate our method on a simpler model, namely the regularized Burger's equation. We also show local well-posedness for the two dimensional KS equation and provide an explicit criteria for (eventual) blow-up in 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 higher order regularity and allows one to get good estimates on the high-frequency tails of the solutions.