Bounds on the Kuramoto-Shivashinsky equation

The Kuramoto--Shivashinsky (KS) equation

u_t + (1/2 u^2)_x + u_{xx} + u_{xxxx} = 0

is a Burgers equation with a destabilizing second and stabilizing fourth order term. One interested in the long--time behavior of periodic mean--zero solutions with period L>>1. Solutions in numerical simulations show a chaotic behavior and are of O(1) in L. Rigorous results are not abundant. Together with Lorenzo Giacomelli, we slightly improve the bound spatial average of |u| \le O(L^{11/10}) by Eckmann et. al. to spatial average of |u| \le o(L) (little o). This is still far from optimal, but at least shows that KS behaves differently than ``Burgers--Shivashinski''

u_t + (1/2 u^2)_x - u - u_{xx} = 0,

which looks similar to KS, but admits periodic solution u of order O(L).