An $L^{5/6}$ bound for the periodic Kuramoto-Sivashinsky equation

We consider the Kuramoto-Sivashinsky equation $u_t+uu_x+u_{xx}+u_{xxxx}=0$ with $L$-periodic boundary conditions and zero spatial mean. We prove that every smooth solution satisfies $\|u(t)\|_{L^2}\leq CL^{5/6}+CL^{3/2}/t$ for $t>0$, where $C$ is a universal constant. In particular, $\|u(t)\|_{L^2}\leq CL^{5/6}$ for all $t\geq L^{2/3}$, independently of the initial norm.