Construction of an optimal background profile for the Kuramoto–Sivashinsky equation using semidefinite programming

•Optimal background profiles are constructed for the Kuramoto–Sivashinsky equation.•Analytical L2L2 bounds for the solution are found using convex optimisation.•The optimal background profile is a double shock profile.•Results attest that L1.5L1.5 scaling is optimal within the classic Lyapunov argument.•We improve the proportionality constant of the scaling law for the attracting set.

A method to construct systematically an optimal background profile for the Kuramoto–Sivashinsky equation is developed by formulating the classical problem as an optimisation problem. In particular, we show that the infinite-dimensional problem can be rewritten as a finite-dimensional convex semidefinite problem, which is solved to construct a background profile and to obtain an upper bound on the energy of the solution ‖u‖‖u‖ that applies to the infinite-dimensional PDE. The results are compared to existing analytical results, and support the fact that limsupt→∞‖u‖≤O(L3/2) is the optimal estimate achievable with the background profile method and a quadratic Lyapunov function.