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.