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We present methods for bounding infinite-time averages in dynamical systemsgoverned by nonlinear PDEs. The methods rely on auxiliary functionals, whichare similar to Lyapunov functionals but satisfy different inequalities. Theinequalities are enforced by requiring certain expressions to be sums ofsquares of polynomials, and the optimal choice of auxiliary functional is posedas a semidefinite program (SDP) that can be solved computationally. Toformulate these SDPs we approximate the PDE by truncated systems of ODEs andproceed in one of two ways. The first approach is to compute bounds for the ODEsystems, increasing the truncation order until bounds converge numerically. Thesecond approach incorporates the ODE systems with analytical estimates on theirdeviation from the PDE, thereby using finite truncations to produce bounds forthe full PDE. We apply both methods to the Kuramoto-Sivashinsky equation, wherewe compute upper bounds on the spatiotemporal average of energy by employingpolynomial auxiliary functionals up to degree six. The first approach is usedfor most computations, but a subset of results are checked using the secondapproach, and the results agree to high precision. These bounds apply to allodd solutions of period $2\pi L$, where $L$ is varied. Sharp bounds areobtained for $L\le10$, and trends suggest that more expensive computationswould yield sharp bounds at larger $L$ also. The bounds are known to be sharp(to within 0.1% numerical error) because they are saturated by the simplestnonzero steady states, which apparently have the largest mean energy among allodd solutions. Prior authors have conjectured that mean energy remains $O(1)$for $L\gg1$ since no particular solutions with larger energy have been found.Our bounds constitute the first positive evidence for this conjecture, albeitup to finite $L$, and they offer some guidance for analytical proofs.

Bounds on mean energy in the Kuramoto-Sivashinsky equation computed using semidefinite programming