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

TL;DR

This paper develops computational methods using semidefinite programming to bound the long-term average energy in the Kuramoto-Sivashinsky equation, providing evidence that mean energy remains bounded for large domain sizes.

Contribution

It introduces a novel SDP-based approach to compute bounds on PDE solutions' energy, combining polynomial auxiliary functionals with PDE truncation and analytical estimates.

Findings

Computed sharp bounds for energy for L ≤ 10

Results agree with analytical estimates and previous conjectures

Provides first positive computational evidence for bounded mean energy as L increases

Abstract

We present methods for bounding infinite-time averages in dynamical systems governed by nonlinear PDEs. The methods rely on auxiliary functionals, which are similar to Lyapunov functionals but satisfy different inequalities. The inequalities are enforced by requiring certain expressions to be sums of squares of polynomials, and the optimal choice of auxiliary functional is posed as a semidefinite program (SDP) that can be solved computationally. To formulate these SDPs we approximate the PDE by truncated systems of ODEs and proceed in one of two ways. The first approach is to compute bounds for the ODE systems, increasing the truncation order until bounds converge numerically. The second approach incorporates the ODE systems with analytical estimates on their deviation from the PDE, thereby using finite truncations to produce bounds for the full PDE. We apply both methods to the Kuramoto-Sivashinsky equation, where we compute upper bounds on the spatiotemporal average of energy by employing polynomial auxiliary functionals up to degree six. The first approach is used for most computations, but a subset of results are checked using the second approach, and the results agree to high precision. These bounds apply to all odd solutions of period $2\pi L$, where $L$ is varied. Sharp bounds are obtained for $L\le10$, and trends suggest that more expensive computations would 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 simplest nonzero steady states, which apparently have the largest mean energy among all odd 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, albeit up to finite $L$, and they offer some guidance for analytical proofs.