I will discuss the task of computing bounds on average quantities in dissipative dynamical systems, including time averages inÌý
finite-dimensional systems and spatiotemporal averages in PDE systems. In the finite-dimensional case, I will describe computer-assisted methods for computing bounds by constructing nonnegative polynomials with certain properties, similarly to the way nonlinear stability can be proven by constructing Lyapunov functions. Nonnegativity of these polynomials is enforced by requiring them to be representable as sums of squares, a condition that can be checked computationally using the convex optimization technique of semidefinite programming. These methods will be illustrated using the Lorenz equations, for which they produce novel bounds on various average quantities. I will then discuss work in progress on extending these computer-assisted methods to dissipative PDEs, using the Kuramoto-Sivashinky equation as an example.
