Infinite dimensional dynamical systems and the Navier-Stokes equations: a rigorous computational approach

Jean-Philippe Lessard
Département de Mathématiques et de Statistique, Université Laval, Québec

Nonlinear dynamics shape the world around us. It shapes biology, from the electrophysiological properties of neurons, via the spiralling waves in contracting heart muscles, to gene regulatory networks. It shapes physics, from the swirling motions in fluid flows, via the creation of complex patterns in materials, to the harmonious motions of celestial bodies. It shapes chemistry, from the rich reaction kinetics phenomena, via the chemical basis of morphogenesis at the origin of patterns on animals, to the complicated biochemistry in the living cell. Mathematically these beautiful and highly complex phenomena are described by nonlinear dynamical systems in the form of ODEs, PDEs and DDEs. Unfortunately, the presence of nonlinearities in the models often obstructs the mathematicians and the scientists from obtaining analytic formulas for the solutions. In particular, the difficulties are even greater for PDEs and DDEs, which are naturally defined on infinite dimensional function spaces. As a consequence of these challenges and with the recent availability of powerful computers and sophisticated software, numerical simulations are quickly becoming the primary tool used by scientists to study the complicated dynamics arising in the models. However, while the pace of progress increases, sometimes we need to take a step back and pose the question, just how reliable are our computations?
In this talk, we introduce and present the recent field of rigorous computing in dynamical systems which emerged to address this fundamental scientific issue in the context of nonlinear dynamics. More specifically, we will ask the following questions and partially answer some of them. Can we mathematically demonstrate the reliability of the solutions computed using the forced Navier-Stokes equations? Can we rigorously control the errors made when computing the solutions of Cauchy problems of parabolic PDEs? If so, can we show that the 3D Navier-Stokes equations do not develop singularities as time evolves for a large class of initial conditions? Can we develop rigorous computations to understand properties of materials? Can we use rigorous numerics as a tool for reliable predictions and computations in astrodynamics?