New Tools in the Study of Wave Propagation: Dynamical Systems for Kinetic Equations, Inviscid Limits for Modulated Periodic Waves, and Rigorous Numerical Stability Analysis

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The PI will study a selection of key open problems in the theory of wave propagation arising in a variety of settings including thin film flow, pattern formation, detonation, and the kinetic theory of gases. The problems considered share the features of computational complexity, multiple length/time scales, and genuine physical interest in applications. Many concern questions that current numerics and experiment are not adequate to resolve. Several of the planned subprojects involve numerically assisted proof using scientific computation with guaranteed error bounds, up to and including rigorous numerical proof. An integral part of the project is the simultaneous development of a user-friendly numerical platform, STABLAB, for numerical stability investigation, and the systematic exploration with this platform of physical behavior in gas and fluid dynamics in the delicate situations of reacting or ionized flow. The problems addressed involve issues in dynamical systems, singular perturbation theory, spectral theory of linear operators, nonlinear partial differential equations, and rigorous scientific computation, and should result in the development of new mathematical tools of general application. In particular, development of dynamical systems tools for kinetic shock and boundary layer problems would unify and extend results obtained for Boltzmann phenomena by the "Kyoto School" of Sone et al using a variety of formal and analytic methods. Likewise, the introduction of new inviscid stability criteria for roll waves and of Kreiss symmetrizer techniques for analysis of modulated fronts open new directions in the study of periodic modulation. The problem on galloping detonations, if solved, will answer a longstanding question, while associated rigorous Wensel,Kramers, and Brillouin method developments will be of wide general use. Determination of simple stability criteria for roll waves in shallow water flow are of practical interest in hydraulic engineering. Finally, the development of rigorous numerical proof and error estimate techniques is potentially transformative, having broader implications for standards in scientific computing.