Frontiers in Modulation, Dynamics, and Pattern Formation for Hyperbolic, Kinetic, and Convection-Reaction-Diffusion Systems

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Roll waves are large-scale periodic pulses that form in a rapidly moving inclined flow, such as on a canal or dam spillway. As potentially destructive phenomena, it is important from a hydroengineering standpoint to understand the conditions and the characteristics for their occurrence. This translates into questions on the stability, or persistence under small disturbances, of such waves. Until recently such questions were out of reach of existing mathematical tools. The PI and collaborators have developed a substantial toolkit for this study, which will be brought to bear on practical applications. A second, equally challenging topic, concerns the study of many-particle systems via Boltzmann's equation, again using recently developed technical tools. Finally, the study of modern biomorphology models promises to give new insights into initiation and emergent dynamics phases of vasculogenesis and related biological processes. The project will also provide research training opportunities for graduate students. The PI will investigate a selection of key open questions on relaxation, kinetic equations, and biomechanical pattern formation. Of particularly interest are open questions on nonlinear time-asymptotic stability of discontinuous inviscid periodic waves and multi-dimensional hydraulic shocks, invariant manifolds for steady Boltzmann’s equation, and bifurcation and stability of Turing patterns in biomorphology models possessing conservation laws. The objective of the project is the development of new theoretical approaches to unresolved questions of practical interest in shallow-water flow, gas dynamics, and morphogenesis. Methods include a blend of finite- and infinite-dimensional dynamical systems tools with specialized techniques coming from shocks and hyperbolic conservation laws: in particular, Kreiss symmetrizer and pseudodifferential techniques.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.