Stability and Behavior of Large Amplitude Relaxation Waves

The investigator studies a selection of key open problems regarding the stability and behavior of shallow water waves arising in applications to hydraulic engineering and coastal flow in the scenario of large-amplitude waves that are sufficiently strong to develop shock-type discontinuities. The interest of such waves ranges from the large-scale understanding of destructive "rogue wave" type solutions, to the small, explaining the structure of the familiar "herringbone" or "crosshatched" flow seen in narrow fast-moving streams. The interest of the mathematical tools needed to understand them is also broad, extending to the general class of "relaxation systems" appearing in a variety of physical applications from plasma dynamics to many-particle systems, in which complex physical systems move toward much simpler equilibrium dynamics under the influence of physical mechanisms. Though these have been widely studied, the treatment of shock discontinuities is still largely undeveloped; the techniques developed in this project are hoped to remove this bottleneck in the general theory. The planned activities have both theoretical and physical components, and involve collaboration with domestic and foreign colleagues, undergraduates, current and former graduate students, and postdocs. This is expected to strengthen and extend existing networks of cooperation across fields and institutions, and to aid in the training of students and postdocs. It will also aid in the dissemination of results and techniques, which will be further accomplished by frequent presentations. The project seeks to resolve the outstanding problems of nonlinear time-asymptotic stability of discontinuous inviscid periodic waves and multi-dimensional hydraulic shocks. Objectives are the development of new theoretical tools in order to treat unresolved questions of practical interest in shallow-water flow, extended thermodynamics, and other models involving relaxation. Methods include a blend of finite- and infinite-dimensional dynamical systems tools with specialized techniques coming from the theory of shock waves and hyperbolic conservation laws. The problems investigated as part of the project involve interesting and nonstandard mathematical issues addressing puzzles from physical applications. For example, successful treatment of nonlinear stability of discontinuous periodic waves would represent a major theoretical advance in both shock and modulation theory while resolving long-standing questions on roll wave behavior in shallow water flow/hydraulic engineering. Likewise, the rigorous mathematical description of herringbone patterns and general hyperbolic bifurcation would mark important theoretical advances. The use of variable-coefficient Kreiss symmetrizers/pseudodifferential damping estimates for the treatment of turning points and regularity in multiple dimensions may be particularly consequential. 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.