Regularity and Approximation of Solutions to Conservation Laws

Hyperbolic conservation laws provide basic mathematical models for continuum physics, and are widely used by scientists and engineers, for instance, in the study of traffic flows and flame propagation fronts. There is a general expectation that these equations should be deterministic: knowing an initial configuration one should be able to uniquely predict the future evolution. However, recent mathematical advances point to the fact that this is not always true. A major goal of this project is to better understand in which situations the uniqueness of solutions can be guaranteed, compared with examples where multiple solutions occur, in one or more space dimensions. Based on these theoretical advances, the investigator will then provide new error bounds for a wide class of computational schemes, which are used in applications as predictive tools. A further research direction will be the accurate description of how solutions can lose regularity. In other words: what happens at the first instant of time when a new shock wave, such as a sudden alteration in pressure, is formed. The project will provide research training opportunities for graduate students and postdoctoral associates. The project will address some fundamental issues at the frontier of the current theory of hyperbolic conservation laws. New uniqueness or non-uniqueness results will be sought, in a wider class of weak solutions, possibly with unbounded variation. For one-dimensional hyperbolic conservation laws endowed with a strictly convex entropy, the investigator aims at establishing universal error estimates, valid for all approximation schemes which are compatible with the conservation equations and the entropy conditions. In addition, for various classes of nonlinear wave equations, a local asymptotic description of generic solutions will be provided, in a neighborhood of a point where a new singularity emerges. 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.