The object of this research is threefold. First, we seek a better understanding of solutions to systems of hyperbolic (inviscid) conservation laws that are large in either amplitude or variation. New examples of explosive behavior will be considered, as well as conditions preventing singular behavior. Next, we will consider the multi-dimensional Navier-Stokes equations for a compressible fluid and establish global existence of large solutions with spherical or cylindrical symmetry. Finally, we will consider flow describing combustion. This is modeled by the Navier-Stokes equations augmented by equations describing the chemical processes. In this case we are particularly interested in the stability of wave patterns.
The work deals with mathematical analysis of solutions to nonlinear partial differential equations. The research will investigate systems of conservation laws, compressible fluid flow, and equations describing reactive flow. Much of the existing theory for such nonlinear equations applies only to small solutions. However, large solutions are of great interest in applications such as gas flow, combustion, and detonations, and study of these solutions requires new techniques that will be developed in this project.
