This project is a study of systems of conservation laws, focussed on the equations of fluid dynamics in one space dimension. The main problem under consideration is the existence and structure of solutions having large BV data. The P.I. will prove two major results: the first is the existence of solutions to the equations of isentropic gas dynamics with large data (including the vacuum), and the second is the construction of time-periodic shockless solutions of the Euler equations of gas dynamics.
This project extends mature results for solutions having small data to the physically in- teresting regime of large data, and makes accessible many fundamental questions for systems with large data. These include the structure of solutions, decay, uniqueness and continuous dependence, and convergence of viscous and/or relaxation approximations. Moreover, the periodic solutions developed here contain a number of new features not previously known in hyperbolic systems, such as a group velocity, sustainability of solutions, and soliton-like traveling waves. The results of this project also raise challenging questions for numerical analysts and experimentalists.
