Damped Euler-Poisson equations and nonlinear diffusion waves

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Compressible flow through porous media with a dissipative external force field is usually described as a hyperbolic p-system with damping, a kind of Euler system of partial differential equations. The damping effect makes the system behave as a set of nonlinear diffusion equations, and the solutions possessing diffusion characters are known as nonlinear diffusion waves. Similar phenomena also occur in the hydrodynamic system for the models of semiconductor devices (Euler-Poisson equations). To study if the solutions of the original complex system behave as those diffusion waves and how to converge to the diffusion waves in the subsonic case, and to investigate the existence of the weak entropy solutions and their large-time behavior in the transonic case, the supersonic case, the case with vacuum, are very significant and challenging from both mathematical and physical points of view, and have been the hot research spots in applied partial differential equations The main purpose in this proposal is to investigate systematically the existence, uniqueness, of the solutions to the damped hydrodynamic Euler-Poisson system, and the construction of the solutions, the asymptotic behavior of the solutions to converge to nonlinear diffusion waves, stationary waves, and so on. The adopted methods will be the combination of the technical weighted energy method, the method of compensated compactness, the construction of entropic pairs, the method of shock wave structure. We are also interested in numerical computations to the large-time behavior of the solutions. We expect to develop some new techniques and methodologies to solve these long-time open questions, which will be a significant contribution to the theory of nonlinear partial differential equations and the application to fluid dynamics.