Multi-Dimensional Problems For Systems Of Conservation Laws

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Fluid free boundary problems arise in many physical, medical, and engineering models. In contrast with fixed boundaries, free boundaries are determined by the dynamics of the problem itself, a typical toy example being the boundary of a piece of ice melting in a glass of water. Problems of fluid-vacuum interfaces arise in the study of dynamics of boundaries of stars in astrophysics or in multi-phase flows. In multi-phase fluids (like blood) fluid-fluid interfaces problems arise. Frequently they involve the study of fluid-deformable structure interfaces such as in cell deformation. Physical vacuum is a natural medium (or rather the lack of any medium) where fluids may spread, such as in the boundary motion of gaseous stars, or propagation of the interface between the gas or liquid and vacuum in flows through porous media. This research project deals with vacuum free boundary problems for some systems of conservation laws in multi-dimensions arising in fluid dynamics and astrophysics. Some new analytic and geometric methods will be developed in this project to achieve the following goals: 1) to establish the long time well-posedness theory and understand long time dynamics for the vacuum free boundary problems for the three spatial dimensional Euler-Poisson and Navier-Stokes-Poisson equations of gaseous stars, capturing the physical vacuum singularity, 2) to establish the vanishing viscosity limit theory for the vacuum free boundary problems for the three spatial dimensional Navier-Stokes-Poisson equations of gaseous stars, capturing the physical vacuum singularity, 3) to elucidate the role of the heat conductivity to the dynamics of physical vacuum boundary of gaseous stars, 4) to establish the long time well-posedness theory for solutions to the vacuum free boundary problem for the three spatial dimensional compressible Euler equations with damping and understand the interface behavior of solutions related to the celebrated Barenblatt self-similar solutions to porous media equations. New ideas and techniques to be developed in this project will contribute to a general theory of degenerate hyperbolic, coupled hyperbolic-parabolic and hyperbolic-elliptic free boundary problems. This project will also provide training opportunities to some graduate and undergraduate students in applied mathematics.