This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
The first goal of this project is to better understand physical vacuum as a free boundary for solutions to compressible Euler and Navier-Stokes equations with or without forcing and to develop a satisfactory nonlinear stability theory. The physical vacuum naturally appears in astrophysics and gas dynamics: a mathematical study of its behavior will help understand dynamics of gaseous stars. Furthermore, this investigation will lead to a general theory of degenerate hyperbolic equations. The second goal is to derive, with rigorous justification, various fluid equations from Boltzmann equations. Study of hydrodynamic limits for Boltzmann equations is important in connecting the two different subjects of kinetic theory and fluid dynamics. A rigorous study will help understand how large-scale behaviors emerge from dynamics or structures on small scales. In particular, for the Vlasov-Maxwell-Boltzmann system, it will open a new line of research.
The compressible Euler and Navier-Stokes fluid equations and the Boltzmann kinetic equations are the basic, challenging models in gas dynamics; they have broad applications in many other areas of science and engineering as well. For example, Euler and Navier-Stokes equations are useful to study shallow water waves, and Boltzmann equations are important for semiconductor theory. The mathematical problems addressed in this project are not only important for the study of these equations, but also of interest in physics and other sciences. A better understanding of them would prompt the study of models for more complex, realistic flows.
