PDE Methods in Kinetic Theory and Their Applications

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Kinetic theory is at the center of multi-scale modeling, which connects the microscopic particle models to macroscopic fluid models. There are many challenging open problems in kinetic theory which are of great importance from both mathematical and physical standpoints. The main goal of this research is to continue developing new mathematical methods to resolve open problems in partial differential equations arising in the kinetic theory and other fields in mathematical physics. The investigations will include: boundary effects in the Boltzmann theory for dilute gases, derivation of various macroscopic fluid models from the kinetic theory, and nonlinear stability and instability of steady states in a wide range of applied problems. These research projects will have important impacts in many areas of physical sciences. The study of stable equilibria in the Vlasov theory (collisionless Boltzmann theory) will shed new light on plasma control in nuclear fusion and on galaxy evolution; the study of the Stefan problem will build a mathematical foundation for morphological stability of crystal growth and many other problems arising in materials sciences; and the study of phase-transitions in the Vlasov-Boltzmann model will lead to better understanding of phase segregation in binary fluids. An important objective of the project is to provide training for students and junior scientists involved in carrying out this research.