Long time dynamics of compressible fluids and kinetic theory with boundaries

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Fluids and gases are ubiquitous in nature and exhibit a wide range of interesting phenomena observed in laboratories, daily life, or space. Their dynamics is governed by nonlinear partial differential equations (PDE). The objective of this project is to investigate long time dynamics and singularities for solutions to PDEs arising from compressible fluid dynamics and kinetic theory with emphasis on physically important objects in the presence of boundaries. They have rich applications in mathematical sciences and engineering and offer a great deal of mathematical challenges. The advances from this project will enhance the understanding of highly nontrivial phenomena and bring novel mathematical methods that could resolve unsolved questions. The research project will be integrated with education and outreach activities through student research projects, mentoring activities, course development and recruitment. Specifically, the following topics will be pursued: (i) stabilization mechanisms and long time dynamics of compressible Euler and MHD flows (ii) stability of gravitational collapses and stellar rotation for self-gravitating stars modeled by the Euler-Poisson system (iii) a new well-posedness theory for the Vlasov-Poisson system and stability of equilibrium galaxies (iv) boundary collisions for the kinetic transport and Fokker-Planck equations. The goal is to find mathematical frameworks that capture the underlying physical phenomena and to develop mathematical theories by applied analysis and PDE methods.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.