CAREER: Hilberts Sixth Problem in the Boltzmann equation

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This mathematics research project aims to develop a unified theory of many body systems (such as gas dynamics) by connecting mesoscopic models (Boltzmann equation) to macroscopic models (fluid) and microscopic models (many particles). The investigator and collaborator will study these connections in models of various physical systems such as tornados, shock waves, water-oil mixtures, and plasma in tokamak fusion reactors. Results of the research are intended to lead to advances in understanding of these and other important systems. In connection with the research, the investigators plan to develop graduate topics courses, undergraduate directed study courses, research activities for undergraduate students, and workshops for practicing mathematicians. One of the fundamental questions in partial differential equations is the development of a unified theory of gas dynamics connecting the mesoscopic Boltzmann equations to the macroscopic fluid models (hydrodynamic limits) that arise in formal limits, and the Boltzmann equations to microscopic N-body problems (Grad-Boltzmann limit). In this project, the investigator and collaborator study the rigorous passage to both limits mathematically in various physical situations. The first part of the project concerns the hydrodynamic limits when free interfaces occur naturally: (i) vortex patch solutions of the incompressible Euler equation, (ii) two-species Vlasov-Boltzmann systems modeling binary fluid mixtures with sharp interface and surface tension, and (iii) shock formation in the compressible Euler equations and its kinetic approximation. The second part of the project concerns the Grad-Boltzmann limit in the presence of a thermal stochastic boundary. The investigators study the passage from the N-body Newtonian system interacting with stochastic boundary toward the Boltzmann equation with diffuse reflection boundary. They will investigate the mixing effect of such boundaries and aim to establish a rigorous passage for a time interval greater than a mean free time. They will also study the link of generic steady non-equilibrium states between the mesoscopic and microscopic levels.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.