Zheng 9703711 Yuxi Zheng proposes to study some nonlinear partial differential equations from fluid dynamics (Euler equations), plasma physics (Vlasov-Poisson system), and liquid crystal physics. These equations are the laws of motion of their respective physics. The turbulent nature and/or defects in the materials show up in the form of singularities and instabilities in the solutions of the equations. Yuxi Zheng plans to use advanced analytical tools to study the structures of the singular solutions. In the case of compressible Euler equations in two space dimensions, for example, Yuxi Zheng plans to isolate typical singularities (hurricanes, tornadoes, shocks, etc.) and investigate their individual structures. The result of the investigation will be a clear understanding of the worst possible solutions, and thereby quantify our knowledge of the physics and offer guidance in high-performance numerical computations of general solutions. Yuxi Zheng proposes to study some applied mathematical problems in the fields of fluid dynamics (which includes motion of the air and water), plasma physics (which includes motion of the material that makes up the Sun, and also more than 90% of the universe), and liquid crystal physics in material science. Scientists and engineers have used mathematical equations, called partial differential equations, to model the motions. The turbulent nature and/or defects in the materials show up in the form of singularities and instabilities in the solutions of the equations. It is these singularities and instabilities that often spoil accurate numerical computations of the solutions. Yuxi Zheng plans to use the state of the art analytical tools to study the structures of the singular solutions. In the case of a compressible gas such as air, for example, Yuxi Zheng plans to isolate typical singularities (hurricanes, tornadoes, shocks, etc.) and investigate their individual structures. The result of the investigation will be a clear understanding of the worst possible solutions, and thereby quantify our knowledge of the physics and offer guidance in high-performance numerical computations of general solutions.
