This project is devoted to the study of large and/or multi-dimensional solutions to dissipative balance laws arising in modeling of flows, including gases, fluids, traffics, and semi-conductor devices. The research focuses on the following particular objects: a) global behavior of compressible Navier-Stokes equations in one and several space dimensions, b) vanishing viscosity limit from compressible Navier-Stokes equations toward Euler equations with physical viscosity coefficient, c) global stability or instability of relaxation system, d) global BV theory for a class of dissipative balance laws, and e) large time behavior for certain fluid dynamics systems including viscous Boussinesq system in two dimension and compressible Navier-Stokes equations in two and three dimensions. The goals of this project are developing novel analytic approaches to constructing large solutions, investigating singular behavior of the solutions, identifying global stability criterion, understanding the large time behavior of multi-dimensional solutions.
The objective of this research is to investigate several open problems for dissipative hyperbolic balance laws. Hyperbolic balance laws are important nonlinear partial differential equations modeling the motion of fluids, gas, and waves. The research will concentrate on problems for large solutions, strong waves, and realistic models in multiple spatial dimensions. These situations often involve complicated but interesting phenomena resulting from nonlinearity and resonance and thus underdeveloped. The results are expected to have applications to dynamics of compressible fluids, elastic material mechanics, traffic control systems, semi-conductor devices modeling, porous medium flows, and geophysical dynamics. The advance of mathematical understanding of these problems plays important role in the design of effective numerical schemes for scientific computation in these fundamental areas. The project will also provide education and training to students and young researchers in this challenging field.