Large and Multi-Dimensional Solutions to Hyperbolic Balance Laws

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Motion of many disturbances in the air, from ordinary noise to shock waves about an airfoil (that cause the familiar sonic boom), is described by equations of compressible fluids. Even liquids at high enough speeds (close to the sound speed) are compressible. This project is devoted to the mathematical study of the different versions of the Navier-Stokes equations that describe dynamics of compressible fluids. Similar equations provide mathematical models for phenomena in related scientific areas, including elastic material mechanics, porous medium flows, and geophysical dynamics. To determine the validity of a particular equation as a model for a given phenomenon, it can be helpful to determine whether the equation has a solution, and if it does, how this solution depends upon the physical parameters. If in particular regimes these equations either lack solutions, or have solutions which depend in a discontinuous way upon the data, this will indicate that these mathematical descriptions of the phenomena have only limited validity for the particular applications. The current project will focus on challenging problems for large solutions modeling strong waves, which often appear in realistic situations. The research will bring fresh insights and develop new tools to characterize complicated large solutions driven by nonlinearity and resonances. The progress in this project will facilitate the design of high-performance computational methods. This project also involves international collaborations and training of graduate students and junior researchers in this challenging field.This research project is devoted to the analytical study of dynamics of compressible fluids. The project will produce further developments in four inter-related research areas. The first objective is to investigate the global existence and finite time blowup for compressible Euler equations in one dimension with large initial data, and to find sharp density lower-bound estimate for generic large solutions. Second, the research on compressible Navier-Stokes-Fourier system with temperature-dependent transport coefficients that is aimed at a systematic theory including the global existence of weak solutions, uniform lower bound on absolute temperature, and large time behavior of the solutions. The third objective is to study the local theory for isentropic Navier-Stokes equations with density-dependent viscosity allowing vacuum in initial data, with applications to various shallow water models such as Saint-Venant equations for shallow water. The last objective is to study the large time asymptotic behavior of compressible full Euler equations in three dimensions to justify the Darcy's law in this very complicated case, and to study the modeling and analysis for interactions between short waves and long waves in compressible fluids under the influence of magnetic field.