Hyperbolic balance laws are important nonlinear systems of equations utilized in the description of the motion of fluids, gases, plasma, and waves. This project focuses on several fundamental and challenging questions concerning the validity and effectiveness of different fluid flow models, the dynamical stability of fluids and the behavior of large amplitude solutions, and the modeling of strong waves in one or multiple spatial dimensions, which often appear in realistic situations. This research aims to provide fresh insights and develop new tools to characterize complicated but interesting behavior driven by nonlinearity and resonances. Increased understanding of the behavior of solutions to these modeling equations will find important applications to the dynamics of compressible fluids, the mechanics of elastic materials, traffic control systems, porous medium flows, and geophysical dynamics. The project also involves international collaborations and training of graduate students and young researchers in this challenging field.
This research project focuses on four goals: (1) To provide a clear picture of the global existence and finite-time blow-up for compressible Euler equations in one dimension with large initial data, and to find sharp density lower bound estimates for generic large solutions; (2) To establish a reasonable justification of the isentropic approximation in compressible fluids when initial entropy tends to a constant, for both inviscid and viscous fluids; (3) To investigate nonlinear Rayleigh-Taylor type instabilities in compressible fluids, including the compressible Navier-Stokes-Fourier system; (4) To study the large-time asymptotic behavior of compressible full Euler equations with frictional damping in three dimensions to justify Darcy's law, and to undertake the modeling and analysis of interactions between short waves and long waves in compressible fluids under the influence of magnetic fields. These research activities aim to provide a solid mathematical foundation to bridge gaps between existing results and for future studies. It is anticipated that the results will significantly advance basic understanding of nonlinear hyperbolic balance laws.
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.
