Mathematical Sciences: Aspects of Compressible and Incompressible Flows

主要研究人员将利用现代泛函分析和数值分析技术研究不可压缩粘性流体、可压缩流体和流体混合物的某些定性性质。特别是，她将研究二维和三维空间中的Navier-Stokes方程的解的长期渐近行为，三维中的不可压缩磁流体动力学方程，以及一些模拟可压缩流动的积分-微分方程组。她还将研究一组多相方程在时间上整体解的存在性。通常，人们只有在很长一段时间之后才对物理系统的行为感兴趣。例如，一旦进程启动，人们可能会认为单一类型的稳定行为是主要的可观察输出。偏离这种稳定的行为可能是非常不受欢迎的。首席研究人员将致力于利用应用数学和数值分析的现代技术，研究各种真实世界流体系统在很长一段时间内的行为。

The principal investigator will study certain qualitative properties of incompressible viscous fluids, compressible fluids and mixtures of fluids by using techniques of modern functional analysis and numerical analysis. In particular, she will study the long-time asymptotic behavior of solutions of the Navier-Stokes equations in two and three space dimensions, the equations of incompressible magnetohydrodynamics in three dimensions, and some integro-differential equations that model compressible flows. She will also investigate the global existence in time of solutions of a set of multiphase equations. Very often one is interested in the behavior of a physical system only after a long time. For example, once a process is started one might expect that a single type of stable behavior is the dominant observable output. Deviations from this stable behavior might be very much unwanted. The principal investigator will devote most of her efforts to study how various real-world fluid systems behave for a very long time by using modern techniques from applied mathematics and numerical analysis.