Behavior and regularity properties of solutions of fluid equations

研究者研究描述流体运动的方程解的性质。他考虑了纳维-斯托克斯系统，这是粘性不可压缩流体运动的主要模型，普朗特方程（纳维-斯托克斯的边界层近似），欧拉方程（粗略地说，零粘度的纳维-斯托克斯），以及原始方程，它代表了海洋和大气的大规模运动。他分析了流体和弹性固体之间的流体结构相互作用，其他涉及自由边界的问题，黏性流体在黏度趋于零时存在边界时的极限行为，以及解的规律性和长期行为。结果扩展了我们对地球科学、工程和其他领域中流体流动问题的理解，特别是与控制和数值方法有关的问题。研究生也参与了该项目的工作。该项目的主要目标是研究流体动力学中出现的方程解的定性性质和规律性。虽然主要的焦点放在纳维-斯托克斯方程，研究者也考虑相关的模型，如欧拉系统，普朗特方程，以及海洋和大气的原始方程。特别强调的是涉及流体（可压缩或不可压缩）和弹性固体的系统。这里的主要目的是研究解的局部和全局适定性，以及它们存在时的大时间行为。利用局部渐近的性质，研究了具有边界的域上的黏性消失问题。这里的主要目标是更好地理解普朗特方程的解，并确定它们在消失的粘度极限中的作用。项目的最后一部分涉及模拟流体的偏微分方程解的局部性质，特别是它们的解析正则性和Gevrey正则性、唯一延拓性和长时间行为。研究生也参与了该项目的工作。

Kukavica1615239 The investigator studies the properties of solutions of equations that describe the motion of fluids. He considers the Navier-Stokes system, which is the principal model for motion of a viscous incompressible fluid, the Prandtl equations (a boundary-layer approximation for Navier-Stokes), the Euler equations (roughly speaking, Navier-Stokes with zero viscosity), and the primitive equations, which represent large-scale motion of the ocean and atmosphere. He analyzes fluid-structure interactions between a fluid and an elastic solid, other problems involving free boundaries, the limiting behavior of viscous fluids in the presence of boundaries as the viscosity goes to zero, and the regularity and long-time behavior of solutions. Results expand our understanding of fluid flow problems arising in geosciences, engineering, and other areas, especially in connection with control and numerical methods. Graduate students are involved in the work of the project. The main goal of the project is the study of the qualitative properties and regularity of solutions of equations arising in fluid dynamics. While the primary focus is placed on the Navier-Stokes equations, the investigator considers also related models such as the Euler system, Prandtl equations, and the primitive equations of the ocean and the atmosphere. A special emphasis is systems involving a fluid (compressible or incompressible) and an elastic solid. Here the main aim is to investigate the local and global well-posedness of solutions as well as their large-time behavior when they exist. The investigator also studies the vanishing viscosity problem in domains with boundaries, using the properties of the local asymptotics. The main goal here is to better understand the solutions of the Prandtl equation and to identify their role in the vanishing viscosity limit. The last part of the project concerns the local properties of solutions of the partial differential equations modeling fluids, in particular their analytic and Gevrey regularity, unique continuation, and their long time behavior. Graduate students are involved in the work of the project.