Behavior and regularity properties of solutions of fluid equations

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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.