Analysis of Models in Fluid Dynamics

Oklahoma State University, Stillwater OK

Investigators

Abstract

Rusin DMS-1613831 The problems addressed in the project involve mathematics of an interdisciplinary character. Analytical study of the nonlinear phenomena in equations describing flows of fluids is essential to areas that range from weather prediction, climate and environmental studies, turbulent and combustion (as for instance in engines) to the cosmological question of mass distribution in the universe. The aim of the project is to develop applicable mathematics theory that yields progress in understanding of fluids and that facilitates efficient computational tools. The main objective of the project is the investigation of well-posedness of partial differential equations arising in fluid dynamics. The primary emphasis is placed on the incompressible three-dimensional Navier-Stokes equations in the context of large initial data. In particular, the analysis focuses on the existence and properties of solutions with axial symmetry and time-periodic solutions. The research extends to related models such as a class of active scalar equations that arises in the geophysical fluid dynamics. Here, the main aim is to investigate ill/well-posedness phenomena of nonlinear problems, where the nonlinearity is based on even or singular symbols, additionally equipped with some intrinsic anisotropy. The project concerns furthermore the problem of uniqueness of solutions to the primitive equation of the ocean. The last objective addresses a local model of the Euler equations and its analysis, in particular related to singular limits and vorticity stretching in the three-dimensional Euler equations.