Topics in fluid dynamics

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The topics addressed in the project concern partial differential equations arising in the study of fluid dynamics. In particular, we study the incompressible Navier-Stokes equations for the motion of a viscous liquid, the Euler equations (and their models) for the motion of an inviscid liquid, and a class of scalar equations arising naturally from the mentioned systems through approximations in physically important settings. In the first part, regarding the Navier-Stokes equations, we intend to explore the anisotropy introduced by the oscillatory structure of solutions, in order to construct classes of large data leading to global well-posedness. The second part of the project is geared towards gaining a deeper understanding of the ill-posedness of a class of active scalar equations, based on the properties of the second iterate. The analysis focuses particularly on the equations where the nonlinearity is given by a Fourier multiplier whose symbol is anisotropic and/or even (the porous media equation and the magneto-geostrophic equation). The third objective of this project is to investigate a system of PDEs simulating the fundamental features of the vorticity-stretching phenomena in the three-dimensional incompressible Euler equations. The project seeks to advance the understanding of the dynamics described by the equations of fluid dynamics by methods of rigorous mathematical analysis. These partial differential equations represent a basic tool for modeling many natural phenomena and technological applications. For instance, the equations describing fluid flow (on which an important part of this project concentrates) are used in weather prediction, climate research, and various technological applications, such as the design of optimal aerodynamical and hydrodynamical shapes. The equations of fluid dynamics are notoriously difficult to solve, even with the help of the largest computers. Some of these difficulties constitute part of the problem, but some are due to the fact that our mathematical understanding of the equations themselves is incomplete. From the analytical point of view, these equations account for interactions of a broad range of space- and time-scales in a highly non-linear fashion. Due to this intrinsic complexity, the accuracy needed to fully resolve the underlying phenomena via numerical computations is out of reach for the foreseeable future. Advances in theoretical understanding of these partial differential equations are vital for the validation of the models and for an accurate interpretation of numerical results. In the final analysis, the main task of the theoretical investigation is to find simple and natural parameters that control the behavior of the solutions. Once a good set of such parameters is known, it becomes much easier to design practical methods for calculating the solutions numerically.