Book Description
Among the most important and most difficult open problems in the field of analysis are questions about the behavior of solutions to differential equations modeling the dynamics of fluids. The main issues that one must overcome in addressing them are frequently the nonlinearity and nonlocality of these equations. In this thesis we study these and related models, focusing on the possibility of singularity formation for their solutions as well as on ways such singular behavior can be suppressed. In the first chapter of this thesis, we discuss the small scale creation and possible singularity formation in PDEs of fluid mechanics, especially the Euler equations and the related models. Recently, Tom Hou and Guo Luo proposed a new scenario, so called the hyperbolic flow scenario, for the development of a finite time singularity in solutions to 3D incompressible Euler equation. We first give a clear and understandable picture of hyperbolic flow restricted in 1D. Then, based on the recent work by Alexander Kiselev and Vladimir \v{S}ver\'{a}k, we look into the hyperbolic geometry in 2D. Finally, we go back to 3D problem, and analyze a simplified 1D model for the potential singularity of the 3D Euler equation by Tom Hou and Guo Luo. In the second chapter of this thesis, we investigate the problem about how to suppress the blowup. At the end of the second chapter, we demonstrate that incompressible mixing flow can indeed arrest the finite time blow up phenomenon. We first concentrate on understanding the mechanisms involved in mixing, studying mixing properties of the flows with different structure, and finding most efficient mixing flows. We resolve the problem of finding the optimal lower bound of the ``mixing norm'' under an enstrophy constraint on the velocity field. On the basis of this result, we evaluate the role of mixing in systems where chemotaxis is present. We prove the result that the presence of fluid flow can affect singularity formation by mixing the density thus making concentration harder to achieve. This is an example to show that the fluid advection can regularize singular nonlinear dynamics.