Browsing by Author "Hardt, Robert M."

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    On Singularity Formation of Monotone Flows
    (2019-05-08) Yang, Hang; Hardt, Robert M.; Kiselev, Alexander
    The well-posedness problem of Euler equations is one of the most intriguing yet difficult mathematical problems in fluids. The global existence of classical solutions of 2D Euler equations has been solved by H older, Wolibner and the global existence of weak solutions by Yudovich. Yet in 3D, due to of quadratic non-linearity and non-locality, the global well-posedness of Euler's equations remains unclear still. In 2013, Hou-Luo investigated 3D Euler's equations under the axisymmetric assumptions and observed numerical blow up on a ring of hyperbolic points on the boundary of the fluid domain. Their numerical simulation has shed important light on studying the evolution of vorticity of Euler equations. In this thesis, we propose and discuss a few models of 3D Euler equations. In particular, with the modi cations to the original models, we are able to gain uniform control in the direction of the flows for the modi ed models, which will then create a mathematical rigorous scenario reminiscent of Hou-Luo's numerical work. In these models, we show that solutions blow up in finite time for a wide range of initial data. The content of Chapter 2 includes a joint work with V. Hoang, M. Radosz and B. Orcan and content of Chapter 3 is a joint work with A. Kiselev. Both of these works are published and can be found in [1] and [2]. The content of Chapter 4 is being revised for publication.