Browsing by Author "Kiselev, Alexander"

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    Global Regularity and Finite-time Blow-up in Model Fluid Equations
    (2017-04-19) Do, Tam; Kiselev, Alexander
    Determining the long time behavior of many partial differential equations modeling fluids has been a challenge for many years. In particular, for many of these equations, the question of whether solutions exist for all time or form singularities is still open. The structure of the nonlinearity and non-locality in these equations makes their analysis difficult using classical methods. In recent years, many models have been proposed to study fluid equations. In this thesis, we will review some new result in regards to these models as well as give insight into the relation between these models and the true equations. First, we analyze a one-dimensional model for the two-dimensional surface quasi-geostrophic equation and vortex sheets. The model gained prominence due to the work of Cordoba, Cordoba, and Fontelos and is often referred to as the CCF model. We will show that solutions are globally regular in the presence of logarithmically supercritcal dissipation and that solutions eventually gain regularity in the presence of supercritical dissipation. Finally, by analyzing a dyadic model of the equation, we will gain insight into how certain possible singularities in the CCF model can be supressed by dissipation. For the second part of this thesis, we study some one-dimensional model equations for the Euler equations. These models are influenced by the recent numerical simulations of Tom Hou and Guo Luo. They observed possible singularity formation for the three-dimensional Euler equation at the boundary of a cylindrical domain under certain symmetry assumptions. Under these assumptions, a singularity was observed numerically and the solution was observe to have hyperbolic structure near the singularity. Hou and Luo proposed a one-dimensional model system to study singularity formation theoritically. We will study a family of one-dimensional models generalizing their model. The results in chapter 2 are the results of joint work with A. Kiselev, V. Hoang, M. Radosz, and X. Xu.
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    Global Regularity for Euler Vortex Patch in Bounded Smooth Domains
    (2018-04-18) Li, Chao; Kiselev, Alexander; Hardt, Robert
    It is well known that the Euler vortex patch in two dimensional plane will remain regular if it is regular enough initially. In bounded domains, the regularity theory for patch solutions is less complete. In this thesis, I study the Euler vortex patch in a general smooth bounded domain. I prove global in time regularity by providing the upper bound of the growth on curvature of the patch boundary. For a special symmetric scenario, I construct an example of double exponential curvature growth, showing that our upper bound is qualitatively sharp.
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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.