On Singularity Formation of a Nonlinear Nonlocal System |
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Authors: | Thomas Y Hou Congming Li Zuoqiang Shi Shu Wang Xinwei Yu |
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Institution: | (1) Department of Mathematics, Sungkyunkwan University, Suwon, 440-746, South Korea |
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Abstract: | We investigate the singularity formation of a nonlinear nonlocal system. This nonlocal system is a simplified one-dimensional
system of the 3D model that was recently proposed by Hou and Lei (Comm Pure Appl Math 62(4):501–564, 2009) for axisymmetric 3D incompressible Navier–Stokes equations with swirl. The main
difference between the 3D model of Hou and Lei and the reformulated 3D Navier–Stokes equations is that the convection term
is neglected in the 3D model. In the nonlocal system we consider in this paper, we replace the Riesz operator in the 3D model
by the Hilbert transform. One of the main results of this paper is that we prove rigorously the finite time singularity formation
of the nonlocal system for a large class of smooth initial data with finite energy. We also prove global regularity for a
class of smooth initial data. Numerical results will be presented to demonstrate the asymptotically self-similar blow-up of
the solution. The blowup rate of the self-similar singularity of the nonlocal system is similar to that of the 3D model. |
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