Archive for Rational Mechanics and Analysis

, Volume 199, Issue 1, pp 117–144

On Singularity Formation of a Nonlinear Nonlocal System

  • Thomas Y. Hou
  • Congming Li
  • Zuoqiang Shi
  • Shu Wang
  • Xinwei Yu
Article

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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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Thomas Y. Hou
    • 1
  • Congming Li
    • 2
  • Zuoqiang Shi
    • 1
  • Shu Wang
    • 3
  • Xinwei Yu
    • 4
  1. 1.Applied and Computational MathematicsCaltech, PasadenaUSA
  2. 2.Department of Applied MathematicsUniversity of ColoradoBoulderUSA
  3. 3.College of Applied SciencesBeijing University of TechnologyBeijingChina
  4. 4.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada

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