Stability of Blowup for a 1D Model of Axisymmetric 3D Euler Equation

J. Nonlinear Science(2016)

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The question of the global regularity versus finite- time blowup in solutions of the 3D incompressible Euler equation is a major open problem of modern applied analysis. In this paper, we study a class of one-dimensional models of the axisymmetric hyperbolic boundary blow-up scenario for the 3D Euler equation proposed by Hou and Luo (Multiscale Model Simul 12:1722–1776, 2014 ) based on extensive numerical simulations. These models generalize the 1D Hou–Luo model suggested in Hou and Luo Luo and Hou ( 2014 ), for which finite-time blowup has been established in Choi et al. (arXiv preprint. arXiv:1407.4776 , 2014 ). The main new aspects of this work are twofold. First, we establish finite-time blowup for a model that is a closer approximation of the three-dimensional case than the original Hou–Luo model, in the sense that it contains relevant lower-order terms in the Biot–Savart law that have been discarded in Hou and Luo Choi et al. ( 2014 ). Secondly, we show that the blow-up mechanism is quite robust, by considering a broader family of models with the same main term as in the Hou–Luo model. Such blow-up stability result may be useful in further work on understanding the 3D hyperbolic blow-up scenario.
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Key words
Effusion,Biot-Savart Law,Finite Time Blow-up,Multiscale Model Simul,Real Line Case
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