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Published January 2011 | public
Journal Article Open

On Singularity Formation of a Nonlinear Nonlocal System


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.

Additional Information

© 2010 Springer-Verlag. Received November 17, 2009. Accepted March 8, 2010. Published online April 20, 2010. Dr. T. Hou would like to acknowledge the NSF for their generous support through the Grants DMS-0713670 and DMS-0908546. The work of Drs. Z. Shi and S. Wang was supported in part by the NSF grant DMS-0713670. The research of Dr. C. Li was in part supported by the NSF grant DMS-0908546. The research of Dr. S. Wang was supported by the Grants NSFC 10771009 and PHR-IHLB 200906103. The research of Dr. X. Yu was in part supported by the Faculty of Science start-up fund of University of Alberta, and the research grant from NSERC. This work was done during Drs. Li, Wang, and Yu's visit to ACM at Caltech. They would like to thank Prof. T. Hou and Caltech for their hospitality during their visit. Finally, we would like to thank the anonymous referee for the valuable comments and suggestions.

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August 19, 2023
August 19, 2023