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Published June 1, 2012 | public
Journal Article Open

On singularity formation of a 3D model for incompressible Navier–Stokes equations


We investigate the singularity formation of a 3D model that was recently proposed by Hou and Lei (2009) in [15] 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. This model shares many properties of the 3D incompressible Navier–Stokes equations. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the 3D inviscid model for a class of initial boundary value problems with smooth initial data of finite energy. We also prove the global regularity of the 3D inviscid model for a class of small smooth initial data.

Additional Information

© 2012 Elsevier Inc. Received 3 February 2010; accepted 3 February 2012. Available online 24 March 2012. Communicated by Charles Fefferman. Dr. T. Hou would like to acknowledge 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. S. Wang was also supported by China 973 Program (Grant No. 2011CB808002), the Grants NSFC 11071009 and PHR-IHLB 200906103. This work was done during Dr. Shu Wang's visit to ACM at Caltech. He would like to thank Prof. T. Hou and Caltech for their hospitality during his visit. We would like to thank Profs. Joseph Keller, Congming Li, Xinwei Yu, and the referee for their comments which helped to improve the quality of this work.

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August 22, 2023
August 22, 2023