# American Institute of Mathematical Sciences

May  2012, 32(5): 1449-1463. doi: 10.3934/dcds.2012.32.1449

## Dynamic growth estimates of maximum vorticity for 3D incompressible Euler equations and the SQG model

 1 Caltech, Applied and Comput. Math, 9-94, Pasadena, CA 91125, United States, United States

Received  March 2011 Revised  May 2011 Published  January 2012

By performing estimates on the integral of the absolute value of vorticity along a local vortex line segment, we establish a relatively sharp dynamic growth estimate of maximum vorticity under some assumptions on the local geometric regularity of the vorticity vector. Our analysis applies to both the 3D incompressible Euler equations and the surface quasi-geostrophic model (SQG). As an application of our vorticity growth estimate, we apply our result to the 3D Euler equation with the two anti-parallel vortex tubes initial data considered by Hou-Li [12]. Under some additional assumption on the vorticity field, which seems to be consistent with the computational results of [12], we show that the maximum vorticity can not grow faster than double exponential in time. Our analysis extends the earlier results by Cordoba-Fefferman [6, 7] and Deng-Hou-Yu [8, 9].
Citation: Thomas Y. Hou, Zuoqiang Shi. Dynamic growth estimates of maximum vorticity for 3D incompressible Euler equations and the SQG model. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1449-1463. doi: 10.3934/dcds.2012.32.1449
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