Dynamic growth estimates of maximum vorticity for 3Dincompressible Euler equations and the SQG model

Thomas Y. Hou; Zuoqiang Shi

doi:10.3934/dcds.2012.32.1449

Article Contents

2012, Volume 32, Issue 5: 1449-1463. Doi: 10.3934/dcds.2012.32.1449

This issue Previous Article Next Article Sharp upperbounds for the number of large amplitude limit cycles in polynomial Lienard systems

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

Thomas Y. Hou^1, and
Zuoqiang Shi^1,

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

Received: February 28, 2011

Revised: April 30, 2011

Published: April 2012

Abstract Related Papers Cited by

Abstract

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].

Keywords:

Mathematics Subject Classification: Primary: 76B03; Secondary: 35L60, 35M10.

Citation:

Related Papers

Cited by

References

[1]	J. T. Beale, T. Kato and A. J. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys., 94 (1984), 61-66.doi: 10.1007/BF01212349.
[2]	P. Constantin, C. Fefferman and A. Majda, Geometric constraints on potentially singular solutions for the 3-D Euler equation, Commun. PDE, 21 (1996), 559-571.
[3]	P. Constantin, A. J. Majda and E. G. Tabak, Singular front formation in a model for quasigeostrophic flow, Phys. Fluids, 6 (1994), 9-11.doi: 10.1063/1.868050.
[4]	P. Constantin, Q. Nie and N. Schörghofer, Nonsingular surface quasi- geostrophic flow, Phys. Lett. A, 241 (1998), 168-172.doi: 10.1016/S0375-9601(98)00108-X.
[5]	D. Cordoba, Nonexistence of simple hyperbolic bolw-up for the quasi-geostrophic equation, Ann. of Math. (2), 148 (1998), 1135-1152.doi: 10.2307/121037.
[6]	D. Cordoba and C. Fefferman, On the collapse of tubes carried by 3D incompressible flows, Commun. Math. Phys., 222 (2001), 293-298.doi: 10.1007/s002200100502.
[7]	D. Cordoba and C. Fefferman, Growth of solutions for QG and 2D Euler equations, J. Amer. Math. Soc., 15 (2002), 665-670.doi: 10.1090/S0894-0347-02-00394-6.
[8]	J. Deng, T. Y. Hou and X. Yu, Geometric properties and nonblowup of 3D incompressible Euler flow, Comm. PDE, 30 (2005), 225-243.doi: 10.1081/PDE-200044488.
[9]	J. Deng, T. Y. Hou and X. Yu, Improved geometric conditions for non-blowup of the 3D incompressible Euler equation, Comm. PDE, 31 (2006), 293-306.doi: 10.1080/03605300500358152.
[10]	J. Deng, T. Y. Hou, R. Li and X. Yu, Level set dynamics and non-blowup of the 2D quasi-geostrophic equation, Methods and Applications of Analysis, 13 (2006), 157-180.
[11]	T. Y. Hou, Blow-up or no blow-up? A unified computational and analytic approach to 3D incompressible Euler and Navier-Stokes equations, Acta Numerica, 18 (2009), 277-346.doi: 10.1017/S0962492906420018.
[12]	T. Y. Hou and R. Li, Dynamic depletion of vortex stretching and non-blowup of the 3-D incompressible Euler equations, J. Nonlinear Science, 16 (2006), 639-664.doi: 10.1007/s00332-006-0800-3.
[13]	T. Y. Hou and R. Li, Blowup or no blowup? The interplay between theory and numerics, Phisica D, 237 (2008), 1937-1944.doi: 10.1016/j.physd.2008.01.018.
[14]	R. M. Kerr, Evidence for a singularity of the three-dimensional, incompressible Euler equations, Phys. Fluids A, 5 (1993), 1725-1746.doi: 10.1063/1.858849.
[15]	A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Cambridge Texts in Applied Mathematics, 27, Cambridge University Press, Cambridge, 2002.
[16]	K. Ohkitani and M. Yamada, Inviscid and inviscid-limit behavior of a surface quasigeostrophic flow, Phys. Fluids, 9 (1997), 876-882.