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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

Thomas Y. Hou 1, and  Zuoqiang Shi 1,

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].
Keywords: 3D Euler equations, growth rate of maximum vorticity, SQG equation, finite time blow-up, geometric properties..
Mathematics Subject Classification: Primary: 76B03; Secondary: 35L60, 35M1.
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, 2012, 32 (5) : 1449-1463. doi: 10.3934/dcds.2012.32.1449
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Cambridge Texts in Applied Mathematics, 27, Cambridge University Press, Cambridge, 2002.  Google Scholar

[16]

K. Ohkitani and M. Yamada, Inviscid and inviscid-limit behavior of a surface quasigeostrophic flow, Phys. Fluids, 9 (1997), 876-882.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Cambridge Texts in Applied Mathematics, 27, Cambridge University Press, Cambridge, 2002.  Google Scholar

[16]

K. Ohkitani and M. Yamada, Inviscid and inviscid-limit behavior of a surface quasigeostrophic flow, Phys. Fluids, 9 (1997), 876-882.  Google Scholar

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