On the Self-Similar Solutions of the 3D Euler and the Related Equations

被引：0
作者：
Dongho Chae
机构：
[1] Sungkyunkwan University,Department of Mathematics
来源：
Communications in Mathematical Physics | 2011年 / 305卷
关键词：
Vorticity; Harmonic Function; Euler Equation; Global Regularity; Inviscid Case;
D O I：
暂无
中图分类号：
学科分类号：
摘要：
We generalize and localize the previous results by the author on the study of self-similar singularities for the 3D Euler equations. More specifically we extend the restriction theorem for the representation for the vorticity of the Euler equations in a bounded domain, and localize the results on asymptotically self-similar singularities. We also present progress towards relaxation of the decay condition near infinity for the vorticity of the blow-up profile to exclude self-similar blow-ups. The case of the generalized Navier-Stokes equations having the laplacian with fractional powers is also studied. We apply the similar arguments to the other incompressible flows, e.g. the surface quasi-geostrophic equations and the 2D Boussinesq system both in the inviscid and supercritical viscous cases.
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页码：333 / 349
页数：16
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  • [1] Beale J.T.(1984)Remarks on the breakdown of smooth solutions for the 3-D Euler equations Commun. Math. Phys. 94 61-66
  • [2] Kato T.(2010)Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation Ann. Math. 171 1903-1930
  • [3] Majda A.(2007)Nonexistence of self-similar singularities for the 3D incompressible Euler equations Commun. Math. Phys. 273 203-215
  • [4] Caffarelli L.(2007)Nonexistence of asymptotically self-similar singularities in the Euler and the Navier-Stokes equations Math. Ann. 338 435-449
  • [5] Vasseur A.(2006)Global regularity for the 2-D Boussinesq equations with partial viscous terms Adv. in Math. 203 497-513
  • [6] Chae D.(2006)On the continuation principles for the Euler equations and the quasi-geostrophic equation J. Diff. Eqns. 227 640-651
  • [7] Chae D.(2006)On the regularity conditions for the dissipative quasi-geostrophic equations SIAM. J. Math. Anal. 37 1649-1656
  • [8] Chae D.(2007)On the Euler equations of incompressible fluids Bull. Amer. Math. Soc. 44 603-621
  • [9] Chae D.(1994)Geometric Statistics in Turbulence SIAM Rev. 36 73-98
  • [10] Chae D.(1996)Geometric constraints on potential singularity formulation in the 3-D Euler equations Comm. P.D.E 21 559-571
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