Abstract
In (Comm Pure Appl Math 62(4):502–564, 2009), Hou and Lei proposed a 3D model for the axisymmetric incompressible Euler and Navier–Stokes equations with swirl. This model shares a number of properties of the 3D incompressible Euler and Navier–Stokes equations. In this paper, we prove that the 3D inviscid model with an appropriate Neumann-Robin or Dirichlet-Robin boundary condition will develop a finite time singularity in an axisymmetric domain. We also provide numerical confirmation for our finite time blowup results. We further demonstrate that the energy of the blowup solution is bounded up to the singularity time, and the blowup mechanism for the mixed Dirichlet-Robin boundary condition is essentially the same as that for the energy conserving homogeneous Dirichlet boundary condition. Finally, we prove that the 3D inviscid model has globally smooth solutions for a class of large smooth initial data with some appropriate boundary condition. Both the analysis and the results we obtain here improve the previous work in a rectangular domain by Hou et al. (Adv Math 230:607–641, 2012) in several respects.
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References
Bardos C., Titi E.: Euler equations for incompressible ideal fluids. Uspekhi Mat. Nauk 623(375), 5–46 (2007)
Beale J.T., Kato T., Majda A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm. Math. Phys. 94(1), 61–66 (1984)
Boratav O.N., Pelz R.B.: Direct numerical simulation of transition to turbulence from a high-symmetry initial condition. Phys. Fluids 6, 2757–2784 (1994)
Caffarelli L., Kohn R., Nirenberg L.: Partial regularity of suitable weak solutions of the Navier–Stokes equations. Comm. Pure Appl. Math. 35, 771–831 (1982)
Caflisch R., Siegel M.: A semi-analytic approach to Euler singularities. Methods Appl. Anal. 11, 423–430 (2004)
Chae D.: On the finite-time singularities of the 3D incompressible Euler equations. Comm. Pure Appl. Math. 60, 597–617 (2007)
Constantin P.: Note on loss of regularity for solutions of the 3D incompressible Euler and related equations. Commun. Math. Phys. 104, 311–326 (1986)
Constantin P.: On the Euler equations of incompressible fluids. Bull. Am. Math. Soc. 44, 603–621 (2007)
Constantin P., Fefferman C., Majda A.: Geometric constraints on potentially singular solutions for the 3-D Euler equation. Comm. PDEs. 21, 559–571 (1996)
Deng J., Hou T.Y., Yu X.: Geometric properties and non-blowup of 3-D incompressible Euler flow. Comm. PDEs. 30, 225–243 (2005)
Deng J., Hou T.Y., Yu X.: Improved geometric conditions for non-blowup of 3D incompressible Euler equation. Comm. PDEs. 31, 293–306 (2006)
Evans, L.C.: Partial Differential Equations. American Mathematical Society Publ. (1998)
Fefferman, C.: http://www.claymath.org/millennium/Navier-Stokesequations
Foland G.B.: Introduction to Partial Differential Equations. Princeton University Press, Princeton (1995)
Grauer R., Sideris T.: Numerical computation of three dimensional incompressible ideal fluids with swirl. Phys. Rev. Lett. 67, 3511–3514 (1991)
Grafke T., Grauer R.: Finite-time Euler singularities: a Lagrangian perspective. Appl. Math. Lett. 26, 500–505 (2013)
Grauer R., Marliani C., Germaschewski K.: Adaptive mesh refinement for singular solutions of the incompressible Euler equations. Phys. Rev. Lett. 80, 4177–4180 (1998)
Hou T.Y.: Blow-up or no blow-up? A unified computational and analytic approach to study 3-D incompressible Euler and Navier–Stokes equations. Acta Numer. 18, 277–346 (2009)
Hou T.Y., Li R.: Dynamic depletion of vortex stretching and non-blowup of the 3-D incompressible Euler equations. J. Nonlinear Sci. 16(6), 39–664 (2006)
Hou T.Y., Li R.: Blowup or no blowup? The interplay between theorey and numerics. Physica D 237, 1937–1944 (2008)
Hou T.Y., Li C.M.: Dynamic stability of the 3D axi-symmetric Navier–Stokes equations with swirl. Comm. Pure Appl. Math. 61(5), 661–697 (2008)
Hou T.Y., Lei Z.: On the stabilizing effect of convection in three-dimensional incompressible flows. Comm. Pure Appl. Math. 62(4), 502–564 (2009)
Hou T.Y., Lei Z.: On partial regularity of a 3D model of Navier–Stokes equations. Comm. Math. Phys. 287, 281–298 (2009)
Hou T.Y., Shi Z., Wang S.: On singularity formation of a 3D model for incompressible Navier–Stokes equations. Adv. Math. 230, 607–641 (2012)
Kerr R.M.: Evidence for a singularity of the three dimensional, incompressible Euler equations. Phys. Fluids 5(7), 1725–1746 (1993)
Lin F.-H.: A new proof of the Caffarelli–Kohn–Nirenberg theorem. Comm. Pure Appl. Math. 51(3), 241–257 (1998)
Luo, G., Hou, T.Y.: Potential singular solutions of the 3D incompressible Euler equations. arXiv:1310.0497v1 [physics.flu-dyn] (2013)
Majda A.J., Bertozzi A.L.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge (2002)
Orlandi P., Carnevale G.F.: Nonlinear amplification of vorticity in inviscid interaction of orthogonal Lamb dipoles. Phys. Fluids 19, 057106 (2007)
Prodi G.: Un teorema di unicità à per le equazioni di Navier–Stokes. Ann. Mat. Pura Appl. 48, 173–182 (1959)
Pumir A., Siggia E.D.: Development of singular solutions to the axisymmetric Euler equations. Phys. Fluids A 4, 1472–1491 (1992)
Serrin, J.: The initial value problem for the Navier–Stokes equations. In: Nonlinear Problems, pp. 69–98. University of Wisconsin Press, Madison (1963)
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Hou, T.Y., Lei, Z., Luo, G. et al. On Finite Time Singularity and Global Regularity of an Axisymmetric Model for the 3D Euler Equations. Arch Rational Mech Anal 212, 683–706 (2014). https://doi.org/10.1007/s00205-013-0717-6
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DOI: https://doi.org/10.1007/s00205-013-0717-6