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Blow-up or no blow-up? A unified computational and analytic approach to 3D incompressible Euler and Navier–Stokes equations

Published online by Cambridge University Press:  08 May 2009

Thomas Y. Hou
Thomas Y. Hou
Affiliation:
Applied and Computational Mathematics, 217-50, Caltech, Pasadena, CA 91125, USAE-mail:[email protected]
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Abstract

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Whether the 3D incompressible Euler and Navier–Stokes equations can develop a finite-time singularity from smooth initial data with finite energy has been one of the most long-standing open questions. We review some recent theoretical and computational studies which show that there is a subtle dynamic depletion of nonlinear vortex stretching due to local geometric regularity of vortex filaments. We also investigate the dynamic stability of the 3D Navier–Stokes equations and the stabilizing effect of convection. A unique feature of our approach is the interplay between computation and analysis. Guided by our local non-blow-up theory, we have performed large-scale computations of the 3D Euler equations using a novel pseudo-spectral method on some of the most promising blow-up candidates. Our results show that there is tremendous dynamic depletion of vortex stretching. Moreover, we observe that the support of maximum vorticity becomes severely flattened as the maximum vorticity increases and the direction of the vortex filaments near the support of maximum vorticity is very regular. Our numerical observations in turn provide valuable insight, which leads to further theoretical breakthrough. Finally, we present a new class of solutions for the 3D Euler and Navier–Stokes equations, which exhibit very interesting dynamic growth properties. By exploiting the special nonlinear structure of the equations, we prove nonlinear stability and the global regularity of this class of solutions.

Type
Research Article
Information
Acta Numerica , Volume 18 , May 2009 , pp. 277 - 346
Copyright
Copyright © Cambridge University Press 2009

References

Babin, A., Mahalov, A. and Nicolaenko, B. (2001), ‘3D Navier–Stokes and Euler equations with initial data characterized by uniformly large vorticity’, Indiana Univ. Math. J. 50, 1–35.Google Scholar
Beale, J. T., Kato, T. and Majda, A. (1984), ‘Remarks on the breakdown of smooth solutions of the 3-D Euler equations’, Comm. Math. Phys. 96, 61–66.CrossRefGoogle Scholar
Boratav, O. N. and Pelz, R. B. (1994), ‘Direct numerical simulation of transition to turbulence from a high-symmetry initial condition’, Phys. Fluids 6, 2757–2784.CrossRefGoogle Scholar
Boyd, J. P. (2000), Chebyshev and Fourier Spectral Methods, 2nd edn, Dover.Google Scholar
Caffarelli, L., Kohn, R. and Nirenberg, L. (1982), ‘Partial regularity of suitable weak solutions of the Navier–Stokes equations’, Commun. Pure Appl. Math. 35, 771–831.Google Scholar
Caflisch, R. (1993), ‘Singularity formation for complex solutions of the 3D incompressible Euler equations’, Physica D 67, 1–18.Google Scholar
Cao, C. and Titi, E. S. (2007), ‘Global well-posedness of the three-dimensional primitive equations of large scale ocean and atmosphere dynamics’, Ann. of Math. 166, 245–267.Google Scholar
Chae, D. (2006), ‘Global regularity of the 2d Boussinesq equation with partial viscous terms’, Adv. Math. 203, 497–513.CrossRefGoogle Scholar
Chae, D. and Lee, J. (2002), ‘On the regularity of the axisymmetric solutions of the Navier–Stokes equations’, Math. Z. 239, 645–671.Google Scholar
Chen, C., Strain, R. M., Tsai, T. and Yau, H. T. (2008), ‘Lower bound on the blow-up rate of the axisymmetric Navier–Stokes equations’, Int. Math. Res. Not., article ID rnn016.Google Scholar
Chen, C., Strain, R. M., Tsai, T. and Yau, H. T. (2009), ‘Lower bound on the blow-up rate of the axisymmetric Navier–Stokes equations II’, to appear in Comm. Partial Differential Equations.CrossRefGoogle Scholar
Chorin, A. (1982), ‘The evolution of a turbulent vortex’, Comm. Math. Phys. 83, 517.CrossRefGoogle Scholar
Chorin, A. J. and Marsden, J. E. (1993), A Mathematical Introduction to Fluid Mechanics 3rd edn, Springer.CrossRefGoogle Scholar
Constantin, P. (1986), ‘Note on loss of regularity for solutions of the 3-D incompressible Euler and related equations’, Comm. Math. Phys. 104, 311–326.CrossRefGoogle Scholar
Constantin, P. (1994), ‘Geometric statistics in turbulence’, SIAM Review 36, 73.Google Scholar
Constantin, P. and Foias, C. (1988), Navier–Stokes Equations, Chicago University Press.CrossRefGoogle Scholar
Constantin, P., Fefferman, C. and Majda, A. (1996), ‘Geometric constraints on potentially singular solutions for the 3-D Euler equation’, Comm. Partial Differential Equations 21, 559–571.Google Scholar
Constantin, P., Lax, P. D. and Majda, A. (1985), ‘A simple one-dimensional model for the three-dimensional vorticity equations’, Commun. Pure Appl. Math. 38, 715–724.Google Scholar
Deng, J., Hou, T. Y. and Yu, X. (2005), ‘Geometric properties and non-blow-up of 3-D incompressible Euler flow’, Comm. Partial Differential Equations 30, 225–243.CrossRefGoogle Scholar
Deng, J., Hou, T. Y. and Yu, X. (2006 a), ‘Improved geometric conditions for non-blowup of 3D incompressible Euler equation’, Comm. Partial Differential Equations 31, 293–306.Google Scholar
Deng, J., Hou, T. Y., Li, R. and Yu, X. (2006 b), ‘Level set dynamics and the nonblow-up of the quasi-geostrophic equation’, Methods Appl. Anal. 13, 157–180.Google Scholar
Fernandez, V. M., Zabusky, N. J. and Gryanik, V. M. (1995), ‘Vortex intensification and collapse of the Lissajous-elliptic ring: Single and multi-filament Biot–Savart simulations and visiometrics’, J. Fluid Mech. 299, 289–331.CrossRefGoogle Scholar
Goodman, J., Hou, T. Y. and Tadmor, E. (1994), ‘On the stability of the unsmoothed Fourier method for hyperbolic equations’, Numer. Math. 67, 93–129.Google Scholar
Grauer, R. and Sideris, T. (1991), ‘Numerical computation of three dimensional incompressible ideal fluids with swirl’, Phys. Rev. Lett. 67, 3511.CrossRefGoogle Scholar
Grauer, R., Marliani, C. and Germaschewski, K. (1998), ‘Adaptive mesh refinement for singular solutions of the incompressible Euler equations’, Phys. Rev. Lett. 80, 19.Google Scholar
Hou, T. Y. and Lei, Z. (2009 a), ‘On the partial regularity of a 3D model of the Navier–Stokes equations’, Comm. Math. Phys. 287, 589–612.CrossRefGoogle Scholar
Hou, T. Y. and Lei, Z. (2009 b), ‘On the stabilizing effect of convection for three dimensional incompressible flows’, Commun. Pure Appl. Math. 62, 501–564Google Scholar
Hou, T. Y. and Li, C. (2005), ‘Global well-posedness of the viscous Boussinesq equations’, Discrete Contin. Dyn. Syst. 12, 1–12.CrossRefGoogle Scholar
Hou, T. Y. and Li, C. (2008 a), ‘Dynamic stability of the 3D axi-symmetric Navier–Stokes equations with swirl’, Commun. Pure Appl. Math. 61, 661–697.Google Scholar
Hou, T. Y. and Li, R. (2006), ‘Dynamic depletion of vortex stretching and nonblowup of the 3-D incompressible Euler equations’, J. Nonlinear Sci. 16, 639–664.CrossRefGoogle Scholar
Hou, T. Y. and Li, R. (2007), ‘Computing nearly singular solutions using pseudospectral methods’, J. Comput. Phys. 226, 379–397.CrossRefGoogle Scholar
Hou, T. Y. and Li, R. (2008 b), ‘Blowup or no blowup? The interplay between theory and numerics’, Physica D 237, 1937–1944.Google Scholar
Hou, T. Y., Lei, Z. and Li, C. (2008), ‘Global regularity of the 3D axi-symmetric Navier–Stokes equations with anisotropic data’, Comm. Partial Differential Equations 33, 1622–1637.CrossRefGoogle Scholar
Iskauriaza, L., Seregin, G. and Sverak, V. (2003), ‘L 3,∞-solutions of Navier–Stokes equations and backward uniqueness’, Uspekhi Mat. Nauk 58, 3–44.Google Scholar
Kerr, R. M. (1993), ‘Evidence for a singularity of the three dimensional, incompressible Euler equations’, Phys. Fluids 5, 1725–1746.Google Scholar
Kerr, R. M. (2005), ‘Velocity and scaling of collapsing Euler vorticesz’, Phys. Fluids 17, 075103–114.Google Scholar
Kerr, R. M. and Hussain, F. (1989), ‘Simulation of vortex reconnection’, Physica D 37, 474.CrossRefGoogle Scholar
Koch, G., Nadirashvili, N., Seregin, G. and Sverak, V. (2009), ‘Liouville theorems for the Navier–Stokes equations and applications’, to appear in Acta Mathematica.Google Scholar
Kozono, H. and Taniuchi, Y. (2000), ‘Bilinear estimates in BMO and Navier–Stokes equations’, Math Z. 235, 173–194.CrossRefGoogle Scholar
Ladyzhenskaya, O. (1970), Mathematical Problems of the Dynamics of Viscous Incompressible Fluids, Nauka, Moscow.Google Scholar
LeVeque, R. J. (1992), Numerical Method for Conservation Laws, Birkhäuser.Google Scholar
Lin, F. H. (1998), ‘A new proof of the Caffarelli–Kohn–Nirenberg theorem’, Commun. Pure Appl. Math. 51, 241–257.3.0.CO;2-A>CrossRefGoogle Scholar
Liu, J. G. and Wang, W. C. (2006), ‘Convergence analysis of the energy and helicity preserving scheme for axisymmetric flows’, SIAM J. Numer. Anal. 44, 2456–2480.CrossRefGoogle Scholar
Majda, A. J. and Bertozzi, A. L. (2002), Vorticity and Incompressible Flow, Cambridge University Press.Google Scholar
Pelz, R. B. (1997), ‘Locally self-similar, finite-time collapse in a high-symmetry vortex flament model’, Phys. Rev. E 55, 1617–1626.Google Scholar
Prodi, G. (1959), ‘Un teorema di unicità per el equazioni di Navier–Stokes’, Ann. Mat. Pura Appl. 48, 173–182.CrossRefGoogle Scholar
Pumir, A. and Siggia, E. E. (1990), ‘Collapsing solutions to the 3-D Euler equations’, Phys. Fluids A 2, 220–241.CrossRefGoogle Scholar
Raugel, G. and Sell, G. (1993 a), ‘Navier–Stokes equations on thin 3D domains I: Global attractors and global regularity of solutions’, J. Amer. Math. Soc. 6, 503–568.Google Scholar
Raugel, G. and Sell, G. (1993 b), Navier–Stokes equations on thin 3D domains III: Global and local attractors, in Turbulence in Fluid Flows: A Dynamical Systems Approach, Vol. 55 of IMA Volumes in Mathematics and its Applications, Springer, pp. 137–163.Google Scholar
Raugel, G. and Sell, G. (1994), Navier–Stokes equations on thin 3D domains II: Global regularity of spatially periodic solutions, in Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, Vol. XI of Pitman Research Notes Math, Series 299, Longman, pp. 205–247.Google Scholar
Serrin, J. (1962), ‘On the interior regularity of weak solutions of the Navier–Stokes equations’, Arch. Ration. Mech. Anal. 9, 187–195.Google Scholar
Serrin, J. (1963), The initial value problem for the Navier–Stokes equations, in Nonlinear Problems (Langer, R., ed.), University of Wisconsin Press, pp. 69–98.Google Scholar
Shelley, M. J., Meiron, D. I. and Orszag, S. A. (1993), ‘Dynamical aspects of vortex reconnection of perturbed anti-parallel vortex tubes’, J. Fluid Mech. 246, 613–652.Google Scholar
Struwe, M. (1988), ‘On partial regularity results for the Navier–Stokes equations’, Commun. Pure Appl. Math. 41, 437–458.Google Scholar
Temam, R. (2001), Navier–Stokes Equations, AMS, Providence, RI.Google Scholar