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7 - The Dual Role of Convection in 3D Navier-Stokes Equations

Published online by Cambridge University Press:  05 December 2012

By
T. Y. Hou ,
Z. Shi  and
S. Wang
Edited by
Felipe Cucker ,
Teresa Krick ,
Allan Pinkus  and
Agnes Szanto
T. Y. Hou
Affiliation:
Applied and Comput. Math, Caltech, Pasadena, CA, USA
Z. Shi
Affiliation:
Tsinghua University
S. Wang
Affiliation:
Beijing University of Technology
Felipe Cucker
Affiliation:
City University of Hong Kong
Teresa Krick
Affiliation:
Universidad de Buenos Aires, Argentina
Allan Pinkus
Affiliation:
Technion - Israel Institute of Technology, Haifa
Agnes Szanto
Affiliation:
North Carolina State University
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Summary

Abstract

We investigate the dual role of convection on the large time behavior of the 3D incompressible Navier-Stokes equations. On the one hand, convection is responsible for generating small scales dynamically. On the other hand, convection may play a stabilizing role in potentially depleting nonlinear vortex stretching for certain flow geometry. Our study is centered around a 3D model that was recently proposed by Hou and Lei in [23] for axisymmetric 3D incompressible Navier-Stokes equations with swirl. This model is derived by neglecting the convection term from the reformulated Navier-Stokes equations and shares many properties with the 3D incompressible Navier-Stokes equations. In this paper, we review some of the recent progress in studying the singularity formation of this 3D model and how convection may destroy the mechanism that leads to singularity formation in the 3D model.

Introduction

Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data with finite energy is one of the seven Millennium problems posted by the Clay Mathematical Institute [16]. This problem is challenging because the vortex stretching nonlinearity is super-critical for the 3D Navier-Stokes equation. Conventional functional analysis based on energy type estimates fails to provide a definite answer to this problem. Global regularity results are obtained only under certain smallness assumptions on the initial data or the solution itself. Due to the incompressibility condition, the convection term seems to be neutrally stable if one tries to estimate the Lp (1 < p≤∞) norm of the vorticity field. As a result, the main effort has been to use the diffusion term to control the nonlinear vortex stretching term by diffusion without making use of the convection term explicitly.

Type
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Information
Foundations of Computational Mathematics, Budapest 2011 , pp. 129 - 164
Publisher: Cambridge University Press
Print publication year: 2012

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References

[1] J. T., Beale, T., Kato and A., Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm. Math. Phys. 94, 61–66, 1984.Google Scholar
[2] L., Caffarelli, R., Kohn and L., Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math., 35, 771–831, 1982.Google Scholar
[3] R., Caflisch and M., Siegel, A semi-analytic approach to Euler singularities. Methods and Appl. of Analysis, 11, 423–430, 2004.Google Scholar
[4] D., Chae, A., Cordoba, D., Cordoba and M. A., Fontelos, Finite time singularities in a 1D model of the quasi-geostrophic equation. Adv. Math., 194, 203–223, 2005.Google Scholar
[5] P., Constantin, Note on loss of regularity for solutions of the 3D incompressible Euler and related equations. Commun. Math. Phys., 104, 311–326, 1986.Google Scholar
[6] P., Constantin, C., Fefferman and A., Majda, Geometric constraints on potentially singular solutions for the 3-D Euler equation. Commun. in PDEs, 21, 559–571, 1996.Google Scholar
[7] D., Cordoba and C., Fefferman, Growth of solutions for QG and 2D Euler equations. J. Amer. Math. Soc., 15, 665–670 (electronic), 2002.Google Scholar
[8] A., Cordoba, D., Cordoba and M. A., Fontelos, Formation of singularities for a transport equation with nonlocal velocity. Ann. of Math., 162, 1–13, 2005.Google Scholar
[9] P., Constantin, P. D., Lax and A. J., Majda, A simple one-dimensional model for the three-dimensional vorticity equation. Comm. Pure Appl. Math., 38, 715–724, 1985.Google Scholar
[10] J., Deng, T. Y., Hou and X., Yu, Geometric properties and non-blowup of 3-D incompressible Euler flow. Commun. PDEs, 30, 225–243, 2005.Google Scholar
[11] J., Deng, T. Y., Hou and X., Yu, Improved geometric conditions for non-blowup of 3D incompressible Euler equation. Commun. PDEs, 31, 293–306, 2006.Google Scholar
[12] S., De Gregorio, On a one-dimensional model for the 3-dimensional vorticity equation. J. Stat. Phys., 59, 1251–1263, 1990.Google Scholar
[13] S., De Gregorio, A partial differential equation arising in a 1D model for the 3D vorticity equation. Math. Method Appl. Sci., 19, 1233–1255, 1996.Google Scholar
[14] R. J., DiPerna and P. L., LionsOn the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. Math., 130, 321–366, 1989.Google Scholar
[15] L. C., Evans, Partial Differential Equations. American Mathematical Society Publ., 1998.
[16] C., Fefferman, http://www.claymath.org/millennium/Navier-Stokes equations.
[17] G. B., Foland, Real Analysis – Modern Techniques and Their Applications. Wiley-Interscience Publ., Wiley and Sons, Inc., 1984.
[18] G. B., Foland, Introduction to Partial Differential Equations. Princeton University Press, Princeton, N.J., 1995.
[19] 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, 639–664, 2006.Google Scholar
[20] T. Y., Hou and R., Li, Computing nearly singular solutions using pseudospectral methods. J. Comput. Phys., 226, 379–397, 2007.Google Scholar
[21] T. Y., Hou and R., Li, Blowup or no blowup? The interplay between theory and numerics. PhysicaD, 237, 1937–1944, 2008.Google Scholar
[22] T. Y., Hou and C., Li, Dynamic stability of the 3D axi-symmetric Navier-Stokes equations with swirl. Comm. Pure Appl. Math., 61, 661–697, 2008.Google Scholar
[23] T. Y., Hou and Z., Lei, On the stabilizing effect of convection in 3D incompressible flows. Comm. Pure Appl. Math., 62, 501–564, 2009.Google Scholar
[24] T. Y., Hou and Z., Lei, On partial regularity of a 3D model of Navier-Stokes equations. Commun. Math Phys., 287, 281–298, 2009.Google Scholar
[25] T. Y., Hou, C., Li, Z., Shi, S., Wang, and X., Yu, On singularity formation of a nonlinear nonlocal system. Arch. Ration. Mech. Anal., 199, 117–144, 2011.Google Scholar
[26] T. Y., Hou, Z., Shi and S., Wang On singularity formation of a 3D model for incompressible Navier-Stokes equations. arXiv:0912.1316v1 [math.AP], submitted to Adv. Math..
[27] T. Y., Hou and Z., Shi, Dynamic growth estimates of maximum vorticity for 3D incompressible Euler equations and the SQG model. DCDS-A, 32, 2011.Google Scholar
[28] R., Kerr, Evidence for a singularity of the three dimensional, incompressible Euler equations. Phys. Fluids, 5, 1725–1746, 1993.Google Scholar
[29] R., Kerr, Velocity and scaling of collapsing Euler vortices. Phys. Fluids, 17, 075103-114, 2005.Google Scholar
[30] D., Li and J., Rodrigo, Blow up for the generalized surface quasigeostrophic equation with supercritical dissipation. Comm. Math. Phys., 286, 111–124, 2009.Google Scholar
[31] D., Li and Y. G., Sinai, Blow ups of complex solutions of the 3D Navier-Stokes system and renormalization group method. J. Europ. Math. Soc., 10, 267–313, 2008.Google Scholar
[32] A. J., Majda and A. L., Bertozzi, Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, 2002.
[33] T., Matsumotoa, J., Bec and U., Frisch, Complex-space singularities of 2D Euler flow in Lagrangian coordinates. PhysicaD, 237, 1951–1955, 2007.Google Scholar
[34] G., Prodi, Un teorema di unicità per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl., 48, 173–182, 1959.Google Scholar
[35] J., Serrin, The initial value problem for the Navier-Stokes equations. In Nonlinear Problems, Univ. of Wisconsin Press, Madison, 69–98, 1963.
[36] R., Temam, Navier-Stokes Equations. Second Edition, AMS Chelsea Publishing, Providence, RI, 2001.

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