Global regularity of the 3D axi-symmetric Navier-Stokes equations with anisotropic data

Communications in Partial Differential Equations

Volumn 33, Issue 9, 2008, Pages 1622-1637

  Hou, Thomas Y ^a   Lei, Zhen ^a   Li, Congming ^b  

^a CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

^b UNIVERSITY OF COLORADO   (United States)

Author keywords

Large anisotropic data; Navier Stokes equations

Indexed keywords

EID: 49949114046     PISSN: 03605302     EISSN: 15324133     Source Type: Journal    
DOI: 10.1080/03605300802108057     Document Type: Article

References (27)

1. Boratav, O. N., Pelz, R. B. (1994). Direct numerical simulation of transition to turbulence from a high-symmetry initial condition. Phys. Fluids 6:2757-2784.
2. Caffarelli, L., Kohn, R., Nirenberg, L. (1982). Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35:771-831.
3. Chae, D. (2006). Global regularity of the 2d Boussinesq equation with partial viscous terms. Advances in Math 203:789-801.
4. Chae, D., Lee, J. (2002). On the regularity of the axisymmetric solutions of the Navier-Stokes Equations. Math. Z. 239:645-671.
5. Chen, C. C., Strain, R. M., Tsai, T. P., Yau, H. T. (2008). Lower Bound on the Blow-up Rate of Axisymmetric Navier-Stokes Equations. International Mathematics Research Notices 2008:rnn016.
6. Fefferman, C. http://www.claymath.org/millenium/Navier-Stokes_Equations/.
7. Greene, J. M., Pelz, R. B. (2000). Stability of postulated, self-similar, hydrodynamic blowup solutions. Phys. Rev. E 62:7982-7986.
8. Hou, T. Y., Li, C. (2005). Global well-posedness of the viscous Boussinesq equations. DCDS 12:1-12.
9. Hou, T. Y., Li, C. (2008). Dynamic stability of the 3D axi-symmetric Navier-Stokes equations with swirl. Comm. Pure Appl. Math. 61:661-697.
10. Hou, T. Y., Li, R. (2006). Dynamic depletion of vortex stretching and non-blowup of the 3-D incompressible Euler equations. J. Nonlinear Science 16:639-664.
11. Hou, T. Y., Li, R. (2008). Computing nearly singular solutions using pseudospectral methods. J. Comput. Phys. 226:379-397.
12. Hou, T. Y., Li, R. (2007). Nonexistence of local self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete and Continuous Dynamical Systems 18:637-642.
13. 3,infin;-solutions of Navier-Stokes equations and backward uniqueness. Uspekhi Mat. Nauk 58:3-44. [14] Kerr, R. M. (1993). Evidence for a singularity of the three dimensional, incompressible Euler equations. Phys. Fluids 5:1725-1746.
14. Kerr, R. M. (2005). Velocity and scaling of collapsing Euler vortices. Phys. Fluids 17:075103-114.
15. Ladyzhenskaya, O. A. (1970). Mathematica Problems of the Dynamics of Viscous Incompressible Fluids. Moscow: Nauka.
16. Lin, F. H. (1998). A new proof of the Caffarelli-Korn-Nirenberg theorem. Comm. Pure Appl. Math. 51:241-257.
17. Liu, J. G., Wang, W. C. (2006). Convergence analysis of the energy and helicity preserving scheme for axisymmetric flows. SINUM 44:2456-2480.
18. Majda, A. J., Bertozzi, A. L. (2002). Vorticity and Incompressible Flow. Cambridge, UK: Cambridge University Press.
19. Necas, J., Ruzicka, M., Sverak, V. (1996). On Leray's self-similar solutions of the Navier-Stokes equations. Acta Math. 176:283-294.
20. Pelz, R. B. (1997). Locally self-similar, finite-time collapse in a high-symmetry vortex filament model. Phys. Rev. E 55:1617-1626.
21. Pelz, R. B. (2001). Symmetry and hydrodynamic blow-up problem. J. Fluid Mech. 444:299-320.
22. Raugel, G., Sell, G. (1993). Navier-Stokes equations on thin 3D domains I: Global attractors and global regularity of solutions. J. Amer. Math. Soc. 6:503-568.
23. Raugel, G., Sell, G. (1993). Navier-Stokes equations on thin 3D domains III: Global and local attractors. In: Turbulence in Fluid Flows: A Dynamical Systems Approach, IMA Volumes in Mathematics and Its Applications. Vol. 55, pp. 137-163. Berlin: Springer-Verlag.
24. Raugel, G., 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, College de France Seminar, Pitman Research Notes Math, Series 299, pp. 205-247. Vol. XI, London: Longman.
25. Stein, E. M. (1993). Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton, NJ: Princeton University Press.
26. Temam, R. (2001). Navier-Stokes Equations. Providence, RI: American Mathematical Society.
27. Tsai, T. P. (1998). On Leray's self-similar solutions of the Navier-Stokes equations satisfying local energy estimates. Arch. Rational Mech. Anal. 143:29-51.