Blow-up or no blow-up? A unified computational and analytic approach to 3D incompressible Euler and Navier–Stokes equations
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.
© 2009 Cambridge University Press. Published online: 08 May 2009. I would like to thank my collaborators, Drs Jian Deng, Zhen Lei, Congming Li, Ruo Li, and Xinwei Yu, who have contributed significantly to the results presented in this survey paper. We would like to thank Prof. Lin-Bo Zhang from the Institute of Computational Mathematics in Chinese Academy of Sciences (CAS) for providing us with the computing resources to perform this large-scale computational project. Additional computing resources were provided by the Center of High Performance Computing in CAS. We also thank Prof. Robert Kerr for providing us with his Fortran subroutine generating his initial data. This work was in part supported by the NSF under the NSF grant DMS-0713670.
Published - Hou2009p5565Acta_Numer_2.pdf