Courses / ActivitiesBlow-up or No Blow-up? A Unified Computational and Analytic Approach to Study 3-D Incompressible Euler and Navier-Stokes Equations
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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. Our studies show that convection has a surprising stabilizing effect. 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 peudo-spectral method on some of the most promising blow-up candidates. Our results show that there is tremendous dynamic depletion of vortex stretching. 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 property. By exploiting the special nonlinear structure of the equations, we prove nonlinear stability and the global regularity of this class of solutions.
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Yizhao Hou
2008-12-23
11:10:00 - 12:00:00
405 , Mathematics Research Center Building (ori. New Math. Bldg.)
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. Our studies show that convection has a surprising stabilizing effect. 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 peudo-spectral method on some of the most promising blow-up candidates. Our results show that there is tremendous dynamic depletion of vortex stretching. 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 property. By exploiting the special nonlinear structure of the equations, we prove nonlinear stability and the global regularity of this class of solutions.