The Interplay between Local Geometric Properties and the Global Regularity for 3D Inocompressible Flows.
Analytical and Stochastic Fluid Dynamics
October 11, 2005 04:00 PM to 04:45 PM
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Abstract: |
Whether the 3D incompressible Euler equation can develop a finite time singularity from smooth initial data has been an outstanding open problem. It has been believed that a finite singularity of the 3D Euler equation could be the onset of turbulence. Here we review some existing computational and theoretical work on possible finite blow-up of the 3D Euler equation. Further, we show that there is a sharp relationship between the geometric properties of the vortex filament and the maximum vortex stretching. By exploring this local geometric property of the vorticity field, we have obtained a global existence of the 3D incompressible Euler equations provided that the unit vorticity vector and the velocity field have certain mild regularity property in a very localized region containing the maximum vorticity. Our assumption on the local geometric regularity of the vorticity field and the velocity field seems consistent with recent numerical experiments. Further, we discuss how viscosity may help preventing singularity formation for the 3D Navier-Stokes equations, and prove the global existence of the 2D dissipative Boussinesq equations. We will also present some recent global existence results for a model of the 3D Navier-Stokes equations and the 3D alpha-averaged Euler equations. |
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Lecture #12203
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