Dynamic growth estimates of maximum vorticity for 3D
incompressible Euler equations and the SQG model
Abstract: By performing estimates on the integral of the absolute value of vorticity along a local vortex line segment, we establish a relatively sharp dynamic growth estimate of maximum vorticity under some assumptions on the local geometric regularity of the vorticity vector. Our analysis applies to both the 3D incompressible Euler equations and the surface quasi-geostrophic model (SQG). As an application of our vorticity growth estimate, we apply our result to the 3D Euler equation with the two anti-parallel vortex tubes initial data considered by Hou-Li . Under some additional assumption on the vorticity field, which seems to be consistent with the computational results of , we show that the maximum vorticity can not grow faster than double exponential in time. Our analysis extends the earlier results by Cordoba-Fefferman [6, 7] and Deng-Hou-Yu [8, 9].
Keywords: 3D Euler equations, SQG equation, finite time blow-up,
growth rate of maximum vorticity, geometric properties.
Received: March 2011; Revised: May 2011; Published: January 2012.
2011 Impact Factor.913