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CATEGORIES:Monday Mechanics Seminars (DAMTP)
SUMMARY:3D Euler fluid equations and ideal MHD mapped to r
egular fluids: Probing the finite-time blowup hypo
thesis - Miguel Bustamante (Dublin)
DTSTART;TZID=Europe/London:20110516T130000
DTEND;TZID=Europe/London:20110516T140000
UID:TALK30980AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/30980
DESCRIPTION:We prove by an explicit construction that solution
s to incompressible 3D Euler equations\, defined i
n the periodic cube\, can be mapped bijectively to
a new system of equations whose solutions are glo
bally regular. We establish that the usual Beale-K
ato-Majda criterion for finite-time singularity (o
r blowup) of a solution to the 3D Euler system is
equivalent to a condition on the corresponding reg
ular solution of the new system. In the hypothetic
al case of Euler finite-time singularity\, we prov
ide an explicit formula for the blowup time in ter
ms of the regular solution of the new system. The
new system is amenable to being integrated numeric
ally using similar methods as in Euler equations.
We propose a method to simulate numerically the ne
w regular system and describe how to use this to d
raw robust and reliable conclusions on the finite-
time singularity problem of Euler equations\, base
d on the conservation of quantities directly relat
ed to energy and circulation. The method of mappin
g to a regular system can be extended to any fluid
equation that admits a Beale-Kato-Majda type of t
heorem\, e.g. 3D Navier-Stokes\, 2D and 3D magneto
hydrodynamics\, and 1D inviscid Burgers. We discus
s briefly the case of 2D ideal magnetohydrodynamic
s. In order to illustrate the usefulness of the ma
pping\, we provide a thorough comparison of the an
alytical solution versus the numerical solution in
the case of 1D inviscid Burgers equation.\n
LOCATION:MR5\, CMS
CONTACT:Dr Ed Brambley
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