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SUMMARY:3D Euler fluid equations and ideal MHD mapped to regular fluids: P
 robing the finite-time blowup hypothesis - Miguel Bustamante (Dublin)
DTSTART:20110516T120000Z
DTEND:20110516T130000Z
UID:TALK30980@talks.cam.ac.uk
CONTACT:Dr Ed Brambley
DESCRIPTION:We prove by an explicit construction that solutions to incompr
 essible 3D Euler equations\, defined in the periodic cube\, can be mapped 
 bijectively to a new system of equations whose solutions are globally regu
 lar. We establish that the usual Beale-Kato-Majda criterion for finite-tim
 e singularity (or blowup) of a solution to the 3D Euler system is equivale
 nt to a condition on the corresponding regular solution of the new system.
  In the hypothetical case of Euler finite-time singularity\, we provide an
  explicit formula for the blowup time in terms of the regular solution of 
 the new system. The new system is amenable to being integrated numerically
  using similar methods as in Euler equations. We propose a method to simul
 ate numerically the new regular system and describe how to use this to dra
 w robust and reliable conclusions on the finite-time singularity problem o
 f Euler equations\, based on the conservation of quantities directly relat
 ed to energy and circulation. The method of mapping to a regular system ca
 n be extended to any fluid equation that admits a Beale-Kato-Majda type of
  theorem\, e.g. 3D Navier-Stokes\, 2D and 3D magnetohydrodynamics\, and 1D
  inviscid Burgers. We discuss briefly the case of 2D ideal magnetohydrodyn
 amics. In order to illustrate the usefulness of the mapping\, we provide a
  thorough comparison of the analytical solution versus the numerical solut
 ion in the case of 1D inviscid Burgers equation.\n
LOCATION:MR5\, CMS
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