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M. Oliver and S. Shkoller,
The vortex blob method as a second-grade non-Newtonian fluid,
Comm. Partial Differential Equations 26 (2001), 295-314.
Abstract:
We show that a certain class of vortex blob approximations for ideal
hydrodynamics in two dimensions can be rigorously understood as
solutions to the equations of second-grade non-Newtonian fluids with
zero viscosity and initial data in the space of Radon measures
M(R2). The solutions of this regularized PDE,
also known as the isotropic Lagrangian averaged Euler or Euler-alpha
equations, are geodesics on the volume preserving diffeomorphism group
with respect to a new weak right invariant metric. We prove global
existence of unique weak solutions (geodesics) for initial vorticity
in M(R2) such as point-vortex data, and show
that the associated co-adjoint orbit is preserved by the flow.
Moreover, solutions of this particular vortex blob method converge to
solutions of the Euler equations with bounded initial vorticity,
provided that the initial data is approximated weakly in measure, and
the total variation of the approximation also converges. In
particular, this includes grid-based approximation schemes as are
common in practical vortex computations.
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