8,599 research outputs found
A Nonlinear Analysis of the Averaged Euler Equations
This paper develops the geometry and analysis of the averaged Euler equations
for ideal incompressible flow in domains in Euclidean space and on Riemannian
manifolds, possibly with boundary. The averaged Euler equations involve a
parameter ; one interpretation is that they are obtained by ensemble
averaging the Euler equations in Lagrangian representation over rapid
fluctuations whose amplitudes are of order . The particle flows
associated with these equations are shown to be geodesics on a suitable group
of volume preserving diffeomorphisms, just as with the Euler equations
themselves (according to Arnold's theorem), but with respect to a right
invariant metric instead of the metric. The equations are also
equivalent to those for a certain second grade fluid. Additional properties of
the Euler equations, such as smoothness of the geodesic spray (the Ebin-Marsden
theorem) are also shown to hold. Using this nonlinear analysis framework, the
limit of zero viscosity for the corresponding viscous equations is shown to be
a regular limit, {\it even in the presence of boundaries}.Comment: 25 pages, no figures, Dedicated to Vladimir Arnold on the occasion of
his 60th birthday, Arnold Festschrift Volume 2 (in press
Relative commutants of strongly self-absorbing C*-algebras
The relative commutant of a strongly self-absorbing
algebra is indistinguishable from its ultrapower . This
applies both to the case when is the hyperfinite II factor and to the
case when it is a strongly self-absorbing C*-algebra. In the latter case we
prove analogous results for and reduced powers
corresponding to other filters on . Examples of algebras with
approximately inner flip and approximately inner half-flip are provided,
showing the optimality of our results. We also prove that strongly
self-absorbing algebras are smoothly classifiable, unlike the algebras with
approximately inner half-flip.Comment: Some minor correction
Automorphism groups of randomized structures
We study automorphism groups of randomizations of separable structures, with
focus on the -categorical case. We give a description of the
automorphism group of the Borel randomization in terms of the group of the
original structure. In the -categorical context, this provides a new
source of Roelcke precompact Polish groups, and we describe the associated
Roelcke compactifications. This allows us also to recover and generalize
preservation results of stable and NIP formulas previously established in the
literature, via a Banach-theoretic translation. Finally, we study and classify
the separable models of the theory of beautiful pairs of randomizations,
showing in particular that this theory is never -categorical (except
in basic cases).Comment: 28 page
Rigidity of gradient Ricci Solitons
We define a gradient Ricci soliton to be rigid if it is a flat bundle where is Einstein. It is known that not all
gradient solitons are rigid. Here we offer several natural conditions on the
curvature that characterize rigid gradient solitons. Other related results on
rigidity of Ricci solitons are also explained in the last section.Comment: 16 page
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