9,193 research outputs found
Polar actions on certain principal bundles over symmetric spaces of compact type
We study polar actions with horizontal sections on the total space of certain
principal bundles with base a symmetric space of compact type. We
classify such actions up to orbit equivalence in many cases. In particular, we
exhibit examples of hyperpolar actions with cohomogeneity greater than one on
locally irreducible homogeneous spaces with nonnegative curvature which are not
homeomorphic to symmetric spaces.Comment: 7 pages; some minor change
Incompressible flows with piecewise constant density
We investigate the incompressible Navier-Stokes equations with variable
density. The aim is to prove existence and uniqueness results in the case of
discontinuous ini- tial density. In dimension n = 2, 3, assuming only that the
initial density is bounded and bounded away from zero, and that the initial
velocity is smooth enough, we get the local-in-time existence of unique
solutions. Uniqueness holds in any dimension and for a wider class of velocity
fields. Let us emphasize that all those results are true for piecewise constant
densities with arbitrarily large jumps. Global results are established in
dimension two if the density is close enough to a positive constant, and in
n-dimension if, in addition, the initial velocity is small. The Lagrangian
formula- tion for describing the flow plays a key role in the analysis that is
proposed in the present paper.Comment: 32 page
The incompressible navier-stokes equations in vacuum
We are concerned with the existence and uniqueness issue for the
inhomogeneous incompressible Navier-Stokes equations supplemented with H^1
initial velocity and only bounded nonnegative density. In contrast with all the
previous works on that topics, we do not require regularity or positive lower
bound for the initial density, or compatibility conditions for the initial
velocity, and still obtain unique solutions. Those solutions are global in the
two-dimensional case for general data, and in the three-dimensional case if the
velocity satisfies a suitable scaling invariant smallness condition. As a
straightforward application, we provide a complete answer to Lions' question in
[25], page 34, concerning the evolution of a drop of incompressible viscous
fluid in the vacuum
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