32 research outputs found
Evolution by the vortex filament equation of curves with a corner
In this proceedings article we shall survey a series of results on the
stability of self-similar solutions of the vortex filament equation. This
equation is a geometric flow for curves in and it is used as a
model for the evolution of a vortex filament in fluid mechanics. The main
theorem gives, under suitable assumptions, the existence and description of
solutions generated by curves with a corner, for positive and negative times.
Its companion theorem describes the evolution of perturbations of self-similar
solutions up to a singularity formation infinite time, and beyond this time. We
shall give a sketch of the proof. These results were obtained in collaboration
with Luis Vega.Comment: 17 pages, 2 pictures, proceedings of the 40th "Journ\'ees EDP" -
Biarritz 201
Weighted Strichartz estimates for radial Schr\"odinger equation on noncompact manifolds
We prove global weighted Strichartz estimates for radial solutions of linear
Schr\"odinger equation on a class of rotationally symmetric noncompact
manifolds, generalizing the known results on hyperbolic and Damek-Ricci spaces.
This yields classical Strichartz estimates with a larger class of exponents
than in the Euclidian case and improvements for the scattering theory. The
manifolds, whose volume element grows polynomially or exponentially at
infinity, are characterized essentially by negativity conditions on the
curvature, which shows in particular that the rich algebraic structure of the
Hyperbolic and Damek-Ricci spaces is not the cause of the improved dispersive
properties of the equation. The proofs are based on known dispersive results
for the equation with potential on the Euclidean space, and on a new one, valid
for C^1 potentials decaying like 1/r^2 at infinity
On scattering for NLS: from Euclidean to hyperbolic space
We prove asymptotic completeness in the energy space for the nonlinear
Schrodinger equation posed on hyperbolic space in the radial case, in space
dimension at least 4, and for any energy-subcritical, defocusing, power
nonlinearity. The proof is based on simple Morawetz estimates and weighted
Strichartz estimates. We investigate the same question on spaces which kind of
interpolate between Euclidean space and hyperbolic space, showing that the
family of short range nonlinearities becomes larger and larger as the space
approaches the hyperbolic space. Finally, we describe the large time behavior
of radial solutions to the free dynamics.Comment: 13 pages. References updated; see Remark 1.
Collisions of vortex filament pairs
We consider the problem of collisions of vortex filaments for a model
introduced by Klein, Majda and Damodaran, and Zakharov to describe the
interaction of almost parallel vortex filaments in three-dimensional fluids.
Since the results of Crow examples of collisions are searched as perturbations
of antiparallel translating pairs of filaments, with initial perturbations
related to the unstable mode of the linearized problem; most results are
numerical calculations. In this article we first consider a related model for
the evolution of pairs of filaments and we display another type of initial
perturbation leading to collision in finite time. Moreover we give numerical
evidence that it also leads to collision through the initial model. We finally
study the self-similar solutions of the model