2,006 research outputs found
A Note on the Regularity of Inviscid Shell Model of Turbulence
In this paper we continue the analytical study of the sabra shell model of
energy turbulent cascade initiated in \cite{CLT05}. We prove the global
existence of weak solutions of the inviscid sabra shell model, and show that
these solutions are unique for some short interval of time. In addition, we
prove that the solutions conserve the energy, provided that the components of
the solution satisfy , for
some positive absolute constant , which is the analogue of the Onsager's
conjecture for the Euler's equations. Moreover, we give a Beal-Kato-Majda type
criterion for the blow-up of solutions of the inviscid sabra shell model and
show the global regularity of the solutions in the ``two-dimensional''
parameters regime
Spatial dependence of the superexchange interactions for transition-metal trimers in graphene
This study examines the magnetic interactions between spatially-variable
manganese and chromium trimers substituted into a graphene superlattice. Using
density functional theory, we calculate the electronic band structure and
magnetic populations for the determination of the electronic and magnetic
properties of the system. To explore the super-exchange coupling between the
transition-metal atoms, we establish the magnetic magnetic ground states
through a comparison of multiple magnetic and spatial configurations. Through
an analysis of the electronic and magnetic properties, we conclude that the
presence of transition-metal atoms can induce a distinct magnetic moment in the
surrounding carbon atoms as well as produce an RKKY-like super-exchange
coupling. It hoped that these simulations can lead to the realization of
spintronic applications in graphene through electronic control of the magnetic
clusters.Comment: 6 pages, 5 Figur
A Blow-Up Criterion for Classical Solutions to the Compressible Navier-Stokes Equations
In this paper, we obtain a blow up criterion for classical solutions to the
3-D compressible Naiver-Stokes equations just in terms of the gradient of the
velocity, similar to the Beal-Kato-Majda criterion for the ideal incompressible
flow. In addition, initial vacuum is allowed in our case.Comment: 25 page
Global well-posedness for a slightly supercritical surface quasi-geostrophic equation
We use a nonlocal maximum principle to prove the global existence of smooth
solutions for a slightly supercritical surface quasi-geostrophic equation. By
this we mean that the velocity field is obtained from the active scalar
by a Fourier multiplier with symbol , where
is a smooth increasing function that grows slower than as
.Comment: 11 pages, second version with slightly stronger resul
Passive tracer in a flow corresponding to a two dimensional stochastic Navier Stokes equations
In this paper we prove the law of large numbers and central limit theorem for
trajectories of a particle carried by a two dimensional Eulerian velocity
field. The field is given by a solution of a stochastic Navier--Stokes system
with a non-degenerate noise. The spectral gap property, with respect to
Wasserstein metric, for such a system has been shown in [9]. In the present
paper we show that a similar property holds for the environment process
corresponding to the Lagrangian observations of the velocity. In consequence we
conclude the law of large numbers and the central limit theorem for the tracer.
The proof of the central limit theorem relies on the martingale approximation
of the trajectory process
Extracting molecular Hamiltonian structure from time-dependent fluorescence intensity data
We propose a formalism for extracting molecular Hamiltonian structure from
inversion of time-dependent fluorescence intensity data. The proposed method
requires a minimum of \emph{a priori} knowledge about the system and allows for
extracting a complete set of information about the Hamiltonian for a pair of
molecular electronic surfaces.Comment: 7pages, no figures, LaTeX2
Interaction of vortices in viscous planar flows
We consider the inviscid limit for the two-dimensional incompressible
Navier-Stokes equation in the particular case where the initial flow is a
finite collection of point vortices. We suppose that the initial positions and
the circulations of the vortices do not depend on the viscosity parameter \nu,
and we choose a time T > 0 such that the Helmholtz-Kirchhoff point vortex
system is well-posed on the interval [0,T]. Under these assumptions, we prove
that the solution of the Navier-Stokes equation converges, as \nu -> 0, to a
superposition of Lamb-Oseen vortices whose centers evolve according to a
viscous regularization of the point vortex system. Convergence holds uniformly
in time, in a strong topology which allows to give an accurate description of
the asymptotic profile of each individual vortex. In particular, we compute to
leading order the deformations of the vortices due to mutual interactions. This
allows to estimate the self-interactions, which play an important role in the
convergence proof.Comment: 39 pages, 1 figur
On the constants in a Kato inequality for the Euler and Navier-Stokes equations
We continue an analysis, started in [10], of some issues related to the
incompressible Euler or Navier-Stokes (NS) equations on a d-dimensional torus
T^d. More specifically, we consider the quadratic term in these equations; this
arises from the bilinear map (v, w) -> v . D w, where v, w : T^d -> R^d are two
velocity fields. We derive upper and lower bounds for the constants in some
inequalities related to the above bilinear map; these bounds hold, in
particular, for the sharp constants G_{n d} = G_n in the Kato inequality | < v
. D w | w >_n | <= G_n || v ||_n || w ||^2_n, where n in (d/2 + 1, + infinity)
and v, w are in the Sobolev spaces H^n, H^(n+1) of zero mean, divergence free
vector fields of orders n and n+1, respectively. As examples, the numerical
values of our upper and lower bounds are reported for d=3 and some values of n.
When combined with the results of [10] on another inequality, the results of
the present paper can be employed to set up fully quantitative error estimates
for the approximate solutions of the Euler/NS equations, or to derive
quantitative bounds on the time of existence of the exact solutions with
specified initial data; a sketch of this program is given.Comment: LaTeX, 39 pages. arXiv admin note: text overlap with arXiv:1007.4412
by the same authors, not concerning the main result
On the (Non)-Integrability of KdV Hierarchy with Self-consistent Sources
Non-holonomic deformations of integrable equations of the KdV hierarchy are
studied by using the expansions over the so-called "squared solutions" (squared
eigenfunctions). Such deformations are equivalent to perturbed models with
external (self-consistent) sources. In this regard, the KdV6 equation is viewed
as a special perturbation of KdV equation. Applying expansions over the
symplectic basis of squared eigenfunctions, the integrability properties of the
KdV hierarchy with generic self-consistent sources are analyzed. This allows
one to formulate a set of conditions on the perturbation terms that preserve
the integrability. The perturbation corrections to the scattering data and to
the corresponding action-angle variables are studied. The analysis shows that
although many nontrivial solutions of KdV equations with generic
self-consistent sources can be obtained by the Inverse Scattering Transform
(IST), there are solutions that, in principle, can not be obtained via IST.
Examples are considered showing the complete integrability of KdV6 with
perturbations that preserve the eigenvalues time-independent. In another type
of examples the soliton solutions of the perturbed equations are presented
where the perturbed eigenvalue depends explicitly on time. Such equations,
however in general, are not completely integrable.Comment: 16 pages, no figures, LaTe
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