3,141 research outputs found
On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations
We consider an active scalar equation that is motivated by a model for
magneto-geostrophic dynamics and the geodynamo. We prove that the non-diffusive
equation is ill-posed in the sense of Hadamard in Sobolev spaces. In contrast,
the critically diffusive equation is well-posed. In this case we give an
example of a steady state that is nonlinearly unstable, and hence produces a
dynamo effect in the sense of an exponentially growing magnetic field.Comment: We have modified the definition of Lipschitz well-posedness, in order
to allow for a possible loss in regularity of the solution ma
On the supercritically diffusive magneto-geostrophic equations
We address the well-posedness theory for the magento-geostrophic equation,
namely an active scalar equation in which the divergence-free drift velocity is
one derivative more singular than the active scalar. In the presence of
supercritical fractional diffusion given by (-\Delta)^\gamma, where 0<\gamma<1,
we discover that for \gamma>1/2 the equations are locally well-posed, while for
\gamma<1/2 they are ill-posed, in the sense that there is no Lipschitz solution
map. The main reason for the striking loss of regularity when \gamma goes below
1/2 is that the constitutive law used to obtain the velocity from the active
scalar is given by an unbounded Fourier multiplier which is both even and
anisotropic. Lastly, we note that the anisotropy of the constitutive law for
the velocity may be explored in order to obtain an improvement in the
regularity of the solutions when the initial data and the force have thin
Fourier support, i.e. they are supported on a plane in frequency space. In
particular, for such well-prepared data one may prove the local existence and
uniqueness of solutions for all values of \gamma \in (0,1).Comment: 24 page
Emergence of fractal behavior in condensation-driven aggregation
We investigate a model in which an ensemble of chemically identical Brownian
particles are continuously growing by condensation and at the same time undergo
irreversible aggregation whenever two particles come into contact upon
collision. We solved the model exactly by using scaling theory for the case
whereby a particle, say of size , grows by an amount over the
time it takes to collide with another particle of any size. It is shown that
the particle size spectra of such system exhibit transition to dynamic scaling
accompanied by the emergence of fractal of
dimension . One of the remarkable feature of this
model is that it is governed by a non-trivial conservation law, namely, the
moment of is time invariant regardless of the choice of the
initial conditions. The reason why it remains conserved is explained by using a
simple dimensional analysis. We show that the scaling exponents and
are locked with the fractal dimension via a generalized scaling relation
.Comment: 8 pages, 6 figures, to appear in Phys. Rev.
"Peeling property" for linearized gravity in null coordinates
A complete description of the linearized gravitational field on a flat
background is given in terms of gauge-independent quasilocal quantities. This
is an extension of the results from gr-qc/9801068. Asymptotic spherical
quasilocal parameterization of the Weyl field and its relation with Einstein
equations is presented. The field equations are equivalent to the wave
equation. A generalization for Schwarzschild background is developed and the
axial part of gravitational field is fully analyzed. In the case of axial
degree of freedom for linearized gravitational field the corresponding
generalization of the d'Alembert operator is a Regge-Wheeler equation. Finally,
the asymptotics at null infinity is investigated and strong peeling property
for axial waves is proved.Comment: 27 page
Hadamard States and Adiabatic Vacua
Reversing a slight detrimental effect of the mailer related to TeXabilityComment: 10pages, LaTeX (RevTeX-preprint style
Quantum mechanics on a circle: Husimi phase space distributions and semiclassical coherent state propagators
We discuss some basic tools for an analysis of one-dimensionalquantum systems
defined on a cyclic coordinate space. The basic features of the generalized
coherent states, the complexifier coherent states are reviewed. These states
are then used to define the corresponding (quasi)densities in phase space. The
properties of these generalized Husimi distributions are discussed, in
particular their zeros.Furthermore, the use of the complexifier coherent states
for a semiclassical analysis is demonstrated by deriving a semiclassical
coherent state propagator in phase space.Comment: 29 page
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