10,664 research outputs found
Uniqueness of bounded solutions for the homogeneous Landau equation with a Coulomb potential
We prove the uniqueness of bounded solutions for the spatially homogeneous
Fokker-Planck-Landau equation with a Coulomb potential. Since the local (in
time) existence of such solutions has been proved by Arsen'ev-Peskov (1977), we
deduce a local well-posedness result. The stability with respect to the initial
condition is also checked
Pinsker estimators for local helioseismology
A major goal of helioseismology is the three-dimensional reconstruction of
the three velocity components of convective flows in the solar interior from
sets of wave travel-time measurements. For small amplitude flows, the forward
problem is described in good approximation by a large system of convolution
equations. The input observations are highly noisy random vectors with a known
dense covariance matrix. This leads to a large statistical linear inverse
problem.
Whereas for deterministic linear inverse problems several computationally
efficient minimax optimal regularization methods exist, only one
minimax-optimal linear estimator exists for statistical linear inverse
problems: the Pinsker estimator. However, it is often computationally
inefficient because it requires a singular value decomposition of the forward
operator or it is not applicable because of an unknown noise covariance matrix,
so it is rarely used for real-world problems. These limitations do not apply in
helioseismology. We present a simplified proof of the optimality properties of
the Pinsker estimator and show that it yields significantly better
reconstructions than traditional inversion methods used in helioseismology,
i.e.\ Regularized Least Squares (Tikhonov regularization) and SOLA (approximate
inverse) methods.
Moreover, we discuss the incorporation of the mass conservation constraint in
the Pinsker scheme using staggered grids. With this improvement we can
reconstruct not only horizontal, but also vertical velocity components that are
much smaller in amplitude
Quantitative lower bounds for the full Boltzmann equation, Part I: Periodic boundary conditions
We prove the appearance of an explicit lower bound on the solution to the
full Boltzmann equation in the torus for a broad family of collision kernels
including in particular long-range interaction models, under the assumption of
some uniform bounds on some hydrodynamic quantities. This lower bound is
independent of time and space. When the collision kernel satisfies Grad's
cutoff assumption, the lower bound is a global Maxwellian and its asymptotic
behavior in velocity is optimal, whereas for non-cutoff collision kernels the
lower bound we obtain decreases exponentially but faster than the Maxwellian.
Our results cover solutions constructed in a spatially homogeneous setting, as
well as small-time or close-to-equilibrium solutions to the full Boltzmann
equation in the torus. The constants are explicit and depend on the a priori
bounds on the solution.Comment: 37 page
Fingering convection in a spherical shell
We use 120 three dimensional direct numerical simulations to study fingering
convection in non-rotating spherical shells. We investigate the scaling
behaviour of the flow lengthscale, mean velocity and transport of chemical
composition over the fingering convection instability domain defined by , being the ratio of density perturbations of thermal
and compositional origins. We show that the horizontal size of the fingers is
accurately described by a scaling law of the form , where is the shell depth,
the thermal Rayleigh number and the flux ratio. Scaling laws
for mean velocity and chemical transport are derived in two asymptotic regimes
close to the two edges of the instability domain, namely
and . For the former, we show that the transport follows
power laws of a small parameter measuring the distance to
onset. For the latter, we find that the Sherwood number , which quantities
the chemical transport, gradually approaches a scaling
when ; and that the P\'eclet number accordingly follows , being the chemical Rayleigh number. When
the Reynolds number exceeds a few tens, a secondary instability may occur
taking the form of large-scale toroidal jets. Jets distort the fingers
resulting in Reynolds stress correlations, which in turn feed the jet growth
until saturation. This nonlinear phenomenon can yield relaxation oscillation
cycles.Comment: 43 pages, 22 figures, 3 tables, submitted to JF
A non-local inequality and global existence
In this article we prove a collection of new non-linear and non-local
integral inequalities. As an example for and we
obtain \int_{\threed} dx ~ u^{p+1}(x) \le (\frac{p+1}{p})^2 \int_{\threed}
dx ~ \{(-\triangle)^{-1} u(x) \} \nsm \nabla u^{\frac{p}{2}}(x)\nsm^2. We
use these inequalities to deduce global existence of solutions to a non-local
heat equation with a quadratic non-linearity for large radial monotonic
positive initial conditions. Specifically, we improve \cite{ksLM} to include
all .Comment: 6 pages, to appear in Advances in Mathematic
Prefix-Projection Global Constraint for Sequential Pattern Mining
Sequential pattern mining under constraints is a challenging data mining
task. Many efficient ad hoc methods have been developed for mining sequential
patterns, but they are all suffering from a lack of genericity. Recent works
have investigated Constraint Programming (CP) methods, but they are not still
effective because of their encoding. In this paper, we propose a global
constraint based on the projected databases principle which remedies to this
drawback. Experiments show that our approach clearly outperforms CP approaches
and competes well with ad hoc methods on large datasets
Bi-defects of Nematic Surfactant Bilayers
We consider the effects of the coupling between the orientational order of
the two monolayers in flat nematic bilayers. We show that the presence of a
topological defect on one bilayer generates a nontrivial orientational texture
on both monolayers. Therefore, one cannot consider isolated defects on one
monolayer, but rather associated pairs of defects on either monolayer, which we
call bi-defects. Bi-defects generally produce walls, such that the textures of
the two monolayers are identical outside the walls, and different in their
interior. We suggest some experimental conditions in which these structures
could be observed.Comment: RevTeX, 4 pages, 3 figure
Modeling planar degenerate wetting and anchoring in nematic liquid crystals
We propose a simple surface potential favoring the planar degenerate
anchoring of nematic liquid crystals, i.e., the tendency of the molecules to
align parallel to one another along any direction parallel to the surface. We
show that, at lowest order in the tensorial Landau-de Gennes order-parameter,
fourth-order terms must be included. We analyze the anchoring and wetting
properties of this surface potential. In the nematic phase, we find the desired
degenerate planar anchoring, with positive scalar order-parameter and some
surface biaxiality. In the isotropic phase, we find, in agreement with
experiments, that the wetting layer may exhibit a uniaxial ordering with
negative scalar order-parameter. For large enough anchoring strength, this
negative ordering transits towards the planar degenerate state
The Generation of Magnetic Fields Through Driven Turbulence
We have tested the ability of driven turbulence to generate magnetic field
structure from a weak uniform field using three dimensional numerical
simulations of incompressible turbulence. We used a pseudo-spectral code with a
numerical resolution of up to collocation points. We find that the
magnetic fields are amplified through field line stretching at a rate
proportional to the difference between the velocity and the magnetic field
strength times a constant. Equipartition between the kinetic and magnetic
energy densities occurs at a scale somewhat smaller than the kinetic energy
peak. Above the equipartition scale the velocity structure is, as expected,
nearly isotropic. The magnetic field structure at these scales is uncertain,
but the field correlation function is very weak. At the equipartition scale the
magnetic fields show only a moderate degree of anisotropy, so that the typical
radius of curvature of field lines is comparable to the typical perpendicular
scale for field reversal. In other words, there are few field reversals within
eddies at the equipartition scale, and no fine-grained series of reversals at
smaller scales. At scales below the equipartition scale, both velocity and
magnetic structures are anisotropic; the eddies are stretched along the local
magnetic field lines, and the magnetic energy dominates the kinetic energy on
the same scale by a factor which increases at higher wavenumbers. We do not
show a scale-free inertial range, but the power spectra are a function of
resolution and/or the imposed viscosity and resistivity. Our results are
consistent with the emergence of a scale-free inertial range at higher Reynolds
numbers.Comment: 14 pages (8 NEW figures), ApJ, in press (July 20, 2000?
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