393 research outputs found
Hilbert Expansion from the Boltzmann equation to relativistic Fluids
We study the local-in-time hydrodynamic limit of the relativistic Boltzmann
equation using a Hilbert expansion. More specifically, we prove the existence
of local solutions to the relativistic Boltzmann equation that are nearby the
local relativistic Maxwellian constructed from a class of solutions to the
relativistic Euler equations that includes a large subclass of near-constant,
non-vacuum fluid states. In particular, for small Knudsen number, these
solutions to the relativistic Boltzmann equation have dynamics that are
effectively captured by corresponding solutions to the relativistic Euler
equations.Comment: 50 page
Rapid response to abalone virus depletion in western Victoria: information acquisition and reef code assessment
Future management of disease-affected abalone must adapt to the changing circumstances, and adopting a precautionary approach will allow maximum potential for stock recovery. This approach is mandated by the observation that no documented examples are known of abalone populations recovering from catastrophic impacts such as have occurred in the abalone fisheries of Victoria's Western and Central zones. Indeed the balance of international evidence points towards the contrary, so these fisheries are in dangerous territory. This need not mean that recovery cannot occur.
However, the modelling results from this project confirm the above precautionary view and suggest that unless it is known with certainty that disease-induced mortalities have been moderate (less than 40%), then any resumption of fishing in the near term risks the future of the fishery. Acquisition of accurate mortality data is the only basis upon which fishing can recommence in the short term (within 5 years) and in many instances, such as for some among those reefs considered in our study, the opportunity has passed. The simulation results provide guidance, but their validity is conditional on myriad assumptions as well as on the accuracy of data employed. We already know that catches early in the fishery’s history were higher than reported officially, but how much higher is conjecture. Growth is highly variable over small spatial scales and feedback effects from reduced abundance together with changed size structure and persistence of habitat will play roles in determining the rate, if any, of recovery. The extent of the contemporary illegal catch is uncertain, particularly given the unprecedented closure of the fisheries. The results show that even small illegal catches can significantly degrade recovery where the viral impact is high, with clear implications for the enforcement aspects of managing these fisheries
Exponential decay for the damped wave equation in unbounded domains
We study the decay of the semigroup generated by the damped wave equation in
an unbounded domain. We first prove under the natural geometric control
condition the exponential decay of the semigroup. Then we prove under a weaker
condition the logarithmic decay of the solutions (assuming that the initial
data are smoother). As corollaries, we obtain several extensions of previous
results of stabilisation and control
The analytic structure of 2D Euler flow at short times
Using a very high precision spectral calculation applied to the
incompressible and inviscid flow with initial condition , we find that the width of its analyticity
strip follows a law at short times over eight decades. The
asymptotic equation governing the structure of spatial complex-space
singularities at short times (Frisch, Matsumoto and Bec 2003, J.Stat.Phys. 113,
761--781) is solved by a high-precision expansion method. Strong numerical
evidence is obtained that singularities have infinite vorticity and lie on a
complex manifold which is constructed explicitly as an envelope of analyticity
disks.Comment: 19 pages, 14 figures, published versio
The Hartree limit of Born's ensemble for the ground state of a bosonic atom or ion
The non-relativistic bosonic ground state is studied for quantum N-body
systems with Coulomb interactions, modeling atoms or ions made of N "bosonic
point electrons" bound to an atomic point nucleus of Z "electron" charges,
treated in Born--Oppenheimer approximation. It is shown that the (negative)
ground state energy E(Z,N) yields the monotonically growing function (E(l N,N)
over N cubed). By adapting an argument of Hogreve, it is shown that its limit
as N to infinity for l > l* is governed by Hartree theory, with the rescaled
bosonic ground state wave function factoring into an infinite product of
identical one-body wave functions determined by the Hartree equation. The proof
resembles the construction of the thermodynamic mean-field limit of the
classical ensembles with thermodynamically unstable interactions, except that
here the ensemble is Born's, with the absolute square of the ground state wave
function as ensemble probability density function, with the Fisher information
functional in the variational principle for Born's ensemble playing the role of
the negative of the Gibbs entropy functional in the free-energy variational
principle for the classical petit-canonical configurational ensemble.Comment: Corrected version. Accepted for publication in Journal of
Mathematical Physic
Internal Anisotropy of Collision Cascades
We investigate the internal anisotropy of collision cascades arising from the
branching structure. We show that the global fractal dimension cannot give an
adequate description of the geometrical structure of cascades because it is
insensitive to the internal anisotropy. In order to give a more elaborate
description we introduce an angular correlation function, which takes into
account the direction of the local growth of the branches of the cascades. It
is demonstrated that the angular correlation function gives a quantitative
description of the directionality and the interrelation of branches. The power
law decay of the angular correlation is evidenced and characterized by an
exponent and an angular correlation length different from the radius of
gyration. It is demonstrated that the overlapping of subcascades has a strong
effect on the angular correlation.Comment: RevteX, 8 pages, 6 .eps figures include
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
Resonance-free Region in scattering by a strictly convex obstacle
We prove the existence of a resonance free region in scattering by a strictly
convex obstacle with the Robin boundary condition. More precisely, we show that
the scattering resonances lie below a cubic curve which is the same as in the
case of the Neumann boundary condition. This generalizes earlier results on
cubic poles free regions obtained for the Dirichlet boundary condition.Comment: 29 pages, 2 figure
A Risk/Benefit Approach to the Application of Iron Nanoparticles for the Remediation of Contaminated Sites in the Environment
The von Neumann Hierarchy for Correlation Operators of Quantum Many-Particle Systems
The Cauchy problem for the von Neumann hierarchy of nonlinear equations is
investigated. One describes the evolution of all possible states of quantum
many-particle systems by the correlation operators. A solution of such
nonlinear equations is constructed in the form of an expansion over particle
clusters whose evolution is described by the corresponding order cumulant
(semi-invariant) of evolution operators for the von Neumann equations. For the
initial data from the space of sequences of trace class operators the existence
of a strong and a weak solution of the Cauchy problem is proved. We discuss the
relationships of this solution both with the -particle statistical
operators, which are solutions of the BBGKY hierarchy, and with the
-particle correlation operators of quantum systems.Comment: 26 page
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