5,289 research outputs found
A Central Limit Theorem for Gibbsian Invariant Measures of 2D Euler Equations
We consider Canonical Gibbsian ensembles of Euler point vortices on the
2-dimensional torus or in a bounded domain of R 2 . We prove that under the
Central Limit scaling of vortices intensities, and provided that the system has
zero global space average in the bounded domain case (neutrality condition),
the ensemble converges to the so-called Energy-Enstrophy Gaussian random
distributions. This can be interpreted as describing Gaussian fluctuations
around the mean field limit of vortices ensembles. The main argument consists
in proving convergence of partition functions of vortices and Gaussian
distributions.Comment: 27 pages, to appear on Communications in Mathematical Physic
Non-negativity preserving numerical algorithms for stochastic differential equations
Construction of splitting-step methods and properties of related
non-negativity and boundary preserving numerical algorithms for solving
stochastic differential equations (SDEs) of Ito-type are discussed. We present
convergence proofs for a newly designed splitting-step algorithm and simulation
studies for numerous numerical examples ranging from stochastic dynamics
occurring in asset pricing theory in mathematical finance (SDEs of CIR and CEV
models) to measure-valued diffusion and superBrownian motion (SPDEs) as met in
biology and physics.Comment: 23 pages, 7 figures. Figures 6.2 and 6.3 in low resolution due to
upload size restrictions. Original resolution at
http://gisc.uc3m.es/~moro/profesional.htm
Vanishing Viscosity Solutions of the Compressible Euler Equations with Spherical Symmetry and Large Initial Data
We are concerned with spherically symmetric solutions of the Euler equations
for multidimensional compressible fluids, which are motivated by many important
physical situations. Various evidences indicate that spherically symmetric
solutions of the compressible Euler equations may blow up near the origin at
certain time under some circumstance. The central feature is the strengthening
of waves as they move radially inward. A longstanding open, fundamental
question is whether concentration could form at the origin. In this paper, we
develop a method of vanishing viscosity and related estimate techniques for
viscosity approximate solutions, and establish the convergence of the
approximate solutions to a global finite-energy entropy solution of the
compressible Euler equations with spherical symmetry and large initial data.
This indicates that concentration does not form in the vanishing viscosity
limit, even though the density may blow up at certain time. To achieve this, we
first construct global smooth solutions of appropriate initial-boundary value
problems for the Euler equations with designed viscosity terms, an approximate
pressure function, and boundary conditions, and then we establish the strong
convergence of the viscosity approximate solutions to a finite-energy entropy
solutions of the Euler equations.Comment: 29 page
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