5,289 research outputs found

    A Central Limit Theorem for Gibbsian Invariant Measures of 2D Euler Equations

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    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

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    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

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    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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