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Feb 2013

Volume 54, Issue 2, Articles (02xxxx)

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J. Math. Phys. 54, 021506 (2013); http://dx.doi.org/10.1063/1.4790887 (24 pages)

Gui-Qiang Chen, Vaibhav Kukreja, and Hairong Yuan
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back to top Partial Differential Equations

Radiative transfer and diffusion limits for wave field correlations in locally shifted random media

Habib Ammari, Emmanuel Bossy, Josselin Garnier, Wenjia Jing, and Laurent Seppecher

J. Math. Phys. 54, 021501 (2013); http://dx.doi.org/10.1063/1.4790409 (18 pages)

Online Publication Date: 5 February 2013

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The aim of this paper is to develop a mathematical framework for opto-elastography. In opto-elastography, a mechanical perturbation of the medium produces a decorrelation of optical speckle patterns due to the displacements of optical scatterers. To model this, we consider two optically random media, with the second medium obtained by shifting the first medium in some local region. We derive the radiative transfer equation for the cross-correlation of the wave fields in the media. Then we derive its diffusion approximation. In both the radiative transfer and the diffusion regimes, we relate the correlation of speckle patterns to the solutions of the radiative transfer and the diffusion equations. We present numerical simulations based on our model which are in agreement with recent experimental measurements.
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05.60.-k Transport processes
02.50.-r Probability theory, stochastic processes, and statistics
02.60.Pn Numerical optimization
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion

The equivalence of the Chern-Simons-Schrödinger equations and its self-dual system

Hyungjin Huh and Jinmyoung Seok

J. Math. Phys. 54, 021502 (2013); http://dx.doi.org/10.1063/1.4790487 (5 pages)

Online Publication Date: 5 February 2013

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In this paper, we discuss the equivalence of the second order Chern-Simons-Schrödinger equations and its first order self-dual system.
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11.15.Yc Chern-Simons gauge theory
03.65.Ge Solutions of wave equations: bound states

Multidimensional Yamada-Watanabe theorem and its applications to particle systems

Piotr Graczyk and Jacek Małecki

J. Math. Phys. 54, 021503 (2013); http://dx.doi.org/10.1063/1.4790507 (15 pages)

Online Publication Date: 15 February 2013

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We prove a multidimensional version of the Yamada-Watanabe theorem, i.e., a theorem giving conditions on coefficients of a stochastic differential equation for existence and pathwise uniqueness of strong solutions. It implies an existence and uniqueness theorem for the eigenvalue and eigenvector processes of matrix-valued stochastic processes, called a “spectral” matrix Yamada-Watanabe theorem. The multidimensional Yamada-Watanabe theorem is also applied to particle systems of squared Bessel processes, corresponding to matrix analogues of squared Bessel processes, Wishart and Jacobi matrix processes. The β-versions of these particle systems are also considered.
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.10.Ud Linear algebra
02.10.Yn Matrix theory
02.30.Gp Special functions
02.30.Hq Ordinary differential equations
02.50.Ey Stochastic processes

New porous medium Poisson-Nernst-Planck equations for strongly oscillating electric potentials

M. Schmuck

J. Math. Phys. 54, 021504 (2013); http://dx.doi.org/10.1063/1.4790656 (21 pages)

Online Publication Date: 20 February 2013

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We consider the Poisson-Nernst-Planck system which is well-accepted for describing dilute electrolytes as well as transport of charged species in homogeneous environments. Here, we study these equations in porous media whose electric permittivities show a strong contrast compared with the electric permittivity of the electrolyte phase. Our main result is the derivation of convenient low-dimensional equations, that is, of effective macroscopic porous media Poisson-Nernst-Planck equations, which reliably describe ionic transport. The contrast in the electric permittivities between liquid and solid phase and the heterogeneity of the porous medium induce strongly oscillating electric potentials (fields). In order to account for this specific physical scenario, we introduce a modified asymptotic multiple-scale expansion which takes advantage of the nonlinearly coupled structure of the ionic transport equations. This allows for a systematic upscaling resulting in a new effective porous medium formulation which shows a new transport term on the macroscale. Solvability of all arising equations is rigorously verified. The emergence of a new transport term indicates promising physical insights into the influence of the microscale material properties on the macroscale. Hence, systematic upscaling strategies provide a source and a prospective tool to capitalize intrinsic scale effects for scientific, engineering, and industrial applications.
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66.30.Dn Theory of diffusion and ionic conduction in solids
77.22.Ch Permittivity (dielectric function)
82.45.Gj Electrolytes

Long-time dynamics for a class of Kirchhoff models with memory

Marcio Antonio Jorge Silva and To Fu Ma

J. Math. Phys. 54, 021505 (2013); http://dx.doi.org/10.1063/1.4792606 (15 pages)

Online Publication Date: 25 February 2013

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This paper is concerned with a class of Kirchhoff models with memory effects utt+αΔ2u−div(|∇u|p−2∇u)− ∫ 0∞μ(s)Δ2u(t−s)ds−Δut+f(u) = h, defined in a bounded domain of mathN. This non-autonomous equation corresponds to a viscoelastic version of Kirchhoff models arising in dynamics of elastoplastic flows and plate vibrations. Under assumptions that the exponent p and the growth of f(u) are up to the critical range, it turns out that the model corresponds to an autonomous dynamical system in a larger phase space, by adding a component which describes the relative displacement history. Then the existence of a global attractor is granted. Furthermore, in the subcritical case, this global attractor has finite Hausdorff and fractal dimensions.
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05.45.-a Nonlinear dynamics and chaos
05.45.Df Fractals

Stability of transonic characteristic discontinuities in two-dimensional steady compressible Euler flows

Gui-Qiang Chen, Vaibhav Kukreja, and Hairong Yuan

J. Math. Phys. 54, 021506 (2013); http://dx.doi.org/10.1063/1.4790887 (24 pages)

Online Publication Date: 27 February 2013

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For a two-dimensional steady supersonic Euler flow past a convex cornered wall with right angle, a characteristic discontinuity (vortex sheet and/or entropy wave) is generated, which separates the supersonic flow from the quiescent gas (hence subsonic). We proved that such a transonic characteristic discontinuity is structurally stable under small perturbations of the upstream supersonic flow in BV. The existence of a weak entropy solution and Lipschitz continuous free boundary (i.e., characteristic discontinuity) is established. To achieve this, the problem is formulated as a free boundary problem for a nonstrictly hyperbolic system of conservation laws; and the free boundary problem is then solved by analyzing nonlinear wave interactions and employing the front tracking method.
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47.40.Ki Supersonic and hypersonic flows
02.30.-f Function theory, analysis
47.32.cd Vortex stability and breakdown
47.35.-i Hydrodynamic waves
47.40.Dc General subsonic flows
47.40.Hg Transonic flows

A dyadic model on a tree

David Barbato, Luigi Amedeo Bianchi, Franco Flandoli, and Francesco Morandin

J. Math. Phys. 54, 021507 (2013); http://dx.doi.org/10.1063/1.4792488 (20 pages)

Online Publication Date: 28 February 2013

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We study an infinite system of nonlinear differential equations coupled in a tree-like structure. This system was previously introduced in the literature and it is the model from which the dyadic shell model of turbulence was derived. It mimics 3D Euler and Navier-Stokes equations in a rough approximation of wavelet decomposition. We prove existence of finite energy solutions, anomalous dissipation in the inviscid unforced case, existence and uniqueness of stationary solutions (either conservative or not) in the forced case.
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47.10.ad Navier-Stokes equations
47.27.-i Turbulent flows
02.10.Ox Combinatorics; graph theory
02.60.Lj Ordinary and partial differential equations; boundary value problems
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