Communications in Mathematical Sciences Articles (Project Euclid)

Minmax variational principle for steady balanced solutions of the rotating shallow water equations

Thu, 05 Aug 2010 15:41 EDT

Visweswaran Nageswaran, Bruce Turkington

Source: Commun. Math. Sci., Volume 8, Number 2, 321--339.

Study of noise-induced transitions in the Lorenz system using the minimum action method

Thu, 05 Aug 2010 15:41 EDT

Xiang Zhou, Weinan E

Source: Commun. Math. Sci., Volume 8, Number 2, 341--355.

Abstract:
We investigate noise-induced transitions in non-gradient systems when complex invariant sets emerge. Our example is the Lorenz system in three representative Rayleigh number regimes. It is found that before the homoclinic explosion bifurcation, the only transition state is the saddle point, and the transition is similar to that in gradient systems. However, when the chaotic invariant set emerges, an unstable limit cycle continues from the homoclinic trajectory. This orbit, which is embedded in a local tube-like manifold around the initial stable stationary point as a relative attractor, plays the role of the most probable exit set in the transition process. This example demonstrates how limit cycles, the next simplest invariant set beyond fixed points, can be involved in the transition process in smooth dynamical systems.

Nonlinear inertia-gravity wave-mode interactions in three dimensional rotating stratified flows

Thu, 05 Aug 2010 15:41 EDT

Mark Remmel, Jai Sukhatme, Leslie M. Smith

Source: Commun. Math. Sci., Volume 8, Number 2, 357--376.

The multidimensional maximum entropy moment problem: a review of numerical methods

Thu, 05 Aug 2010 15:41 EDT

Rafail V. Abramov

Source: Commun. Math. Sci., Volume 8, Number 2, 377--392.

Abstract:
Recently the author developed a numerical method for the multidimensional momentconstrained maximum entropy problem, which is practically capable of solving maximum entropy problems in the two-dimensional domain with moment constraints of order up to 8, in the threedimensional domain with moment constraints of order up to 6, and in the four-dimensional domain with moment constraints of order up to 4, corresponding to the total number of moment constraints of 44, 83 and 69, respectively. In this work, the author brings together key algorithms and observations from his previous works as well as other literature in an attempt to present a comprehensive exposition of the current methods and results for the multidimensional maximum entropy moment problem.

Application of the stochastic mode-reduction strategy and a priori prediction of symmetry breaking in stochastic systems with underlying symmetry

Thu, 05 Aug 2010 15:41 EDT

N. Barlas, I. Timofeyev

Source: Commun. Math. Sci., Volume 8, Number 2, 393--408.

Abstract:
We consider application of the stochastic mode-reduction strategy to a particular class of coupled models where a part of self-interactions of the slow variables is given by a rotationally invariant gradient system. The stochastic mode-reduction strategy is utilized to derive stochastic reduced models which yield a simple description of the phenomena resulting from breaking the original rotational symmetry. It is demonstrated that the direction of the symmetry breaking can be predicted a priori without any knowledge of the statistical behavior of the fast modes.

Hybrid deterministic stochastic systems with microscopic look-ahead dynamics

Thu, 05 Aug 2010 15:41 EDT

M. A. Katsoulakis, A. J. Majda, A. Sopasakis

Source: Commun. Math. Sci., Volume 8, Number 2, 409--437.

Abstract:
We study the impact of stochastic mechanisms on a coupled hybrid system consisting of a general advection-diffusion-reaction partial differential equation and a spatially distributed stochastic lattice noise model. The stochastic dynamics include both spin-flip and spin-exchange type interparticle interactions. Furthermore, we consider a new, asymmetric, single exclusion pro- cess, studied elsewhere in the context of traffic flow modeling, with an one-sided interaction potential which imposes advective trends on the stochastic dynamics. This look-ahead stochastic mechanism is responsible for rich nonlinear behavior in solutions. Our approach relies heavily on first deriving approximate differential mesoscopic equations. These approximations become exact either in the long range, Kac interaction partial differential equation case, or, given sufficient time separation con- ditions, between the partial differential equation and the stochastic model giving rise to a stochastic averaging partial differential equation. Although these approximations can in some cases be crude, they can still give a first indication, via linearized stability analysis, of the interesting regimes for the stochastic model. Motivated by this linearized stability analysis we choose particular regimes where interacting nonlinear stochastic waves are responsible for phenomena such as random switching, convective instability, and metastability, all driven by stochasticity. Numerical kinetic Monte Carlo simulations of the coarse grained hybrid system are implemented to assist in producing solutions and understanding their behavior.

Reduced dynamics of stochastically perturbed gradient flows

Thu, 05 Aug 2010 15:41 EDT

Ibrahim Fatkullin, Gregor Kovacic, Eric Vanden-Eijnden

Source: Commun. Math. Sci., Volume 8, Number 2, 439--461.

Abstract:
We consider stochastically perturbed gradient flows in the limit when the amplitude of random fluctuations is small relative to the typical energy scale in the system and the minima of the energy are not isolated but form submanifolds of the phase space. In this case the limiting dynamics may be described in terms of a diffusion process on these manifolds. We derive explicit equations for this limiting dynamics and illustrate them on a few finite-dimensional examples. Finally, we formally extrapolate the reduction technique to several infinite-dimensional examples and derive equations of the stochastic kink motion in Allen-Cahn-type systems.

Diffusion limit of the Vlasov-Poisson-Fokker-Planck system

Thu, 05 Aug 2010 15:41 EDT

Najoua El Ghani, Nader Masmoudi

Source: Commun. Math. Sci., Volume 8, Number 2, 463--479.

Abstract:
We study the diffusion limit of the Vlasov-Poisson-Fokker-Planck System. Here, we generalize the local in time results and the two dimensional results of Poupaud-Soler and of Goudon to the case of several space dimensions. Renormalization techniques, the method of moments and a velocity averaging lemma are used to prove the convergence of free energy solutions (renormalized solutions) to the Vlasov-Poisson-Fokker- Planck system towards a global weak solution of the Drift-Diffusion-Poisson model.

Two coarse-graining studies of stochastic models in molecular biology

Thu, 05 Aug 2010 15:41 EDT

Peter R. Kramer, Juan C. Latorre, Adnan A. Khan

Source: Commun. Math. Sci., Volume 8, Number 2, 481--517.

Abstract:
We examine stochastic coarse-graining strategies for two biomolecular systems. First, we compute the large-scale transport properties of the basic flashing ratchet mathematical model for (Brownian) molecular motors and consider in this light whether the underlying continuous-space, continuous-time Markovian model can be coarse-grained as a discrete-state, continuous-time Markovian random walk model. Through careful computation of associated statistical signatures of Markovianity, we find that such a discrete coarse-graining is an excellent approximation over much but not all of the parameter regime. In particular, for the parameter values associated with the fastest transport by the flashing ratchet, the discretized model displays non-Markovian features such as waiting times between jumps which are not exponentially distributed. We provide a theoretical framework for understanding the conditions under which Markovianity is to be expected in the discretized model and two mechanisms by which the flashing ratchet model coarse-grains to a non-Markovian discretized model. Next we turn to a basic question of how the dynamics of water molecules near the surface of a solute can be represented by a simple drift-diffusion stochastic model. This question is of most interest for the purpose of accelerating molecular dynamics simulations of proteins, but for simplicity, here we examine the simple case where the solute is a C60 buckyball, which has a homogenous, roughly isotropic form. We compare the mathematical drift-diffusion framework with a statistical quantification of water dynamics near a solute discussed in the biophysical literature. A key concern is the choice of time interval on which to sample the molecular dynamics data to generate estimators for the drift and diffusivity. We use a simple mathematical toy model to establish insight and a strategy, but find for the actual molecular dynamics data that the sampling times which produce the most faithful drift coefficient and the sampling times which produce the most faithful diffusion coefficient do not overlap, so that sacrifice of quality in one or the other parameter appears necessary.

Intraseasonal multi-scale moist dynamics of the tropical atmosphere

Thu, 05 Aug 2010 15:41 EDT

Joseph A. Biello, Andrew J. Majda

Source: Commun. Math. Sci., Volume 8, Number 2, 519--540.

Abstract:
We derive a multi-scale model of moist tropical dynamics which is valid on horizontal synoptic scales, zonal planetary scales, and synoptic and intraseasonal time scales. The Intraseasonal Multi-Scale Moist Dynamics (IMMD) framework builds on the IPESD framework of A.J. Majda and R. Klein, J. Atmos. Sci. , 60, 393–408, 2003. It generalizes the latter by allowing for strong zonal winds (the Trade Winds) and the pressure and stratification variations that they generate. The framework consists of three pieces. The first, called TH, are planetary scale climatology modulation equations which govern the Trade Winds and Hadley Circulation. Self-consistency of the asymptotic theory requires that the meridional component of the Hadley Circulation is an order of magnitude weaker than the zonal component. The second piece, S, is a linear system of equations which govern synoptic scale velocity, temperature, and pressure fluctuations forced by synoptic scale heating fluctuations. Unlike the IPESD theory, these fluctuations are advected by part of the planetary scale climatology from TH. Since the meridional component of TH is an order of magnitude weaker than the zonal component, the synoptic scale fluctuations are only advected by the latter. The third, P, govern the planetary scale anomalies which, like IPESD, are driven both by planetary scale mean heating and by upscale fluxes from the synoptic scales. These planetary scale anomalies are advected both by the zonal component of the Trade Winds and by the meridional component of the Hadley Circulation and, furthermore, respond to an in-scale flux from the mean climatology. We also present an asymptotic analysis of the equations of bulk cloud thermodynamics in order to lay out a self-contained path for incorporating synoptic scale cloud models into the IMMD framework. This framework has potentially important implications for the development of models describing the Madden-Julian Oscillation (MJO) since the MJO manifests itself as planetary scale anomalies from a mean climatology which it modulates on intraseasonal time scales.

Dynamics of current-based, Poisson driven, integrate-and-fire neuronal networks

Thu, 05 Aug 2010 15:41 EDT

Katherine A. Newhall, Gregor Kovacic, Peter R. Kramer, Douglas Zhou, Aaditya V. Rangan, David Cai

Source: Commun. Math. Sci., Volume 8, Number 2, 541--600.

Abstract:
Synchronous and asynchronous dynamics in all-to-all coupled networks of identical, excitatory, current-based, integrate-and-fire (I&F) neurons with delta-impulse coupling currents and Poisson spike-train external drive are studied. Repeating synchronous total firing events, during which all the neurons fire simultaneously, are observed using numerical simulations and found to be the attracting state of the network for a large range of parameters. Mechanisms leading to such events are then described in two regimes of external drive: superthreshold and subthreshold. In the former, a probabilistic argument similar to the proof of the Central Limit Theorem yields the oscillation period, while in the latter, this period is analyzed via an exit time calculation utilizing a diffusion approximation of the Kolmogorov forward equation. Asynchronous dynamics are observed computationally in networks with random transmission delays. Neuronal voltage probability density functions (PDFs) and gain curves—graphs depicting the dependence of the network firing rate on the external drive strength—are analyzed using the steady solutions of the self-consistency problem for a Kolmogorov forward equation. All the voltage PDFs are obtained analytically, and asymptotic solutions for the gain curves are obtained in several physiologically relevant limits. The absence of chaotic dynamics is proved for the type of network under investigation by demonstrating convergence in time of its trajectories.

The exact evolution of the scalar variance in pipe and channel flow

Thu, 05 Aug 2010 15:41 EDT

Roberto Camassa, Zhi Lin, Richard M. McLaughlin

Source: Commun. Math. Sci., Volume 8, Number 2, 601--626.

Stochastic homogenization of Hamilon-Jacobi and "viscous"-Hamilton-Jacobi equations with convex nonlinearities -- Revisited

Thu, 05 Aug 2010 15:41 EDT

Pierre-Louis Lions, Panagiotis E. Souganidis

Source: Commun. Math. Sci., Volume 8, Number 2, 627--637.

Abstract:
In this note we revisit the homogenization theory of Hamilton-Jacobi and “viscous”- Hamilton-Jacobi partial differential equations with convex nonlinearities in stationary ergodic envi- ronments. We present a new simple proof for the homogenization in probability. The argument uses some a priori bounds (uniform modulus of continuity) on the solution and the convexity and coer- civity (growth) of the nonlinearity. It does not rely, however, on the control interpretation formula of the solution as was the case with all previously known proofs. We also introduce a new formula for the effective Hamiltonian for Hamilton-Jacobi and “viscous” Hamilton-Jacobi equations.

Coarsening in high order, discrete, ill-posed diffusion equations

Tue, 02 Nov 2010 15:14 EDT

Catherine Kublik

Source: Commun. Math. Sci., Volume 8, Number 4, 797--834.

Abstract:
We study the discrete version of a family of ill-posed, nonlinear diffusion equations of order $2n$. The fourth order $(n=2)$ version of these equations constitutes our main motivation, as it appears prominently in image processing and computer vision literature. It was proposed by You and Kaveh as a model for denoising images while maintaining sharp object boundaries (edges). The second order equation $(n=1)$ corresponds to another famous model from image processing, namely Perona and Malik’s anisotropic diffusion, and was studied in earlier papers. The equations studied in this paper are high order analogues of the Perona-Malik equation, and like the second order model, their continuum versions violate parabolicity and hence lack well-posedness theory. We follow a recent technique from Kohn and Otto, and prove a weak upper bound on the coarsening rate of the discrete in space version of these high order equations in any space dimension, for a large class of diffusivities. Numerical experiments indicate that the bounds are close to being optimal, and are typically observed.

Some decay estimates of solutions for the 3-D compressible isentropic magnetohydrodynamics

Tue, 02 Nov 2010 15:14 EDT

Jianwen Zhang, Junning Zhao

Source: Commun. Math. Sci., Volume 8, Number 4, 835--850.

A finite time result for vanishing viscosity in the plane with nondecaying vorticity

Tue, 02 Nov 2010 15:14 EDT

Elaine Cozzi

Source: Commun. Math. Sci., Volume 8, Number 4, 851--862.

Heterogeneous multiscale finite element method with novel numerical integration schemes

Tue, 02 Nov 2010 15:14 EDT

Rui Du, Pingbing Ming

Source: Commun. Math. Sci., Volume 8, Number 4, 863--885.

Abstract:
In this paper we introduce two novel numerical integration schemes within the framework of the heterogeneous multiscale method (HMM), when the finite element method is used as the macroscopic solver, to resolve the elliptic problem with a multiscale coefficient. For nonself-adjoint elliptic problems, optimal convergence rate is proved for the proposed methods, which naturally yields a new strategy for refining the macro-micro meshes and a criterion for determining the size of the microcell. Numerical results following this strategy show that the new methods significantly reduce the computational cost without loss of accuracy.

Tailored finite point method for steady-state reaction-diffusion equations

Tue, 02 Nov 2010 15:14 EDT

Houde Han, Zhongyi Huang

Source: Commun. Math. Sci., Volume 8, Number 4, 887--899.

On the energy conservation by weak solutions of the relativistic Vlasov-Maxwell system

Tue, 02 Nov 2010 15:14 EDT

Reinel Sospedra-Alfonso

Source: Commun. Math. Sci., Volume 8, Number 4, 901--908.

A level set approach to modeling general service rules in supply chains

Tue, 02 Nov 2010 15:14 EDT

Christian Ringhofer

Source: Commun. Math. Sci., Volume 8, Number 4, 909--930.

Abstract:
The need for service rules, or policies, in supply chains arises if not all the parts processed in the chain are considered identical, but are distinguished by certain attributes. We develop and analyze a methodology to model arbitrary service rules in large supply chains based on a kinetic (traffic flow like) theory and a level set approach. The final result is a system of hyperbolic conservation laws for the densities of parts, grouped by their attributes. The validity of the model is verified against discrete event simulations for several test cases.

A strategy for non-strictly convex transport costs and the example of $║x-y║^p$ in $R^2$

Tue, 02 Nov 2010 15:14 EDT

Guillaume Carlier, Luigi De Pascale, Filippo Santambrogio

Source: Commun. Math. Sci., Volume 8, Number 4, 931--941.

Abstract:
This paper deals with the existence of optimal transport maps for some optimal transport problems with a convex but non-strictly convex cost. We give a decomposition strategy to address this issue. As a consequence of our procedure, we have to treat some transport problems, of independent interest, with a convex constraint on the displacement. To illustrate possible results obtained through this general approach, we prove existence of optimal transport maps in the case where the source measure is absolutely continuous with respect to the Lebesgue measure and the transportation cost is of the form $h║x-y║$, with h strictly convex increasing and $║.║$ an arbitrary norm in $R^2$.

Small amplitude oscillatory shear permeation flow of cholesteric liquid crystal polymers

Tue, 02 Nov 2010 15:14 EDT

Zhenlu Cui

Source: Commun. Math. Sci., Volume 8, Number 4, 943--963.

Boundary layers in incompressible Navier-Stokes equations with Navier boundary conditions for the vanishing viscosity limit

Tue, 02 Nov 2010 15:14 EDT

Xiao-Ping Wang, Ya-Guang Wang, Zhouping Xin

Source: Commun. Math. Sci., Volume 8, Number 4, 965--998.

Abstract:

Strong convergence of principle of averaging for multiscale stochastic dynamical systems

Tue, 02 Nov 2010 15:14 EDT

Di Liu

Source: Commun. Math. Sci., Volume 8, Number 4, 999--1020.

Global existence and finite dimensional global attractor for a 3D double viscous MHD-α model

Tue, 02 Nov 2010 15:14 EDT

Davide Catania, Paolo Secchi

Source: Commun. Math. Sci., Volume 8, Number 4, 1021--1040.

Abstract:
We consider a magnetohydrodynamic-α model with kinematic viscosity and magnetic diffusivity for an incompressible fluid in a three-dimensional periodic box (torus). Similar models are useful to study the turbulent behavior of fluids in presence of a magnetic field because of the current impossibility to handle non-regularized systems neither analytically nor via numerical simulations. We prove the existence of a global solution and a global attractor. Moreover, we provide an upper bound for the Hausdorff and the fractal dimension of the attractor. This bound can be interpreted in terms of degrees of freedom of the system. In some sense, this result provides an intermediate bound between the number of degrees of freedom for the simplified Bardina model and the Navier–Stokes-α equation.

Gaussian beam approach for the boundary value problem of high frequency Helmholtz equation

Tue, 02 Nov 2010 15:14 EDT

Chunxiong Zheng

Source: Commun. Math. Sci., Volume 8, Number 4, 1041--1066.

Abstract:
We propose an asymptotic numerical method called the Gaussian beam approach for the boundary value problem of high frequency Helmholtz equation. The basic idea is to approximate the traveling waves with a summation of Gaussian beams by the least squares algorithm. Gaussian beams are asymptotic solutions of linear wave equations in the high frequency regime. We deduce the ODE systems satisfied by the Gaussian beams up to third order. The key ingredient of the proposed method is the construction of a finite-dimensional beam space which has a good approximating property. If the exact solutions of boundary value problems contain some strongly evanescent wave modes, the Gaussian beam approach might fail. To remedy this problem, we resort to the domain decomposition technique to separate the domain of definition into a boundary layer region and its complementary interior region. The former is handled by a domain-based discretization method, and the latter by the Gaussian beam approach. Schwarz iterations should then be performed based on suitable transmission boundary conditions at the interface of two regions. Numerical tests demonstrate that the proposed method is very promising.

Periodic homogenization of the inviscid G-equation for incompressible flows

Tue, 02 Nov 2010 15:14 EDT

Jack Xin, Yifeng Yu

Source: Commun. Math. Sci., Volume 8, Number 4, 1067--1078.

Abstract:
G-equations are popular front propagation models in combustion literature and describe the front motion law of normal velocity equal to a constant plus the normal projection of fluid velocity. G-equations are Hamilton-Jacobi equations with convex but non-coercive Hamiltonians. We prove homogenization of the inviscid G-equation for space periodic incompressible flows. This extends a two space dimensional result in "Periodic homogenization of G-equations and viscosity effects," Nonlinearity , to appear. We construct approximate correctors to bypass the lack of compactness due to the non-coercive Hamiltonian. The existence of approximate correctors rely on a local reachability property of the controlled flow trajectory as well as incompressibility of the flow. Homogenization then follows from the comparison principle and the perturbed test function method. The effective Hamiltonian is convex and homogeneous of degree one. It is also coercive if we further assume that the flow is mean zero.

On the Chang and Cooper scheme applied to a linear Fokker-Planck equation

Tue, 02 Nov 2010 15:14 EDT

Christophe Buet, Stéphane Dellacherie

Source: Commun. Math. Sci., Volume 8, Number 4, 1079--1090.

Abstract:
We show that for a particular linear Fokker-Planck operator, the explicit Chang and Cooper scheme is positive and entropy satisfying under a CFL criterion when the initial condition is positive. Then, we deduce that the distribution given by the explicit Chang and Cooper scheme converges toward a discrete Maxwellian equilibrium.

Addendum to: A new median formula with applications to PDE based denoising

Tue, 02 Nov 2010 15:14 EDT

Yingying Li, Stanley Osher

Source: Commun. Math. Sci., Volume 8, Number 4, 1091--.

Kerr-Debye relaxation shock profiles for Kerr equations

Tue, 04 Jan 2011 14:45 EST

Denise Aregba-Driollet, Bernard Hanouzet

Source: Commun. Math. Sci., Volume 9, Number 1, 1--31.

Abstract:
The electromagnetic wave propagation in a nonlinear medium can be described by a Kerr model in the case of an instantaneous response of the material, or by a Kerr-Debye model if the material exhibits a finite response time. Both models are quasilinear hyperbolic, and the Kerr-Debye model is a physical relaxation approximation of the Kerr model. In this paper we characterize the shocks in the Kerr model for which there exists a Kerr-Debye profile. First we consider 1D models for which explicit calculations are performed. Then we determine the plane discontinuities of the full vector 3D Kerr system and their admissibility in the sense of Liu and in the sense of Lax. Finally we characterize the large amplitude Kerr shocks giving rise to the existence of Kerr-Debye relaxation profiles.

Multi-scale methods for wave propagation in heterogeneous media

Tue, 04 Jan 2011 14:45 EST

Björn Engquist, Henrik Holst, Olof Runborg

Source: Commun. Math. Sci., Volume 9, Number 1, 33--56.

Abstract:
Multi-scale wave propagation problems are computationally costly to solve by traditional techniques because the smallest scales must be represented over a domain determined by the largest scales of the problem. We have developed and analyzed new numerical methods for multi-scale wave propagation in the framework of heterogeneous multi-scale method. The numerical methods couple simulations on macro- and micro-scales for problems with rapidly oscillating coefficients. We show that the complexity of the new method is significantly lower than that of traditional techniques with a computational cost that is essentially independent of the micro-scale. A convergence proof is given and numerical results are presented for periodic problems in one, two, and three dimensions. The method is also successfully applied to non-periodic problems and for long time integration where dispersive effects occur.

On the uniqueness for sub-critical quasi-geostrophic equations

Tue, 04 Jan 2011 14:45 EST

Lucas C. F. Ferreira

Source: Commun. Math. Sci., Volume 9, Number 1, 57--62.

On the Cauchy problem for the nonlocal derivative nonlinear Schrodinger equation

Tue, 04 Jan 2011 14:45 EST

Roger Peres de Moura, Ademir Pastor

Source: Commun. Math. Sci., Volume 9, Number 1, 63--80.

Wave propagation in shallow-water acoustic random waveguides

Tue, 04 Jan 2011 14:45 EST

Christophe Gomez

Source: Commun. Math. Sci., Volume 9, Number 1, 81--125.

Abstract:
In shallow-water waveguides a propagating field can be decomposed in three kinds of modes: the propagating modes, the radiating modes and the evanescent modes. In this paper we consider the propagation of a wave in a randomly perturbed waveguide and we analyze the coupling between these three kinds of modes using an asymptotic analysis based on a separation of scales technique. Then, we derive the asymptotic form of the distribution of the mode amplitudes and the coupled power equation for propagating modes. From this equation, we show that the total energy carried by the propagating modes decreases exponentially with the size of the random section and we give an expression of the decay rate. Moreover, we show that the mean propagating mode powers converge to the solution of a diffusion equation in the limit of a large number of propagating modes.

Relaxation to equilibrium in diffusive-thermal models with a strongly varying diffusion length-scale

Tue, 04 Jan 2011 14:45 EST

Paul Clavin, Laurent Masse, Jean-Michel Roquejoffre

Source: Commun. Math. Sci., Volume 9, Number 1, 127--141.

Abstract:
We consider reaction-diffusion equations with a strongly varying diffusion lengthscale. We provide a mathematical study of the relaxation towards the steady planar solution, in the context of infinitesimal disturbances whose wavelength is much shorter than the total thickness of the wave. The models under study are relevant in the description of ablation fronts encountered in inertial confinment fusion, when hydrodynamical effects are neglected.

On the Ostwald ripening of thin liquid films

Tue, 04 Jan 2011 14:45 EST

Shibin Dai

Source: Commun. Math. Sci., Volume 9, Number 1, 143--160.

On the uniqueness of entropy solutions to the Riemann problem for 2x2 hyperbolic systems of conservation laws

Tue, 04 Jan 2011 14:45 EST

Hiroki Ohwa

Source: Commun. Math. Sci., Volume 9, Number 1, 161--185.

Abstract:
In this paper we revisit the Riemann problem for 2×2 hyperbolic systems of conservation laws, which satisfy the condition that the product of non-diagonal elements in the Fréchet derivative (Jacobian) of the flux is positive, the genuine nonlinearity condition, and the Smoller- Johnson condition in one space variable. The first condition implies that the system is strictly hyperbolic. By developing the shock curve approach, we give an alternative shock curve approach and re-prove the uniqueness of self-similar solutions satisfying the Lax entropy condition at discontinuities.

High-order entropy-based closures for linear transport in slab geometry

Tue, 04 Jan 2011 14:45 EST

Cory D. Hauck

Source: Commun. Math. Sci., Volume 9, Number 1, 187--205.

Abstract:
We compute high-order entropy-based $(M_N)$ models for a linear transport equation on a one-dimensional slab geometry. We simulate two test problems from the literature: the two- beam instability and the plane-source problem. In the former case, we compute solutions for systems up to order $N = 6$; in the latter, up to $N = 15$. The most notable outcome of these results is the existence of shocks in the steady-state profiles of the two-beam instability for all odd values of $N$.

Asymptotic stability of rarefaction waves in radiative hydrodynamics

Tue, 04 Jan 2011 14:45 EST

Chunjin Lin

Source: Commun. Math. Sci., Volume 9, Number 1, 207--223.

A parametrix construction for the wave equation with low regularity coefficients using a frame of Gaussians

Tue, 04 Jan 2011 14:45 EST

Alden Waters

Source: Commun. Math. Sci., Volume 9, Number 1, 225--254.

From Boltzmann equation to spherical harmonics expansion model: diffusion limit and Poisson coupling

Tue, 04 Jan 2011 14:45 EST

Mohamed Lazhar Tayeb

Source: Commun. Math. Sci., Volume 9, Number 1, 255--275.

Abstract:
The diffusion approximation of an initial-boundary value problem for a Boltzmann- Poisson system is studied. An elastic operator modeling electron-impurity collision is considered. A relative entropy is used to control the terms coming from the boundary and to prove useful $L^2$−estimates for the renormalized solutions of the scaled Boltzmann equation (coupled to Poisson). A careful analysis of a relative entropy for high velocity allows us to show uniform bounds for the total mass and the kinetic energy which gives the compactness of the self-consistent electrostatic potential. Then, the moment method is used to prove the convergence of the renormalized solutions to a weak solution of a Spherical Harmonics Expansion (or SHE-) model coupled to the Poisson equation.

An improvement of the TYT algorithm for GF(2m) based on reusing intermediate computation results

Tue, 04 Jan 2011 14:45 EST

Yin Li, Gong-liang Chen, Yi-yang Chen, Jian-hua Li

Source: Commun. Math. Sci., Volume 9, Number 1, 277--287.

Abstract:
Multiplicative inversion plays an important role to Elliptic Curve Cryptosystems. This paper presents an efficient inversion algorithm in $GF(2^m)$ using a normal basis which improves the Itoh-Tsujii (IT) algorithm and the Takagi et al. (TYT) algorithm . The proposed algorithm reduces the number of required multiplications by decomposing $m−1$ into several factors plus a remainder and by reusing intermediate computation values. It is proved that the decomposition of $m−1$ can be made simpler, but requires even fewer multiplications. Furthermore, a practical algorithm for finding an optimal decomposition of $m−1$ is investigated.

A singular 1-D Hamilton-Jacobi equation, with application to large deviation of diffusions

Tue, 04 Jan 2011 14:45 EST

Xiaoxue Deng, Jin Feng, Yong Liu

Source: Commun. Math. Sci., Volume 9, Number 1, 289--300.

Abstract:
The comparison principle (uniqueness) for the Hamilton-Jacobi equation is usually established through arguments involving a distance function. In this article we illustrate the subtle nature of choosing such a distance function, using a special example of one dimensional Hamiltonian with coefficient singularly (non-Lipschitz) depending upon the state variable. The standard method of using Euclidean distance as a test function fails in such situation. Once the comparison is established, we apply it to obtain a new result on small noise Freidlin-Wentzell type probabilistic large deviation theorem for certain singular diffusion processes. This article serves to explain basic ideas behind an abstract approach to comparison developed in J. Feng and T.G. Kurtz, American Mathematical Society, Providence, Rhode Island. Mathematical Surveys and Monographs , 131, 2006, J. Feng and M. Katsoulakis, Arch. Ration. Mech. Anal. , 192(2), 275-310, 2009 in a simple manner, removing all technicalities due to infinite dimensionality.

A weak trapezoidal method for a class of stochastic differential equations

Tue, 04 Jan 2011 14:45 EST

David F. Anderson, Jonathan C. Mattingly

Source: Commun. Math. Sci., Volume 9, Number 1, 301--318.

Abstract:
We present a numerical method for the approximation of solutions for the class of stochastic differential equations driven by Brownian motions which induce stochastic variation in fixed directions. This class of equations arises naturally in the study of population processes and chemical reaction kinetics. We show that the method constructs paths that are second order accurate in the weak sense. The method is simpler than many second order methods in that it neither requires the construction of iterated Itô integrals nor the evaluation of any derivatives. The method consists of two steps. In the first an explicit Euler step is used to take a fractional step. The resulting fractional point is then combined with the initial point to obtain a higher order, trapezoidal like, approximation. The higher order of accuracy stems from the fact that both the drift and the quadratic variation of the underlying SDE are approximated to second order.

Crank-Nicolson finite element methods using symmetric stabilization with an application to optimal control problems subject to transient advection-diffusion equations

Tue, 04 Jan 2011 14:45 EST

Erik Burman

Source: Commun. Math. Sci., Volume 9, Number 1, 319--329.

Abstract:
We consider a finite element method with symmetric stabilization for transient advection-diffusion-reaction problems. The Crank-Nicolson finite difference scheme is used for discretization in time. We prove stability of the numerical method both for implicit and explicit treatment of the stabilization operator. The resulting convergence results are given and the results are illustrated by a numerical experiment. We then consider a model problem for pde-constrained optimization. Using discrete adjoint consistency of our stabilized method we show that both the implicit and semi-implicit methods proposed yield optimal convergence for the control and the state variable.

Unique minimizer for a random functional with double-well potential in dimension 1 and 2

Tue, 10 May 2011 09:34 EDT

Nicolas Dirr, Enza Orlandi

Source: Commun. Math. Sci., Volume 9, Number 2, 331--351.

Abstract:
We add a random bulk term, modelling the interaction with the impurities of the medium, to a standard functional in the gradient theory of phase transitions consisting of a gradient term with a double well potential. We show that in $d≤2$ exists, for almost all the realizations of the random bulk term, a unique random macroscopic minimizer. This result is in sharp contrast to the case when the random bulk term is absent. In the latter case there are two minimizers which are (in law) invariant under translations in space.

Entropies for radially symmetric higher-order nonlinear diffusion equations

Tue, 10 May 2011 09:34 EDT

Mario Bukal, Ansgar Jungel, Daniel Matthes

Source: Commun. Math. Sci., Volume 9, Number 2, 353--382.

Abstract:
A previously developed algebraic approach to proving entropy production inequalities is extended to deal with radially symmetric solutions for a class of higher-order diffusion equations in multiple space dimensions. In application of the method, novel a priori estimates are derived for the thin-film equation, the fourth-order Derrida-Lebowitz-Speer-Spohn equation, and a sixth-order quantum diffusion equation.

Corrector theory for elliptic equations in random media with singular Green's function. Application to random boundaries

Tue, 10 May 2011 09:34 EDT

Guillaume Bal, Wenjia Jing

Source: Commun. Math. Sci., Volume 9, Number 2, 383--411.

Abstract:
We consider the problem of the random fluctuations in the solutions to elliptic PDEs with highly oscillatory random coefficients. In our setting, as the correlation length of the fluctuations tends to zero, the heterogeneous solution converges to a deterministic solution obtained by averaging. When the Green’s function to the unperturbed operator is sufficiently singular (i.e., not square integrable locally), the leading corrector to the averaged solution may be either deterministic or random, or both in a sense we shall explain. Our main application is the solution of an elliptic problem with random Robin boundary condition that may be used to model diffusion of signaling molecules through a layer of cells into a bulk of extracellular medium. The problem is then described by an elliptic pseudo-differential operator (a Dirichlet-to-Neumann operator) on the boundary of the domain with random potential. In the physical setting of a three dimensional extracellular medium on top of a two-dimensional surface of cells forming a layer of epithelium, we show that the approximate corrector to averaging consists of a deterministic correction plus a Gaussian field of amplitude proportional to the correlation length of the random medium. The result is obtained under some assumptions on the four-point correlation function in the medium. We provide examples of such random media based on Gaussian and Poisson statistics.

Unconditionally stable schemes for higher order inpainting

Tue, 10 May 2011 09:34 EDT

Carola-Bibiane Schönlieb, Andrea Bertozzi

Source: Commun. Math. Sci., Volume 9, Number 2, 413--457.

A mathematical model for the hard sphere repulsion in ionic solutions

Tue, 10 May 2011 09:34 EDT

YunKyong Hyon, Bob Eisenberg, Chun Liu

Source: Commun. Math. Sci., Volume 9, Number 2, 459--475.

Abstract:
We introduce a mathematical model for the finite size (repulsive) effects in ionic solutions. We first introduce an appropriate energy term into the total energy that represents the hard sphere repulsion of ions. The total energy then consists of the entropic energy, electrostatic potential energy, and the repulsive potential energy. The energetic variational approach derives a boundary value problem that includes contributions from the repulsive term with a no flux boundary condition for charge density which is a consequence of the variational approach, and physically implies charge conservation. The resulting system of partial differential equations is a modification of the Poisson-Nernst-Planck (PNP) equations widely if not universally used to describe the drift-diffusion of electrons and holes in semiconductors, and the movement of ions in solutions and protein channels. The modified PNP equations include the effects of the finite size of ions that are so important in the concentrated solutions near electrodes, active sites of enzymes, and selectivity filters of proteins. Finally, we do some numerical experiments using finite element methods, and present their results as a verification of the utility of the modified system.

Pricing and hedging contingent claims with regime switching risk

Tue, 10 May 2011 09:34 EDT

Robert J. Elliott, Tak Kuen Siu

Source: Commun. Math. Sci., Volume 9, Number 2, 477--498.

Abstract:
We study the pricing and hedging of contingent claims in a Markov regime-switching market with a money market account, a zero-coupon bond, and an ordinary share. General contingent claims with payoffs depending on both the share price and the state of a Markov chain describing regime switching are considered. A general pricing kernel defined by the product of two density processes is used to explicitly take into account regime switching risk. Under some differentiability and boundedness conditions, a martingale representation result is established and the integrands in the representation are explicitly identified with respect to the general pricing kernel. We then determine a pricing kernel and a hedging strategy by minimizing the residual risk due to incomplete hedging. Our analysis is also extended to Asian-style and American-style general contingent claims.

Exponentially-stable steady flow and asymptotic behavior for the magnetohydrodynamic equations

Tue, 10 May 2011 09:34 EDT

Lucas C. F. Ferreira, Elder J. Villamizar-Roa

Source: Commun. Math. Sci., Volume 9, Number 2, 499--516.

Gaussian processes associated to infinite bead-spring networks

Tue, 10 May 2011 09:34 EDT

Michael Taylor

Source: Commun. Math. Sci., Volume 9, Number 2, 517--534.

Approximate solutions to several visibility optimization problems

Tue, 10 May 2011 09:34 EDT

Rostislav Goroshin, Quyen Huynh, Hao-Min Zhou

Source: Commun. Math. Sci., Volume 9, Number 2, 535--550.

Abstract:
The visibility level set function introduced by Tsai et al. allows for gradient based and variational formulations of many classical visibility optimization problems. In this work we propose solutions to two such problems. The first asks where to position $n$-observers such that the area visible to these observers is maximized. The second problem is to determine the shortest route an observer should take through a map such that every point in the map is visible from at least one vantage point on the route. These problems are similar to the "art gallery" and "watchman route" problems, respectively. We propose a greedy iterative algorithm, formulated in the level set framework as the solution to the art gallery problem. We also propose a variational solution to the watchman route problem which achieves complete visibility coverage of the domain while attaining a local minimum of path length.

Frame based segmentation for medical images

Tue, 10 May 2011 09:34 EDT

Bin Dong, Aichi Chien, Zuowei Shen

Source: Commun. Math. Sci., Volume 9, Number 2, 551--559.

Abstract:
Medical image segmentation is an important but difficult problem that attracts tremendous attention from researchers in various fields. In this paper, we propose a frame based model, as well as a fast implementation, for general medical image segmentation problems. Our model combines ideas of the frame based image restoration model of J. Cai, S. Osher, and Z. Shen, Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, 8(2), 337–369, 2009 with ideas of the total variation based segmentation model of T. Chan and L. Vese, Scale-Space Theories in Computer Vision, 141–151, 1999, T. Chan and L. Vese, IEEE Transactions on image processing, 10(2), 266–277, 2001, T. Chan, S. Esedoglu and M. Nikolova, ALGORITHMS, 66(5), 1632–1648, and X. Bresson, S. Esedoglu, P. Vandergheynst, J. Thiran and S. Osher, Journal of Mathematical Imaging and Vision, 28(2), 151–167, 2007. Numerical experiments show that the proposed frame based model outperforms the total variation based model in terms of capturing key features of biological structures. Successful segmentations of blood vessels and aneurysms in 3D CT angiography images are also presented.

Homogenization of the G-equation with incompressible random drift in two dimensions

Tue, 10 May 2011 09:34 EDT

James Nolen, Alexei Novikov

Source: Commun. Math. Sci., Volume 9, Number 2, 561--582.

Abstract:
We study the homogenization limit of solutions to the G-equation with random drift. This Hamilton-Jacobi equation is a model for flame propagation in a turbulent fluid in the regime of thin flames. For a fluid velocity field that is statistically stationary and ergodic, we prove sufficient conditions for homogenization to hold with probability one. These conditions are expressed in terms of travel times for the associated control problem. When the spatial dimension is equal to two and the fluid velocity is divergence-free, we verify that these conditions hold under suitable assumptions about the growth of the random stream function.

Critical thresholds in multi-dimensional Restricted Euler equations

Tue, 10 May 2011 09:34 EDT

Dongming Wei

Source: Commun. Math. Sci., Volume 9, Number 2, 583--596.

Abstract:
Using the spectral dynamics, we study the critical threshold phenomena in the multidimensional restricted Euler (RE) equations. We identify sub-critical and sup-critical initial data for all space dimensions, which extends the previous result for the 3D and 4D restricted Euler equations. Our result suggests that: if the number of dimensions is odd, the finite time blowup is generic; in contrast, if the number of dimensions is even, there is a rich set of initial data which yields global smooth solutions.

Derivation of continuum models for the moving contact line problem based on thermodynamic principles

Tue, 10 May 2011 09:34 EDT

Weiqing Ren, Weinan E

Source: Commun. Math. Sci., Volume 9, Number 2, 597--606.

Abstract:
Contact lines arise as the boundaries of free boundaries in fluids. This problem is interesting and important, not only because it arises in many applications, but also because of the distinct mathematical and physical features it has, such as singularities, hysteresis, instabilities, competing scaling regimes, etc. For a long time, this area of study was plagued with conflicting theories and uncertainties regarding how the problem should be modeled. In the present paper we illustrate how continuum models for the moving contact line problem can be derived using simple thermodynamic considerations. Both the sharp interface models and diffuse interface models are derived.

A sharp bound on the L2 norm of the solution of a random elliptic difference equation

Tue, 10 May 2011 09:34 EDT

Tomasz Komorowski, Lenya Ryzhik

Source: Commun. Math. Sci., Volume 9, Number 2, 607--622.

Kramers' formula for chemical reactions in the context of Wasserstein gradient flows

Tue, 10 May 2011 09:34 EDT

Michael Herrmann, Barbara Niethammer

Source: Commun. Math. Sci., Volume 9, Number 2, 623--635.

Abstract:
We derive Kramers’ formula as singular limit of the Fokker-Planck equation with double-well potential. The convergence proof is based on the Rayleigh principle of the underlying Wasserstein gradient structure and complements a recent result by Peletier, Savaré and Veneroni.