Asymptotic behavior of solutions of the Cauchy problem for Burgers type equations

We show that the solutions of the initial value problems for a large class of Burgers type equations approach with time to the sum of appropriately shifted wave-trains and of diffusion waves.

Résumé

Nous montrons que les solutions du problème de Cauchy pour une grande classe d'équations de type de Burgers sont approchées en temps grand vers des sommes d'ondes de diffusion et d'ondes progressives adéquatement translatées.     

Letter: Electromagnetic Mass-Models in General Relativity Reexamined

The problem of constructing a model of an extended charged particle within the context of general relativity has a long and distinguished history. The distinctive feature of these models is that, in some way or another, they require the presence of negative mass in order to maintain stability against Coulomb's repulsion. Typically, the particle contains a core of negative mass surrounded by a positive-mass outer layer, which emerges from the Reissner-Nordström field. In this work we show how the Einstein-Maxwell field equations can be used to construct an extended model where the mass is positive everywhere. This requires the principal pressures to be unequal inside the particle. The model is obtained by setting the effective matter density, rather than the rest matter density, equal to zero. The Schwarzschild mass of the particle arises from the electrical and gravitational field (Weyl tensor) energy. The model satisfies the energy conditions of Hawking and Ellis. A particular solution that illustrates the results is presented.     

变系数Burgers方程的分类及相似解(英文)

应用经典李群理论考虑了描述非平面冲击波形成和衰减现象的(1 1)维变系数Burgers方程,得到该方程的群分类及相应的对称.对于某些具体形式的色散项系数a(t)和非线性项系数b(t),给出了对应方程的对称约化及相似解.本文在已有基础上给出了方程新的显式解.这些解对于研究某些复杂的物理现象,以及验证数值求解法则的可行性有重要的意义.     

A survey of numerical methods for nonlinear filtering problems

The main goal of filtering is to obtain, recursively in time, good estimates of the state of a stochastic dynamical system based on noisy partial observations of the same. In settings where the signal/observation dynamics are significantly nonlinear or the noise intensities are high, an extended Kalman filter (EKF), which is essentially a first order approximation to an infinite dimensional problem, can perform quite poorly: it may require very frequent re-initializations and in some situations may even diverge. The theory of nonlinear filtering addresses these difficulties by considering the evolution of the conditional distribution of the state of the system given all the available observations, in the space of probability measures. We survey a variety of numerical schemes that have been developed in the literature for approximating the conditional distribution described by such stochastic evolution equations; with a special emphasis on an important family of schemes known as the particle filters. A numerical study is presented to illustrate that in settings where the signal/observation dynamics are non linear a suitably chosen nonlinear scheme can drastically outperform the extended Kalman filter.     

Statistics of “Worms” in Isotropic Turbulence Treated on the Multifractal Basis

As is well known, there appear innumerable, randomly posed, unsteady, vorticity-concentrated slender regions, which may be called worms, moving around in isotropic turbulence with a high Reynolds number. The universal feature of instantaneous spatial distribution of these worms is formulated based on a multifractal structure of the energy dissipation rate previously investigated and on the assumption that a worm can be locally replaced by a Burgers vortex.     

Application of the Augmented Lagrangian-SQP Method to Optimal Control Problems for the Stationary Burgers Equation

In this paper optimal control problems for the stationary Burgers equation are analyzed. To solve the optimal control problems the augmented Lagrangian-SQP method is applied. This algorithm has second-order convergence rate depending upon a second-order sufficient optimality condition. Using piecewise linear finite elements it is proved that the discretized augmented Lagrangian-SQP method is well-defined and has second-order rate of convergence. This result is based on the proof of a uniform discrete Babuka-Brezzi condition and a uniform second-order sufficient optimality condition.     

Stabilization of aqueous nanoscale zerovalent iron dispersions by anionic polyelectrolytes: adsorbed anionic polyelectrolyte layer properties and their effect on aggregation and sedimentation

Nanoscale zerovalent iron (NZVI) particles are 5–40 nm sized Fe⁰/Fe-oxide particles that rapidly transform many environmental contaminants to benign products and are a promising in situ remediation agent. Rapid aggregation and limited mobility in water-saturated porous media limits the ability to deliver NZVI dispersions in the subsurface. This study prepares stable NZVI dispersions through physisorption of commercially available anionic polyelectrolytes, characterizes the adsorbed polymer layer, and correlates the polymer coating properties with the ability to prevent rapid aggregation and sedimentation of NZVI dispersions. Poly(styrene sulfonate) with molecular weights of 70 k and 1,000 k g/mol (PSS70K and PSS1M), carboxymethyl cellulose with molecular weights of 90 k and 700 k g/mol (CMC90K and CMC700K), and polyaspartate with molecular weights of 2.5 k and 10 k g/mol (PAP2.5K and 10K) were compared. Particle size distributions were determined by dynamic light scattering during aggregation. The order of effectiveness to prevent rapid aggregation and stabilize the dispersions was PSS70K(83%) > ≈PAP10K(82%) > PAP2.5K(72%) > CMC700K(52%), where stability is defined operationally as the volume percent of particles that do not aggregate after 1 h. CMC90K and PSS1M could not stabilize RNIP relative to bare RNIP. A similar trend was observed for their ability to prevent sedimentation, with 40, 34, 32, 20, and 5 wt%, of the PSS70K, PAP10K, PAP2.5K, CMC700K, and CMC90K modified NZVI remaining suspended after 7 h of quiescent settling, respectively. The stable fractions with respect to both aggregation and sedimentation correlate well with the adsorbed polyelectrolyte mass and thickness of the adsorbed polyelectrolyte layers as determined by Oshima’s soft particle theory. A fraction of the particles cannot be stabilized by any modifier and rapidly agglomerates to micron sized aggregates, as is also observed for unmodified NZVI. This non-dispersible fraction is attributed to strong magnetic attractions among the larger particles present in the polydisperse NZVI slurry, as the magnetic attractive forces increase as r⁶.     

New periodic wave solutions,localized excitations and their interaction for （2＋l）-dimensional Burgers equation

Based on the B/icklund method and the multilinear variable separation approach （MLVSA）, this paper finds a general solution including two arbitrary functions for the （2＋1）-dimensional Burgers equations. Then a class of new doubly periodic wave solutions for （2＋l）-dimensional Burgers equations is obtained by introducing appropriate Jacobi elliptic functions, Weierstrass elliptic functions and their combination in the general solutions （which contains two arbitrary functions）. Two types of limit cases are considered. Firstly, taking one of the moduli to be unity and the other zero, it obtains particular wave （called semi-localized） patterns, which is periodic in one direction, but localized in the other direction. Secondly, if both moduli are tending to 1 as a limit, it derives some novel localized excitations （two-dromion solution）.     

On a Shock Front in Burgers Turbulence

We investigate how chaos propagates in the solution of Burgers equation _t u+u _x u=0 with initial condition u(,0) distributed as a white noise on and 0 on . We describe the evolution of the shock front that travels to the left. Asymptotics are given for both large and small time t.