Validity of nonlinear geometric optics for entropy solutions of multidimensional scalar conservation laws

Nonlinear geometric optics with various frequencies for entropy solutions only in L^∞ of multidimensional scalar conservation laws is analyzed. A new approach to validate nonlinear geometric optics is developed via entropy dissipation through scaling, compactness, homogenization, and L¹-stability. New multidimensional features are recognized, especially including nonlinear propagations of oscillations with high frequencies. The validity of nonlinear geometric optics for entropy solutions in L^∞ of multidimensional scalar conservation laws is justified.     

Damped wave equation with a critical nonlinearity

We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity

where 0,$"> and space dimensions . Assume that the initial data

where \frac{n}{2},$"> weighted Sobolev spaces are

Also we suppose that

0,\int u_{0}\left( x\right) dx>0, \end{displaymath}">

where

Then we prove that there exists a positive such that the Cauchy problem above has a unique global solution satisfying the time decay property

for all 0,$"> where

     

A comparison of computational strategies for geometric programs

Numerous algorithms for the solution of geometric programs have been reported in the literature. Nearly all are based on the use of conventional programming techniques specialized to exploit the characteristic structure of either the primal or the dual or a transformed primal problem. This paper attempts to elucidate, via computational comparisons, whether a primal, a dual, or a transformed primal solution approach is to be preferred.The authors wish to thank Captain P. A. Beck and Dr. R. S. Dembo for making available their codes. This research was supported in part under ONR Contract No. N00014-76-C-0551 with Purdue University.     

Large time behavior of small solutions to subcritical derivative nonlinear Schrödinger equations

We consider the derivative nonlinear Schrödinger equations

where the coefficient satisfies the time growth condition

is a sufficiently small constant and the nonlinear interaction term consists of cubic nonlinearities of derivative type

where and . We suppose that the initial data satifsfy the exponential decay conditions. Then we prove the sharp decay estimate , for all , where . Furthermore we show that for there exist the usual scattering states, when and the modified scattering states, when

     

Asymptotics for the third‐order nonlinear Schrödinger equation in the critical case

We consider the Cauchy problem for the third‐order nonlinear Schrödinger equation where and is the Fourier transform. Our purpose in this paper is to prove the large time asymptoitic behavior of solutions for the defocusing case λ > 0 with a logarithmic correction under the non zero mass condition Copyright © 2016 John Wiley & Sons, Ltd.     

Impacts of Gaussian noises on the blow-up times of nonlinear stochastic partial differential equations

In this paper, a blow-up problem to nonlinear stochastic partial differential equations driven by Brownian motions is investigated. In particular, the impact of noises on the life span of solutions is studied. It is interesting to know that the noise can be used to postpone the blow-up time of a stochastic nonlinear system.     

关于一个非线性几何光学中的非严格双曲守恒律系统的研究

研究了一个产生于非线性几何光学中的非严格双曲守恒律系统.该系统具有强非线性流函数项,且狄拉克激波可能同时出现在解的两个状态变量中.通过未知函数的一个变换,该系统的非线性流函数项得到弱化,从而其黎曼问题被完全解决.     

Asymptotics for the heat equation in the exterior of a shrinking compact set in the plane via Brownian hitting times

Let and let be a continuous, nonincreasing function on satisfying . Consider the heat equation in the exterior of a time-dependent shrinking disk in the plane:

0.\end{split}\end{displaymath}">

If there exist constants and a constant 0$"> such that , for sufficiently large , then . The same result is also shown to hold when is replaced by , where . Also, a discrepancy is noted between the asymptotics for the above forward heat equation and the corresponding backward one. The method used is probabilistic.

     

The mathieu-hill operator equation with dissipation and estimates of its instability index

The behavior ast→∞ of solutions of the equation

Resonant Leading Term Geometric Optics Expansions with Boundary Layers for Quasilinear Hyperbolic Boundary Problems

We construct and justify leading order weakly nonlinear geometric optics expansions for nonlinear hyperbolic initial value problems, including the compressible Euler equations. The technique of simultaneous Picard iteration is employed to show approximate solutions tend to the exact solutions in the small wavelength limit. Recent work [2 Coulombel, J.-F., Gues, O., and Williams, M., 2011. Resonant leading order geometric optics expansions for quasilinear hyperbolic fixed and free boundary problems, Comm. Part. Diff. Eqs. 36 (2011), pp. 1797–1859.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] by Coulombel et al. studied the case of reflecting wave trains whose expansions involve only real phases. We treat generic boundary frequencies by incorporating into our expansions both real and nonreal phases. Nonreal phases introduce difficulties such as approximately solving complex transport equations and result in the addition of boundary layers with exponential decay. This also prevents us from doing an error analysis based on almost periodic profiles as in [2 Coulombel, J.-F., Gues, O., and Williams, M., 2011. Resonant leading order geometric optics expansions for quasilinear hyperbolic fixed and free boundary problems, Comm. Part. Diff. Eqs. 36 (2011), pp. 1797–1859.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]].     

半线性波动方程的振荡波

对于一类二维和三维的半线性波动方程,当其振荡初值具有一个非退化性相时,给出了双位相振荡解的任意阶的有效几何光学展开,同时给出了层面的清晰结构。     

随机环境中的一类非线性时间序列的伴随几何遍历性分析

本文将AR(m)-ARCH(m)模型推广为REAR(m)-ARCH(m)模型,并给出了REAR(m)-ARCH(m)具有伴随几何遍历性的一个充分条件。     

非线性Black-Scholes模型下几何平均亚式期权定价

在非线性Black-Scholes模型下,本文研究了几何平均亚式期权定价问题.首先利用单参数摄动方法,将亚式期权适合的偏微分方程分解成一系列常系数抛物方程.其次通过计算这些常系数抛物型方程的解,给出了几何平均亚式期权的近似定价公式.最后利用Green函数分析了近似结论的误差估计.     

The nonlinear Schrödinger equation with white noise dispersion

Under certain scaling the nonlinear Schrödinger equation with random dispersion converges to the nonlinear Schrödinger equation with white noise dispersion. The aim of this work is to prove that this latter equation is globally well posed in L² or H¹. The main ingredient is the generalization of the classical Strichartz estimates. Additionally, we justify rigorously the formal limit described above.     

Explicit through-thickness integration schemes for geometric nonlinear analysis of laminated composite shells

Degenerated shell elements were found to be attractive in solving homogeneous shell problems. Direct extension of the same to layered shells becomes computationally inefficient as, in the computation of element matrices, 3-D numerical integration in each layer and summation over the layers have to be carried out. In order to make the formulation efficient, explicit through-thickness schemes have been devised for linear problems. The present paper deals with the extension of the same to geometric nonlinear problems with options of small and large rotations. The explicit through-thickness integration becomes possible due to the assumption on the variation of inverse Jacobian through the thickness. Depending on the assumptions, three different schemes under large and small rotation cases have been presented and their relative numerical accuracy and computational efficiency have been evaluated. It has been observed that there is no sacrifice on the numerical accuracy due to the assumptions leading to the explicit through-thickness integration, but at the same time, there is considerable saving in the computational time. The computational efficiency improves as the number of layers in the laminate increases. The small rotation formulation with the assumption of linear variation of Jacobian inverse across the thickness and based on further approximation regarding certain submatrices is seen to be computationally efficient, as applied to geometric nonlinear layered shell problems.     

The cover time of random geometric graphs

We study the cover time of random geometric graphs. Let $I(d)=[0,1]^{d}$ denote the unit torus in d dimensions. Let $D(x,r)$ denote the ball (disc) of radius r. Let $\Upsilon_d$ be the volume of the unit ball $D(0,1)$ in d dimensions. A random geometric graph $G=G(d,r,n)$ in d dimensions is defined as follows: Sample n points V independently and uniformly at random from $I(d)$ . For each point x draw a ball $D(x,r)$ of radius r about x. The vertex set $V(G)=V$ and the edge set $E(G)=\{\{v,w\}: w\ne v,\,w\in D(v,r)\}$ . Let $G(d,r,n),\,d\geq 3$ be a random geometric graph. Let $C_G$ denote the cover time of a simple random walk on G. Let $c>1$ be constant, and let $r=(c\log n/(\Upsilon_dn))^{1/d}$ . Then whp the cover time satisfies © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 38, 324–349, 2011     

Existence of periodic solutions of nonlinear systems with nonlinear boundary conditions

In this paper we consider the periodic solutions of nonlinear parabolic systems with nonlinear boundary conditions. By constructing the Poincare operator, we obtain the existence of -periodic weak solutions under some reasonable assumptions.     

On the construction of a nonlinear recursive predictor

In this paper, we present a novel approach for constructing a nonlinear recursive predictor. Given a limited time series data set, our goal is to develop a predictor that is capable of providing reliable long-term forecasting. The approach is based on the use of an artificial neural network and we propose a combination of network architecture, training algorithm, and special procedures for scaling and initializing the weight coefficients. For time series arising from nonlinear dynamical systems, the power of the proposed predictor has been successfully demonstrated by testing on data sets obtained from numerical simulations and actual experiments.     

A blow-up result in a system of nonlinear viscoelastic wave equations with arbitrary positive initial energy

In this paper we consider a system of nonlinear viscoelastic wave equations. Under arbitrary positive initial energy and standard conditions on the relaxation functions, we prove a finite-time blow-up result.