二维弱噪声随机Burgers方程的奇摄动解
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摘要
讨论了一类无界区域上具有有色噪声干扰的二维随机Burgers方程的奇摄动解。其波动率服从弱噪声Ornstein-Uhlenbeck(O-U)过程,应用奇摄动方法构造了相应的形式渐近解,得到了波期望和边界条件的渐近分析,并证明了渐近解的一致有效性。
The singular perturbation solution for a class of two-dimensional stochastic Burgers equation with colored noise in an unbounded area is discussed,and its volatility is subject to the weak noise Ornstein-Uhlenbeck(O-U)process.By the perturbation method,the corresponding asymptotic solution is constructed and the wave expectation and boundary condition is obtained.And the uniformly valid estimate for the asymptotic solution of the system is obtained.
The singular perturbation solution for a class of two-dimensional stochastic Burgers equation with colored noise in an unbounded area is discussed,and its volatility is subject to the weak noise Ornstein-Uhlenbeck(O-U)process.By the perturbation method,the corresponding asymptotic solution is constructed and the wave expectation and boundary condition is obtained.And the uniformly valid estimate for the asymptotic solution of the system is obtained.
引文
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[2]CHEKHLOV A,YAKHOT V.Kolmogorov turbulence in a random-force-driven Burgers equation[J].Physical Review E Statistical Physics Plasmas Fluids Related Interdisciplinary Topics,1995,51(4):5681-5684.
[3]WEINAN E,ERIC V E.Statistical theory for the stochastic burgers equation in the inviscid limit[J].Communications on Pure&Applied Mathematics,2000,53(7):852-901.
[4]VILLARROEL J.The stochastic Burger's equation in ito's sense[J].Studies in Applied Mathematics,2004,112(1):87-100.
[5]AUDUSES E,BOYAVAL S,GAO Y,et al.Numerical simulations of the inviscid Burgers equation with periodic boundary conditions and stochastic forcing[J].Esaim Proceedings&Surveys,2015,48:308-320.
[6]LV G,DUAN J.Martingale and weak solutions for a stochastic nonlocal Burgers equation on finite intervals[J].Journal of Mathematical Analysis and Applications,2017,449(1):176-194.
[7]DONG Z,SUN X,XIAO H,et al.Averaging principle for one dimensional stochastic Burgers equation[J].ArXiv Preprint ArXiv:1701.05920,2017:1-43.
[8]XIU D,EM K G.Supersensitivity due to uncertain boundary conditions[J].International Journal for Numerical Methods in Engineering,2010,61(12):2114-2138.
[9]OLIVIER P.LE MA,TREA,KNIOB O M.A stochastic projection method for fluid flow I.basic formulation[J].Journal of Computational Physics,2001,173(2):481-511.
[10]WAN X,XIU D,KARNIADAKIS G E.Stochastic solutions for the two-dimensional advection-diffusion equation[J].Siam Journal on Scientific Computing,2004,26(2):578-590.
[11]GARBEY,MARC,KAPER,HANS G.Asymptotic-numerical study of supersensitivity for generalized Burgers'equations[J].Siam Journal on Scientific Computing,2000,22(1):368-385.
[12]魏云云.二维Burgers方程的有限元数值解法[D].西安:长安大学,2016.
[13]魏云云,史峰,张引娣.算子分裂有限元方法求解二维Burgers方程[J].应用数学进展,2017,6(2):174-182.
[14]付新刚.广义Burgers方程的随机超敏感现象的数值研究[D].青岛:中国海洋大学,2009.
[15]高飞.随机Burgers方程的格子Boltzmann模拟[D].武汉:华中科技大学,2013.
[16]董光昌.非线性二阶偏微分方程[M].北京:清华大学出版社,1988:117-130.
[17]董光昌.线性二阶偏微分方程[M].杭州:浙江大学出版社,1987:20-25.
[18]伍卓群,尹景学.椭圆与抛物型方程引论[M].北京:科学出版社,2003:1-5.