Singular Perturbation Solutions to 1D Stochastic Burgers Equations Under Weak Noises

BAO Liping; HONG Wenzhen

doi:10.21656/1000-0887.380068

Volume 39 Issue 1

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BAO Liping, HONG Wenzhen. Singular Perturbation Solutions to 1D Stochastic Burgers Equations Under Weak Noises[J]. Applied Mathematics and Mechanics, 2018, 39(1): 113-122. doi: 10.21656/1000-0887.380068

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Singular Perturbation Solutions to 1D Stochastic Burgers Equations Under Weak Noises

doi: 10.21656/1000-0887.380068

School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, P.R.China

Funds: The National Natural Science Foundation of China(51175134)

Received Date: 2017-03-24
Rev Recd Date: 2017-06-11
Publish Date: 2018-01-15

Abstract

Abstract

The singular perturbation solutions to a class of bounded stochastic Burgers equations under colored noises were discussed, of which the volatility followed the weak noise Ornstein-Uhlenbeck (O-U) process. With the Kolmogorov equation satisfied by the probability density function of wave motion, the Kolmogorov equation satisfied by the expectation of the random Burgers equation was obtained. Since the initial boundary conditions for the Kolmogorov equation relate to a class of deterministic solutions to the Burgers equation, this problem is actually a simultaneous form of the Burgers equation and the Kolmogorov equation. Firstly, the regular asymptotic expansion of a class of deterministic Burgers equations was given. Based on the Schauder estimates and the Ascoli-Arzela theorem, boundedness and existence of the asymptotic solutions to the nonlinear parabolic equations were proved; moreover, according to the Lax-Milgram theorem, boundedness and existence of the asymptotic solutions to the linear parabolic equations were proved. The formal asymptotic solution of wave expectation was obtained. Secondly, with the singular perturbation theory, the asymptotic expansion of singular perturbation and the boundary layer correction of a class of expected equations were got. The existence and boundedness of the asymptotic solutions to the boundary layer functions were obtained according to the theory of linear partial differential equations. By means of the extremum principle and the De-Giorgi iterative techniques, the boundedness of the remainder terms of the asymtotic solutions of wave velocity and wave expectation was proved respectively, and the uniformly valid estimate for the asymptotic solution of the system was obtained.
- singular perturbation,
- random Burgers equation,
- average velocity,
- Ornstein-Uhlenbeck (O-U) process,
- uniformly valid estimate

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References

[1]	KREISS G, KREISS H O. Convergence to steady state of solution of Burgers’ equation[J]. Applied Numerical Mathematics,1986,2(3): 161-179.
[2]	LAFORGUE J G L, O’MALLEY R E Jr. On the motion of viscous shocks and the supersensitivity of their steady-state limits[J]. Methods in Applied Analysis,1994,1(4): 465-487.
[3]	LAFORGUE J G L, O’MALLEY R E Jr. Shock layer movement for Burgers’ equation[J]. SIAM Journal on Applied Mathematics,1994,55(2): 332-347.
[4]	LAFORGUE J G L, O’MALLEY R E Jr. Viscous shock motion for advection-diffusion equations[J]. Studies in Applied Mathematics,1995,95(2): 147-170.
[5]	LAFORGUE J G L, O’MALLEY R E Jr. Exponential asymptotics, the viscous Burgers’ equaton, and standing wave solutions for reaction-advection-diffusion model[J]. Studies in Applied Mathematics ,1999,102(2): 137-172.
[6]	VILLARROEL J. The stochastic Burger’s equation in Ito’s sense[J]. Blackwell Publishing,2004,112(1): 87-100.
[7]	XIU Dongbin, KARANIADAKIS G E. Supersensitivity due to uncertain boundary conditions[J]. Int J Numer Meth Engng,2004,61(12): 2114-2138.
[8]	LE MAITRE O P, KNIOB O M, NAJM H N. A stochastic projection method for fluid flow: I. basic formulation[J].Journal of Computational Physics,2001,173(2): 481-511.
[9]	高飞. 随机Burgers方程的格子Boltzmann模拟[D]. 硕士学位论文. 武汉: 华中科技大学, 2013.(GAO Fei. Lattice Boltzmann simulation of stochastic Burgers equation[D]. Master Thesis. Wuhan: Huazhong University of Science and Technology, 2013.(in Chinese))
[10]	付新刚. 广义Burgers方程的随机超敏感现象的数值研究[D]. 硕士学位论文. 青岛: 中国海洋大学, 2009: 483-486.(FU Xingang. Numerical study of stochastic supersensitivity of generalized Burgers equation[D]. Master Thesis. Qingdao: Ocean University of China, 2009: 483-486.(in Chinese))
[11]	VILLARROEL J. Stochastic perturbations of line solitons of KP[J]. Theoretical and Mathematical Physics,2003,137(3): 1753-1765.
[12]	XUE J K. A spherical KP equation for dust acoustic waves[J]. Physics Letters A,2003,314(5/6): 479-483.
[13]	YERMAKOU V, SUCCI S. A fluctuation lattice Boltzman scheme for the one-dimensional KPZ equation[J]. Phys A: Statistical Mechanics Its Applications,2012,391(20): 4557-4563.
[14]	GHANMI I, JEBARI R, BOUKRICHA A. Numerical solution of the (3+1)-dimensional KP equation with initial condition by homotopy perturbation method[J]. International Journal of Contemporary Mathematical Sciences,2012(41/44): 2089-2098.
[15]	CHAKRAVARTY S, KODAMA Y. Line-soliton solutions of the KP equation[C]// AIP Conference Proceedings 1212.2010: 312-341.
[16]	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: 44-56, 80-85.(WU Zhuoqun, YIN Jingxue. Introduction of Ellipse and Parabolic Equation [M]. Beijing: Science Press, 2003: 44-56, 80-85.(in Chinese))