一维弱噪声随机Burgers方程的奇摄动解
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摘要
讨论了一类有界区域上具有有色噪声干扰的随机Burgers方程奇摄动解,其波动率服从弱噪声Ornstein-Uhlenbeck(O-U)过程.由波运动的转移概率密度函数满足的后向Kolmogorov方程,得到随机Burgers的期望所满足的后向Kolmogorov方程.由于期望满足的后向Kolmogorov方程的初边值问题条件涉及到一类确定性Burgers方程的解,因此该问题实际上是Burgers方程和Kolmogorov方程的联立形式.首先,应用奇摄动方法,对一类确定性Burgers方程进行了正则渐近展开,由Schauder估计、Ascoli-Arzela定理证明了非线性抛物方程渐近解的有界性与存在性,由Lax-Milgram定理证明了线性抛物方程渐近解的有界性与存在性,得到波速率的形式渐近解.其次,由奇摄动理论,对期望满足的方程进行了奇摄动渐近展开和边界层矫正,由二阶线性偏微分方程理论,得到边界层函数渐近解存在且有界.应用极值原理、De-Giorgi迭代技术分别证明了波速率和波期望渐近解的余项有界,得到渐近解的一致有效性.
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 LaxMilgram 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.
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 LaxMilgram 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.
引文
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[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 A pplied A nalysis,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 A pplied 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 A pplied Mathematics,1995,95(2):147-170.
[5]LAFORGUE J G L,O’MALLEY R E Jr.Exponential asymptotics,the viscous Burgers’equaton,and standing w ave solutions for reaction-advection-diffusion model[J].Studies in A pplied 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 N umer 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 C omputational 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 U niversity of Science and T echnology,2013.(in C hinese))
[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].M aster T hesis.Q ingdao:O cean U niversity of C hina,2009:483-486.(in C hinese))
[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 A pplications,2012,391(20):4557-4563.
[14]GHANMI I,JEBARI R,BOUKRICHA A.Numerical solution of the(3+1)-dimensional KP equation w ith initial condition by homotopy perturbation method[J].International Journal of C ontemporary 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 Z huoqun,YIN Jingxue.Introduction of Ellipse and Parabolic Equation[M].Beijing:Science Press,2003:44-56,80-85.(in C hinese))