Singularly Perturbed Stochastic Ultrasonic Equation

People usually described fluid motion by Navi-Stokes equation in imperfect medium,where Burgers equation is a typical form of the Navi-Stokes equation,so for the study of the random ultrasonic equation,we mainly discuss stochastic Burgers model.In the view of the viscous heat conduction effect of medium,it found that when the Reynolds number R is large enough,the finite amplitude wave gradually became distorted and eventually formed shock wave;when the Reynolds number is small,which the fluid dissipation cannot be ignored,there is no shock wave.Hence,applying the asymptotic expansion method of singular perturbation,we respectively discuss the asymptotic solutions of onedimensional random Burgers model and two-dimensional random Burgers model under different Reynolds number.Study on the stochastic Burgers model is discussed by predecessors that most results involved numerical solutions and stochastic hypersensitivity under disturbance by numerical analysis and computer simulation.And it is difficult to solve the analytical solution of stochastic model.So far,the application of singular perturbation related asymptotic analysis method to solve the stochastic Burgers model solution is rarely seen.In the paper,we firstly present the random Burgers model in one dimension,where the Reynolds number R is small.Based on the backward Kolmogorov formula satisfied by the diffusion process,we construct the corresponding expectation equation.According to the singular perturbation method and maximum principle,we prove the uniform validity of the asymptotic solution of nonlinear Burgers equation and the linear expectation equation.Then we discuss the stochastic Burgers model with colored noise in one-dimensional bounded domain.Combining the deleting of law with the singular perturbation theory,the regular part and the left boundary layer are obtained.When the Reynolds number R ? 1,we demonstrate in the sense of weak noise,that one-dimensional random Burgers model with shock wave in bounded domain.And the position and the arrival time of shock wave are obtained.Connected with the singular perturbation analysis method,we get the regular part and middle layer function,and the multiple solutions and attenuation of the correction function fully display the detailed characteristics of the shock wave.The uniform validity of formal asymptotic solutions is proved by applying the extremum principle.Then the one-dimensional problem is extended to two dimensions.Then we major in the stochastic Burgers model with small Reynolds number on two-dimensional unbounded domains,and construct two-dimensional stochastic Burgers and expectation equation model.Due to the interference of weak noise,we obtain the corresponding formal asymptotic solution carried out using singular perturbation method.Based on one-dimensional extremum principle,we construct a modified maximum principle,and applied to the estimation of the remainder.When the Reynolds number is large enough,we study a twodimensional stochastic Burgers model with shock wave and establish the corresponding expectation equation.Using the local transformation and stretching transformation,we get the regular part and correction function,and the multiple solutions and attenuation of correction function fully express the characteristics of shock wave.It is proved that the uniform validity of the formal asymptotic solution by the modified maximum principle.The main contents are as follows:First,A one-dimensional random Burgers model is studied.Firstly,we present onedimensional stochastic Burgers equation at low Reynolds number in unbounded domain,which the weak noise is subjected to Ornstein-Uhlenbeck(O-U)process.Based on the backward Kolmogorov equation satisfying the transition probability density function of diffusion process,the expected equation of wave motion is established.Since the definite condition of the expectation equation involves the solution of the Burgers equation,this problem is actually a simultaneous problem of nonlinear Burgers equation and linear expectation equation.We obtain the closed form solutions of two order quasilinear partial differential equations through the singular perturbation asymptotic expansion.Connected with the differential equation theory,the asymptotic solutions of the wave motion and expectation equations are got.The existence and boundedness of formal asymptotic solution is verified by the extremum principle.Then we show the one-dimensional stochastic Burgers equation in bounded domain,and establish the expectation equation of wave motion.By the deleting of law and the singular perturbation analysis method,the expectation equation will produce the boundary layer at the left boundary.The application of asymptotic expansion method to obtain regular part and correction function.The boundedness and existence of solutions of nonlinear parabolic equations are proved by Ascoli-Arzela theorem.The boundedness and existence of solutions of linear parabolic equations are verified by the Lax-Milgram theorem.Applying the De-Giorgi iterative technique to prove the uniform validity of formal asymptotic solution.Second,A one-dimensional stochastic Burgers model with shock waves is studied,and the nonlinear Burgers equation and linear expectation equation are constructed.The position and arrival time of the shock wave are obtained.From the singular perturbation method,the equation is carried out before and after the shock wave formation,then we obtain regular part and the middle layer correction function of the asymptotic solution.Because of the multiplicity and attenuation of the correction function,We find that under certain conditions,the correction function are arbitrary solutions;and in other cases,the correction function of the left boundary is rising exponentially,while the correction function on the right boundary decays in exponential form or in power-law form.Similarly,the boundary correction function of expectation equation is exponential form.It is proved that the uniformly valid estimate of the asymptotic solution by the extremum principle.Third,the two-dimensional random Burgers model is studied,where the two dimensional stochastic Burgers equation is a class of unbounded domains with colored noise,and the weak noise is subjected to Ornstein-Uhlenbeck(O-U)process.The simultaneous form of the nonlinear Burgers equation and the linear expectation equation is established by the Kolmogorov formula.The asymptotic solution of wave motion and expectation equation are derived by the singular perturbation method.And combing the modified extremum principle to prove the uniform validity of the formal asymptotic solution.Forth,the two-dimensional stochastic Burgers model with shock wave is studied,which is class of Burgers equation in the sense of weak noise on the two-dimensional unbounded domain.The arrival time and characteristic face of the two-dimensional shock wave are obtained.Applying local transformation and stretching transformation,the singular perturbation asymptotic expansion is carried out on both sides of the shock wave characteristic face,and we obtain the regular part and the middle layer correction function of the wave motion equation and expectation equation.Then we respectively discuss the correction function solution.The results imply that the wave motion equations under certain conditions,the correction function is arbitrary multiple solutions;and in other cases,the left boundary correction function always presents exponential rising,while the right boundary function is decay exponentially or power-law form.Similarly,we find that the correction function of expectation equation is exponential form.The uniformly valid estimate of the formal asymptotic solution is proved by the modified extremum principle.In the research process,we synthetically apply constant ordinary differential equations,nonlinear acoustic partial differential equation,stochastic process,stochastic differential equation,shock wave theory,the physical fluid wave,singular perturbation theory,and so on.Not only enrich the research of stochastic Burgers equation,but also depth the discussion on ultrasound.