The Singularly Perturbed Solutions Of Burgers And Navier-Stokes Equations With Discontinuous Initial Values And Their Application In Plasmas

Firstly,the problem of small-amplitude acoustic waves generated by laser pulse signals propagating in weakly damped medium is discussed,and the singularly perturbed linear mixed-wave equation with intermittent initial values is obtained.Secondly,the model of finite amplitude wave propagation generated by laser plasma is discussed,and the singularly perturbed Burgers equation with the initial value is the step function.Then the model of finite amplitude wave propagation generated by laser plasma was further discussed,and the singularly perturbed variable coefficient Burgers equation with an initial value of a discontinuous function was obtained.Finally,the singularly perturbed NavierStokes equations whose initial values are discontinuous functions are discussed.We solve the above problem,the main contents are as follows:1.The singularly perturbed linear mixed wave equation with discontinuous initial values is discussed.For the problem of small amplitude acoustic wave propagation in weakly damped medium,it can be described by a class of singularly perturbed linear mixed wave equations with discontinuous initial values.The asymptotic solutions of linear mixed wave equation with discontinuous initial values are constructed by singular perturbation method.The asymptotic solution contains two parts of the outer solution and the inner layer correction.Outer solution produces angular phenomena at the boundary of the affected area.Corrected by internal layer,and the residual estimate is used to obtain the consistent validity result of the asymptotic solution in the L2 sense.It can be obtained that the singularly perturbed solution is continuously,whose frist-order derivative is obtained also.The results show that the regularity of the asymptotic solution is improved Compared to the undamped wave equation.2.The singularly perturbed Burgers equation with the initial value as a step function is discussed.The ultrasonic model generated by laser plasma is studied to form the singularly perturbed Burgers equation with the initial value as a step function,the singularly perturbed asymptotic solution of the Burgers equation with initial value as a step function is obtained by singularly perturbed expansion method.It is divided into two parts:the outer solution and tne inner layer correction term.Since the initial condition is a step function,the wave generates characteristic boundary in the process of propagation,andthe correction term is parabolic characteristic boundary.The external solution is internally layer corrected on the feature lines.The existence and uniqueness of the asymptotic solution is proved by Hopf-Cole transform,Fourier transform and extremum principle.Then asymptotic expansion is obtained.The uniform validity of the asymptotic expansion is obtained.3.The singularly perturbed variable coefficient Burgers equation with discontinuous initial value function is discussed.The ultrasonic model produced by laser plasma is studied,the problem of Burgers equation with singularly perturbed variable coefficients and discontinuous initial values is formed.Using the singular perturbation method,the asymptotic expansion is obtained.The asymptotic solution contains two parts: the outer solution and the inner solution.The correction term is expressed as a parabolic equation.The existence and uniqueness of the asymptotic solution are proved by the method of trial function and extremum,and the form asymptotic expansion is obtained.Finally,the residual estimation is carried out by the extremum principle,and the uniform validity of the form asymptotic solution is obtained.4.The singularly perturbed Navier-Stokes equation with discontinuous initial value function is discussed.The ultrasonic model produced by laser plasma is studied,and the problem of singularly perturbed Navier-Stokes equation with discontinuous initial value is formed.Using the singular perturbation method,the asymptotic expansion is obtained.The asymptotic solution contains two parts: the outer solution and the inner solution.The inner solution is a differential equation group,the first term is directly solved,and the high-order term is solved by the constant variable method,and the form asymptotic expansion is obtained.Finally,the uniform validity of the formal asymptotic solution is obtained through the remainder estimation.In the course of the study,we have synthetically applied the knowledge of ordinary differential equations,partial differential equations,nonlinear acoustics,mathematical analysis,the singular perturbation theory and so on,which not only enriches the study of discontinuous initial value problems,but also further applies to plasma and ultrasonic problems.