Convergence Rates Of Vanishing Diffusion Limit On Conservative Form Of Hsieh's Equation

The aim of this thesis is to study the global unique solvability on Sobolev solution perturbated around diffusion waves to the Cauchy prob-lem of conservative form of Hsieh's equations(?)with initial data(?~?,?~?)(x,0)=((?)(x),(?)(x))?((?),(?)),as x?±?, where ?,?,? and v are all constants.Furthermore,convergence rates are also obtained as one ? of the diffusion parameters goes to zero.The dif-ficulty is created due to conservative nonlinearity in order to enclose the uniform(in diffusion parameter)higher-order energy estimates.However this kind of difficulty will not occur for both the nonconservative non-linearity and fixed diffusion parameter.The more subtle mathematical analysis need to be introduced to overcome the difficulties.Numerical experiments demonstrated and found drastically different behavior between conservative form and nonconservative form of Hsieh's equations.We try to find the difference and relationship on theoretical results from various properties between conservative form and noncon-servative form.The thesis consists of four chapters:In Chapter one,we summarize the background of the related literature,we also give some preliminary results and notations.In Chapter two,we prove the local existence of solutions to the Cauchy problems of Hsieh's equations by the method of induction.In Chapter three,we give the proof of Theorem 1.2.The key point is to establish series of Uniform-in-?a priori estimates on perturbation system.In Chapter four,we give some preliminary Lemmas which is con-cerned with the properties of diffusion waves,and correct function that we frequently used in the thesis.