Global Existence Of Classical Solutions For Quasilinear Hyperbolic Equations And Plasma Model

This paper is consist of two parts. The first part investigates the existence and stability of traveling wave solutions to Cauchy problem for first-order quasilinear hy-perbolic equations. We proved the existence and stability of traveling wave solutions with appropriate small W1,1?L? norm to Cauchy problem for quasilinear hyperbol-ic equations under weakly linearly degenerate condition, and the stability results can be applied to the diagonalizable quasilinear hyperbolic equations and Chaplygin gas dynamic equations. The second part considers the global existence and convergence limits of solutions to period problem for Euler-Poisson equations. By converting the equations satisfying the variables (nu,uv) to the symmetrizable hyperbolic equations and establishing uniform energy and time dissipation estimates with respect to time t and the parameters ?,? based on energy estimates, we prove the global existence of smooth solutions with small initial data. Furthermore, we discuss the zero-relaxation limit ??0 and the zero-electron-mass limit ?? 0 of the equations by using these energy estimates. The structure of this paper as follows:In the first chapter, we briefly give the history and the present situation on the Cauchy problem for quasilinear hyperbolic equations and two-fluid Euler-Poisson equa-tions, then show a survey of our results in the thesis.In the second chapter, by introducing the local normalized coordinates, we give the corresponding formulas of wave decomposition and get some estimates of the C1 solu-tion. Then, we prove the stability of of traveling wave solutions under weakly linearly degenerate condition. Finally, we apply our results to the diagonalizable quasilinear hyperbolic equations and Chaplygin gas dynamic equations.In the third chapter, by establishing uniformly prior estimates with respect to parameters ?,? and time t, we obtain the uniformly global existence. Furthermore, by using the prior estimates, we discuss the convergence limits of the solutions as the parameter ?,? tend to zero. A basic assumption on the initial data is (n0,u0,v)? Hs(Td) which is uniformly small with respect to the corresponding parameter.In the forth chapter, we make a conclusion for this paper, and point out the shortcomings of the paper and next work in the future.