Estimation Of Solutions Of Initial Value Problem For A Class Of Quasilinear Hyperbolic Systems And The Subsonic Euler Flows In One-demensional Ducts With Friction

This paper mainly studies the initial value problem and initial boundary value prob-lem of quasilinear hyperbolic systems.In general,for nonlinear hyperbolic systems,no matter how smooth the initial value is,the classical solution to the Cauchy problem only exists locally and explodes in finite time.There are classical solutions for quasilinear hyperbolic systems under certain conditions,and the solution of the systems decays exponentially with respect to time.We prove by the method of characteristics that when the coefficient matrix of the zero-order term of the quasilinear hyperbolic equations are strictly row(or column)-diagonally dominant matrix,the C~0norm of the solution of the equations decays expo-nentially with respect to time.Finally,we study the stability of steady subsonic Euler flow in a one-dimensional pipeline with friction.The above problem is actually the initial boundary value problem of quasilinear hyperbolic systems.We use the charac-teristics method and the mathematical induction to prove the stability of global solution of the systems under given dissipative boundary conditions.