Goursat Problems Of Euler Equations In Gas Dynamics And Cauchy Problems Of Variational Wave Equations

This paper is concerned with two families of Goursat problems for the two–dimensionalisothermal pseudo–steady Euler equations and the Cauchy problems for two class of vari-ational wave equations.Chapter 2 considers the characteristic decomposition of the general 2×2 quasilin-ear strictly hyperbolic systems and the two–dimensional isothermal pseudo–steady Eulerequations. A suffcient condition for the existence of characteristic decompositions to thegeneral 2×2 quasilinear strictly hyperbolic systems is derived. These decompositionsallow us to extend the well-known result on reducible equations in Courant and Friedrichs([26]), that any hyperbolic state adjacent to a constant state must be a simple wave,despite the fact that the coeffcients here depend on the independent variables. By in-troducing inclination angle variables of characteristics, we establish a set of characteristicdecompositions for the two–dimensional isothermal pseudo–steady Euler equations and itsa diagonal form.Chapter 3 is devoted to investigating two families of Goursat problems for the two–dimensional isothermal pseudo–steady Euler equations: the gas expansion problem andthe semi-hyperbolic patch problem. We establish a priori C0 and C1 norms estimates bythe characteristic decompositions derived in Chapter 2. The priori C1,1 norm estimatescan be obtained by the C1 norm estimates and the diagonal form. Based on these prioriestimates and the method employed by Dai and Zhang ([29]), we extend the local solutionsto the entire domain. Furthermore, for the semi–hyperbolic problem, we verify that thesolutions are uniform Ho¨lder continuous and that the characteristics starting from thesonic curve will form an envelope before their sonic points.Chapter 4 studies the Cauchy problems of the variational sine-Gordon equation anda system of variational wave equations. The main diffculty of these problems arises fromthe possible concentration of energy in finite time. To deal with this diffculty, we in-troduce a new set of variables depending on the energy and then obtain an equivalentsemilinear system, whereby all singularities are resolved. The global solution of the newequivalent semilinear system is obtained by the standard fixed point method and a priorestimates. Returning to the original variables, we establish the global existence of con-servative weak solutions to the Cauchy problems for initial data of finite energy. The continuous dependence of the solutions upon the initial data follows directly from theconstructive procedure. It is worthwhile to point out that the solutions we obtained areconservative, in the sense that the total energy represented by a measure equals a con-stant, for almost every time, and this energy may only be concentrated on a set of timesof zero measure under some conditions.