Existence Of Shock Front Solution To 2-Dimensional Axially Symmetric Piston Problem

The present Ph.D. dissertation is concerned with existence of shock front solution to axially symmetric piston problem in 2-dimentional compressible flows.Axially symmetric piston problem is an important mathematical model in the study of conservation laws. In ([6]), the author first considered existence and uniqueness of shock front solution in the case when the velocity of the piston is some constant Then the author proved that there exists a shock front surface which moves outward with some velocity S0(S0 > b0). Moreover, the author also considered the case that the piston is not symmetric.The main work of this dissertation is to make some further study in piston problem, i.e., the existence of shock front solution when the velocity of the axially symmetric piston is not constant. Because there is no essential difference between 2-dimensioal case and multi-dimensional case (the difference lies only in one coefficient), we only take the 2-dimensional case as an example. The main results of the dissertation are as follows. In the case of isentropic compressible flow, we give the local and global existence of shock front solutions under some assumptions. For the case of non-isentropic case, i.e., the whole Euler equations, we give the local existence of shock front solution.The whole contents are organized as follows.Chapter I, Preface. This chapter is devoted to describing of the physical background and the mathematical model. The main results of this Ph.D. dissertation are briefly introduced in the last section of this chapter.Chapter II is devoted to the derivation of several mathematical models of piston problem (including 1-d isentropic flow, 2-d isentropic flow and 2-d non-isentropic flow). Some properties that concerned are also given here.Chapter III deals with the local existence of 2-d piston problem in isentropic com-pressible flow. The method we adopted here is Newton interation procedure. Namely, first we construct approximate solution as the first term of iteration, then we linearize the problem near the approximate solution. Next we construct energy estimate for the linearized problem. At last, with the help of the energy estimate, we prove the convergence of iteration. With the procedure above, we get the local existence under some assumption of smoothness.In Chapter IV, we focused on the global existence of 2-d piston problem in isen-tropic compressible flow. The method we used here is modified Glimm Scheme. First we construct approximate solution by random choice method, then we construct local and global interaction estimate respectively. With the estimate above we proved that there exists a subsequence of the approximate solutions sequence that convergences to the weak solution of this problem. Thus, under some assumptions of the velocity of the piston and the density of the gas outside, we give the global existence of shock front solution.In Chapter V, the local existence of 2-d piston problem for non-isentropic compressible flow is considered. At this place, we studyed the full Euler system. Compared to Chapter III and IV, this model describes the physical model more accurately. We also used the Newton iteration and proved the local existence of shock front solution under some assumption of smoothness.