The Study On The Existence Of Weak Solutions For The Initial Problem Of Isentropic Euler Equations With State Of Chaplygin Type

In this paper,we consider the existence of weak solutions for the initial problem of isen-tropic Euler equations with state of Chaplygin type.Under some conditions,for one-dimensional isentropic Chaplygin gas we obtain the Lloc 1 solution for the Cauchy problem.For two-dimensional isentropic irrotational Chaplygin gas with four piecewise constants,without singularity,we anal-yse the wave structures for piecewise smooth solutions in hyperbolic regions,then we get the conditions on initial data for the corresponding wave structures.In Chapter 1 and Chapter 2,we give the background,the current method and the relevant result for studying the Euler equations.Moreover,we introduce the basic theory of the nonlinear one-dimensional and two-dimensional conservation laws,respectively.In Chapter 3,we present a construction method on the existence of Lloc 1 solution of the Cauchy problem for the Euler system with state of Chaplygin type,in which the initial density ranges in(0,+?).The linearly degenerate property of this model makes it more difficult to study the L1 solution rather than providing some advances as that in the study of the L? solution.The basic idea is to reduce the partial differential equations to the integral equations and construct the weak solutions of the PDEs through that of the integral equations.This method has two distinct features:(i)it is clear to see at least what kind of function space the initial data should be in to study the well-posedness,and(ii)it is natural to employ real analysis to study this kind of problem with lower regularity,through which we develop the previous works.The success of real analysis applied here lies in that we find a sufficient and necessary condition on judging whether a monotone continuous function is absolutely continuous.The assumptions on the initial data in this chapter are also necessary in some sense.In Chapter 4,we study the two-dimensional Riemann problem for isentropic irrotational Chaplygin gas with initial data being four states,and construct the piecewise smooth solutions in the hyperbolic regions.Assuming that the density p satisfying 0<p<+? and there eight elementary waves for the initial discontinuity,we mainly analyse the necessary and sufficient con-ditions on the initial data as the self-similar solution can be constructed uniquely.In particular,if the initial densities are identical,we obtain a necessary and sufficient condition concretely and construct the solution explicitly.In Chapter 5,we mainly study the Riemann problem with a class of initial data for two-dimensional Chaplygin gas.We.prove the existence of the weak solutions under the large data in some sense for this problem.By using the method of generalized characteristics,we consider the initial discontinuity with six elementary waves,and classify the wave structures.Then we derive the necessary and sufficient conditions for two representative wave structures and construct the corresponding piecewise smooth solutions.