Construction And Application Of Radon Measure Solutions For Compressible Euler Equations

In this dissertation,we study the steady hypersonic-limit flow problem and Riemann problem by constructing the Radon measure solution of compressible Euler equations with Dirac measure.Qu,Yuan and Zhao proposed the concept of Radon measure solution for the problem of two-dimensional steady compressible Euler flow passing a straight wedge,and used it to prove that Newton theory is valid when Mach number tends to infinity and adiabatic index goes to 1.Based on their work,we consider the problem of steady compressible Euler flow passing two-dimensional ramp or threedimensional axisymmetric cone,prove the famous Newton-Busemann formula with the definition of Radon measure solution.As studying interaction of the flow field,we propose the singular Riemann problem,which provides a physical background for some Riemann problems with Dirac measure in the initial data.By using the definition of Radon measure solution to study the interaction of flow field,we find that when the steady hypersonic-limit Euler flow interacts with the still gas after passing two-dimensional or three-dimensional axisymmetric finite obstacle,it happens that if the pressure of the downstream still gas is quite large,the free layer will terminate at a finite distance from the obstacle.In this dissertation,we also study the Riemann problem of one-dimensional unsteady isentropic compressible Euler equations for polytropic gases.It is found that:(1)the solution of the Riemann problem is not unique in the class of Radon measures.For example,in the case of two strong shocks connected in the classical Riemann solution,a unique delta shock solution satisfying over-compressing entropy condition can also be constructed.We also discuss possible physical interpretations and applications of these new solutions.(2)For the classical Riemann problem,the delta shock curves are straight lines.However,for the generalized Riemann problem with mass concentration at the initial discontinuity,the delta shock curve is no longer a straight line due to the initial velocity at the mass concentration point.(3)From the study on discontinuous solution by Riemann,it is concluded that the range of shock is a curve,while the study of measure solution in this paper shows that the range of delta shock is a region.This is the first result on the construction of singular measure solutions to the compressible Euler system of polytropic gases,that is strictly hyperbolic,and whose characteristics are both genuinely nonlinear.In this dissertation,we can obtain the explicit solution of delta shock by the detailed analysis of ordinary differential equation.