| In this dissertation,we consider some problems related on the existence of sonicsupersonic solution in 2D steady relativistic Euler equations.These problems consist of the degenerate boundary value problem involved in the sonic curve,the degenerate Cauchy-Goursat problem and the semi-hyperbolic patch problem.Based on the theory of relativistic fluid mechanics,the characteristic decomposition in hyperbolic equations,the hodograph transformation method together with the functional analysis,the existence of sonic-supersonic solution for problems introduced as above are studied,the main contexts of this dissertation are as follows:(Ⅰ)In chapter 2,the existence and regularity of the sonic-supersonic solution for the degenerate sonic boundary value problem are proved.Distinct from the classical Euler equations,in view of the mathematical structure,the relativistic Euler equations are more complex.Moreover,the relationship between the speed of the relativistic fluid and the sonic speed can’t be directly obtained by the Bernoulli’s law.Firstly,using the sonic speed of the relativistic Euler equations,the characteristic decomposition related on the variables R:=?+a/a and S:=?-a/a is derived,we transform the 2D steady irrotational isentropic relativistic Euler equations into the first-order hyperbolic equations.Secondly,utilizing the partial hodograph transformation(x,y)|→((?),Φ)together with the relationships of the thermodynamic quantities,we prove the flow speed and the sonic speed in the relativistic Euler equations are a function of the variable(?).For this reason,it simplifies the computations in this dissertation.Using Arzel’a-Ascoli theorem,in the weight metric space consists of the continuously differentiable function classes,we prove the existence of sonic-supersonic solution for 2D steady irrotational isentropic relativistic Euler equations.Thirdly,in the physical plane,the solution of the 2D steady irrotational isentropic relativistic Euler equations is verified.Lastly,with the help of bootstrap method,the C1,1/6 regularity for sonic-supersonic solution and the sonic curve is proved.(Ⅱ)In chapter 3,the degenerate Cauchy-Goursat problem for 2D steady irrotational isentropic relativistic Euler equations is solved.We obtain the existence and the uniqueness of the sonic-supersonic solution.The degenerate Cauchy-Goursat problem governed by the smooth positive characteristic curve and the smooth sonic curve is set,using the characteristic decomposition and the partial hodograph transformation involved in the angle variables,the first-order hyperbolic equations is transformed into the linear equations.Different from the degenerate Cauchy-Goursat problem for the classical Euler equations with the polytropic gas,based on the monotonicity of the Mach angle with respect to the sonic speed,we obtain the characteristic decompositions involved in the angle variable.In the partial hodograph plane,the iteration consequence is constructed.Employing the monotonicity of the function for variable t=cos ω(x,y)and the iteration method,the existence and uniqueness of the sonic-supersonic solution for the degenerate CauchyGoursat problem are obtained.Using the invertibility of the hodograph transformation,the sonic-supersonic solution for the physical relativistic Euler equations is established.(Ⅲ)In chapter 4,the semi-hyperbolic patch problem for 2D steady irrotational isentropic relativistic Euler equations is studied.The C1,1/6 regularity for the sonic curve and the sonic-supersonic solution are obtained.Utilizing the streamline,characteristic curve and sonic curve,we set the semi-hyperbolic patch problem.Employing the partial hodograph transformation t=cos ω(x,y),r=θ(x,y)-θ(x,y),the streamline is transformed into a straight line and the boundary information is given.Based on the transformation U=1/U,V=-1/V.the linear equations is obtained.Using the classical theory of the linear hyperbolic equations,the local classical solution for the semi-hyperbolic patch problem is obtained.Then the prior estimates for the solution to use,we extend the local solutions to the global solution.Compared with the semi-hyperbolic patch problem in classical Euler equations,we prove that the function with respect to the variable t=cos ω(x,y)is strictly increasing.Using this monotonicity,the C1,1/3 regularity for the sonic-supersonic solution in(t,r)partial hodograph plane is given.In(x,y)physical plane,we finally prove the C1,1/6 regularity for the sonic curve and the sonic-supersonic solution. |
