In this article, we prove the existence and regularity of a smooth solution for a supersonic-sonic patch arising in a modified Frankl problem in the study of three-dimensional axisymmetric steady isentropic relativistic transonic flows over a symmetric airfoil. We consider a general convex equation of state which makes this problem complicated as well as interesting in the context of the general theory for transonic flows. Such type of patches appear in many transonic flows over an airfoil and flow near the nozzle throat. Here the main difficulty is the coupling of nonhomogeneous terms due to axisymmetry and the sonic degeneracy for the relativistic flow. However, using the well-received characteristic decompositions of angle variables and a partial hodograph transformation we prove the existence and regularity of solution in the partial hodograph plane first. Further, by using an inverse transformation we construct a smooth solution in the physical plane and discuss the uniform regularity of solution up to the associated sonic curve. Finally, we also discuss the uniform regularity of the sonic curve.
In this article, we study the gas expansion problem by turning a sharp corner into vacuum for the two-dimensional pseudo-steady compressible Euler equations with a convex equation of state. This problem can be considered as interaction of a centered simple wave with a planar rarefaction wave. In order to obtain the global existence of solution up to vacuum boundary of the corresponding two-dimensional Riemann problem, we consider several Goursat type boundary value problems for 2-D self-similar Euler equations and use the ideas of characteristic decomposition and bootstrap method. Further, we formulate two-dimensional modified shallow water equations newly and solve a dam-break type problem for them as an application of this work. Moreover, we also recover the results from the available literature for certain equation of states which provide a check that the results obtained in this article are actually correct.
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