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This work is concerned with establishing the existence and uniqueness of the solution to a mixed-type degenerate boundary value problem for a relativistic magnetohydrodynamics system. We first consider a full relativistic magnetohydrodynamics system and reduce it to a simplified form under the assumption that the magnetic field vector is orthogonal to the velocity vector. We consider a boundary value problem for the steady part of this reduced system where the boundary data is prescribed on a sonic boundary and a characteristic curve. Here the main difficulty is the consideration of a relativistic system, with a general equation of state while considering the magnetic field effects as well, which we believe has never been analyzed before in the context of the analytical study of sonic-supersonic flows. Also, the degeneracy of the governing equations along the sonic curve is a crucial challenge. However, we employ the iteration method used in the work of Li and Hu \cite{li2019degenerate} to prove the existence and uniqueness of a local classical supersonic solution in the partial hodograph plane first and finally, we recover a local smooth solution to the boundary value problem in the physical plane by applying an inverse transformation.

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.

In this article we obtain exact solutions of (2+1)-dimensional Boiti-Leon-Pempinelli system of nonlinear partial differential equations which describes the evolution of horizontal velocity component of water waves propagating in two directions. We perform the Lie symmetry analysis to the given system and construct one-dimensional optimal subalgebra which involves some arbitrary functions of spatial variables. Several new exact solutions are obtained by symmetry reduction using each of the optimal subalgebra. We then study the physical behavior of some exact solutions by numerical simulations and observed many interesting phenomena such as traveling waves, lump type solitons, kink and anti-kink type solitons, breather solitons, singular kink type solitons and etc. We construct several conservation laws of the system by using multipliers method. As an application, we study the nonlocal conservation laws of the system by constructing potential systems and appending gauge constraints., Comment: 23 pages, 10 figures

This article is concerned in establishing the existence and regularity of solution of semi-hyperbolic patch problem for two-dimensional isentropic Euler equations with van der Waals gas. This type of solution appears in the transonic flow over an airfoil and Guderley reflection and is very common in the numerical solution of Riemann problems. We use the idea of characteristic decomposition and bootstrap method to prove the existence of global smooth solution which is uniformly $C^{1, \frac{1}{2}}$ continuous up to the sonic curve. We also prove that the sonic curve is $C^{1, \frac{1}{2}}$ continuous. Further, we show the formation of shock as an envelope for positive characteristics before reaching their sonic points.