Abstract
This paper focuses on a degenerate boundary value problem arising from the study of the two-dimensional Riemann problem to the nonlinear wave system. In order to deal with the parabolic degeneracy, we introduce a partial hodograph transformation to transform the nonlinear wave system into a new system, which displays a clear regularity–singularity structure. The local existence of classical solutions for the new system is established in a weighted metric space. Returning the solution to the original variables, we obtain the existence of classical solutions to the degenerate boundary value problem for the nonlinear wave system.
Highlights
1 Introduction We are interested in a degenerate boundary value problem arising from the study of the two-dimensional four-constant Riemann problem to the nonlinear wave system
The multidimensional Riemann problem of the quasilinear hyperbolic conservation laws is one of important problems in mathematical fluid dynamics containing in particular the oblique shock reflection problem and the dam collapse problem
(2019) 2019:1 the two-dimensional Riemann problem to the Euler equations was initiated by Zhang and Zheng [32]
Summary
Where ρ, (u, v), are, respectively, the density and the velocity, and p = p(ρ) is a given function of ρ This system is obtained either by ignoring the quadratic terms in the velocity (u, v) to the two-dimensional isentropic compressible Euler equations in the gas dynamics, or by writing the nonlinear wave equation as a first-order system. The equation of state of the Chaplygin gas was introduced by Chaplygin [5] and was taken as a suitable mathematical approximation for calculating the lifting force on a wing of an airplane in aerodynamics by Tsien [28] and von Karman [29]. The purpose of the present paper is to establish the local existence of classical solutions to the two-dimensional self-similar nonlinear wave system (1.3) with degenerate boundary data, which is an essential step for using an iterative process to construct a global solution of mixed type equation. For convenience to deal with our problem, we rewrite (2.1) in terms of the polar coordinates (r, θ ) as
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