Degenerate Goursat-type boundary value problems arising from the study of two-dimensional isothermal Euler equations

Semi-hyperbolic patches of solutions to the two-dimensional compressible magnetohydrodynamic equations

“…For semi-hyperbolic patches of solutions to some other type of hyperbolic conservation laws, the readers can see [9,10,11]. Song et al [24,28] also studied the regularity of semi-hyperbolic patches near sonic lines.…”

Characteristic decompositions and boundary value problems for two‐dimensional steady relativistic Euler equations

“…The characteristic decomposition method was introduced by Dai and Zhang [5] and Li, Zhang and Zheng [6] in investigating the interaction of rarefaction waves of gasdynamic equations. Recently, it was extensively used in constructing global classical solutions of "supersonic" flow problems of gasdynamic equations and other related models, see [7] and [8] for the pressure gradient system, [9] and [10] for the isentropic irrotational steady Euler equations, [10][11][12][13][14][15][16] for the isentropic irrotational pseudo-steady Euler equations and [17] for Chaplygin gas Euler equations. This paper is organized as follows.…”

Simple waves and characteristic decompositions of quasilinear hyperbolic systems in two independent variables

“…For example, for the bi-symmetric four rarefaction wave interaction problems (Figure1), one can take advantage of it to determinate that the regions I-IV are simple waves and then the regions V and VI are two interaction regions of simple waves, see Li and Zheng [9] for details. The method used in [7] is the so-called characteristic decomposition, which is quite effective to deal with some problems for quasilinear hyperbolic systems [9][10][11][12][13][14][15][16]. The idea to use the characteristic decomposition can trace back to the classical one-dimensional wave equation u tt c 2 u xx D 0 with constant speed c, which has an interesting decomposition…”

“…The characteristic decomposition as a powerful tool was first revealed by Dai and Zhang [10] for considering the pressure-gradient system and then was used extensively for studying other systems, for instance, the Euler equations, see [9,11,[13][14][15][16][17]. The concept of the characteristic decomposition indeed has more implications, and it can also apply to high order estimates of solutions, even to numerical schemes [18].…”