This paper is devoted to the four-constant Riemann problem for the two-dimensional (2D) scalar conservation laws involving linear fluxes with discontinuous coefficients. First, under the assumption that each discontinuity ray of initial data outside of the origin emits exactly one elementary wave, by studying the pointwise interactions occurring at the interaction points of waves, we completely solve this Riemann problem in the self-similar plane with 21 nontrivial and different geometric structures. Second, when each discontinuity ray of initial data outside of the origin emits two different kinds of contact discontinuities, by studying the pointwise interactions, we construct an interesting kind of spiral structure in the self-similar plane.

中文翻译：

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涉及具有不连续系数的线性通量的二维标量守恒定律的黎曼问题

本文专门研究涉及具有不连续系数的线性通量的二维 (2D) 标量守恒定律的四常数黎曼问题。首先，假设原点外初始数据的每条不连续射线恰好发射一个基波，通过研究波的相互作用点处发生的逐点相互作用，我们在具有21个非平凡的自相似平面上彻底解决了这个黎曼问题。和不同的几何结构。其次，当原点外初始数据的每条不连续射线发出两种不同的接触不连续性时，通过研究逐点相互作用，我们在自相似平面上构建了一种有趣的螺旋结构。