We consider self-similar solutions to the 1-dimensional isothermal Euler system for compressible gas dynamics. For each\(\beta \in {\mathbb {R}}\), the system admits solutions of the form

where \(\rho \) and u denote the density and velocity fields. The ODEs for \(\Omega \) and U can be solved implicitly and yield the solution to generalized Riemann problems with initial data

where \(R_l,\, R_r>0\) and \(U_l,\ U_r\) are arbitrary constants. For \(\beta \in (-1,0)\), the data are locally integrable but unbounded at \(x=0\), while for \(\beta \in (0,1)\), the data are locally bounded and continuous but with unbounded gradients at \(x=0\). Any (finite) degree of smoothness of the data is possible by choosing \(\beta >1\) sufficiently large and \(U_l=U_r\). (The case \(\beta \le -1\) is unphysical as the initial density is not locally integrable and is not treated in this work.) The case \(\beta =0\) corresponds to standard Riemann problems whose solutions are combinations of backward and forward shocks and rarefaction waves. In contrast, for \(\beta \in (-1,\infty )\smallsetminus \{0\}\), we construct the self-similar solution and show that it always contains exactly two shock waves. These are necessarily generated at time \(0+\) and move apart along straight lines. We provide a physical interpretation of the solution structure and describe the behavior of the solution in the emerging wedge between the shock waves.

我们考虑可压缩气体动力学的一维等温欧拉系统的自相似解。对于每个\（\ beta \ in {\ mathbb {R}} \），系统接受以下形式的解

其中\（\ rho \）和u表示密度场和速度场。\（\ Omega \）和U的ODE可以隐式求解，并给出具有初始数据的广义Riemann问题的解决方案

其中\（R_1，\，R_r> 0 \）和\（U_1，\ U_r \）是任意常量。对于\（\ beta \ in（-1,0）\），数据是本地可积分的，但在\（x = 0 \）处是无界的，而对于\（\ beta \ in（0,1）\），则数据是局部有界和连续的，但在\（x = 0 \）处具有无界的渐变。通过选择足够大的\（\ beta> 1 \）和\（U_l = U_r \），可以实现任何（有限）数据平滑度。（案例\（\ beta \ le -1 \）是非物理的，因为初始密度不可局部积分，因此在本工作中未进行处理。）案例\（\ beta = 0 \）对应于标准Riemann问题，其解决方案是前后冲击和稀疏波的组合。相反，对于\（\ beta \ in（-1，\ infty）\ smallsetminus \ {0 \} \），我们构造了自相似解，并表明它始终恰好包含两个冲击波。它们必定在时间\（0+ \）生成，并沿直线分开。我们提供了溶液结构的物理解释，并描述了在冲击波之间出现的楔形区域中溶液的行为。