In this paper, we are concerned with the Riemann problem for the generalized pressureless Euler equations with a composite source term, which covers the important Eulerian droplet model associated with aircrafts and airways. The Riemann solutions exhibit exactly two types of structures: delta-shocks and vacuum solutions. The generalized Rankine–Hugoniot conditions of the delta shock wave are established and the exact position, propagation speed and strength of the delta shock wave are given explicitly. It is shown that the composite source term makes contact discontinuities and delta shock waves bend into curves and the Riemann solutions are not self-similar anymore, which is a new and interesting phenomenon different from the homogeneous generalized pressureless Euler equations. On the other hand, compared with previous results on the nonhomogeneous generalized pressureless Euler equations, different from the nonhomogeneous case with friction where the state variable u changes linearly with respect to t, here u changes exponentially with respect to t under the influence of composite source term, but with a velocity quite different from the nonhomogeneous case with dissipation. It is also shown that, as the composite source term vanishes wholly or partly, the Riemann solutions converge to the corresponding ones of the homogeneous system or the nonhomogeneous system. These results will give us valuable insights into later research on the (generalized) pressureless Euler equations and other conservation laws with more complicated source terms, such as discontinuous source terms or singular source terms. Finally, two typical examples are given to show the application of our results on the Eulerian droplet model and the nonlinear geometric optics system with a source term.

在本文中，我们关注具有复合源项的广义无压欧拉方程的黎曼问题，其中涵盖了与飞机和航线相关的重要欧拉液滴模型。黎曼解恰好展示了两种类型的结构：δ激波解和真空解。建立了δ激波的广义Rankine-Hugoniot条件，明确给出了δ激波的确切位置、传播速度和强度。结果表明，复合源项使接触不连续，δ激波弯曲成曲线，黎曼解不再自相似，这是不同于齐次广义无压欧拉方程的一个有趣的新现象。另一方面，u相对于t线性变化，这里u 相对于t呈指数变化在复合源项的影响下，但速度与具有耗散的非均匀情况完全不同。还表明，随着复合源项全部或部分消失，黎曼解收敛于齐次系统或非齐次系统的相应解。这些结果将为我们以后对（广义）无压欧拉方程和其他具有更复杂源项（例如不连续源项或奇异源项）的守恒定律的研究提供宝贵的见解。最后，给出了两个典型例子来说明我们的结果在欧拉液滴模型和具有源项的非线性几何光学系统中的应用。