The Exact Riemann Solutions to the Generalized Pressureless Euler Equations with Dissipation
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The Exact Riemann Solutions to the Generalized Pressureless Euler Equations with Dissipation Qingling Zhang1
· Fen He2
Received: 9 September 2019 / Revised: 27 February 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract The Riemann solutions for the generalized pressureless Euler equations with a dissipation term are constructed explicitly. It is shown that the delta shock wave appears in Riemann solutions in some situations. 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. Unlike the homogeneous case, it is shown that the dissipation term makes contact discontinuities and delta shock waves bend into curves and the Riemann solutions are not self-similar anymore. Moreover, as the dissipation term vanishes, the Riemann solutions converge to the corresponding ones of the generalized pressureless Euler equations. Finally, we give the application of our results on two typical examples. Keywords Generalized pressureless Euler equations · Riemann problem · Delta shock wave · Vacuum state · Dissipation Mathematics Subject Classification 35L65 · 35L67 · 35B30 · 76N10
1 Introduction In this paper, we consider the following generalized pressureless Euler equations with dissipation
Communicated by Yong Zhou.
B
Qingling Zhang [email protected]
1
School of Mathematics and Computer Sciences, Jianghan University, Wuhan 430056, People’s Republic of China
2
Center for Mathematical Sciences and Department of Mathematics, Wuhan University of Technology, Wuhan 430070, People’s Republic of China
123
Q. Zhang, F. He
vt + (v f (u))x = 0, (vu)t + (vu f (u))x = −αvu,
(1.1)
with initial data (v, u)(x, 0) = (v± , u ± ),
±x > 0,
(1.2)
where f (u) is given to be a smooth and strictly monotone function, the sign of v is assumed to be unchanging and α > 0 is the constant dissipation coefficient. The dissipation term first appeared in [8] to reflect the clustering mechanism. If α = 0, namely the dissipation vanishes, then system (1.1) becomes the so-called generalized pressureless Euler equations, whose Riemann problem was solved by Yang [25] in 1999. Then Yang and Sun solved the Riemann problem with delta initial data in [26] and obtained solutions with four kinds of different structures. Furthermore, Huang [9] solved the Cauchy problem by generalized potential. Mitrovic and Nedeljkov [14] studied its delta shock waves obtained as a limit of two shock waves. Recently, some research has been done on the generalized pressureless Euler equations with source term (see [29] for its Riemann problem with friction). While if α = 0, and f (u) = u, v ≥ 0, then (1.1) becomes the noted pressureless Euler equations, which are also called transport equations [1,2]. It can be used to describe some important physical phenomena, such as the motion of free particles sticking together under collision and the formation of large scale structures in the u
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