https://doi.org/10.1140/epjp/s13360-021-02158-5
Regular Article
The perturbed Riemann problem for the Eulerian droplet model
1
School of Artificial Intelligence, Jianghan University, 430056, Wuhan, People’s Republic of China
2
College of Science, China Jiliang University, 310018, Hangzhou, People’s Republic of China
Received:
25
April
2021
Accepted:
8
November
2021
Published online:
27
November
2021
In this paper, we are concerned with a general Riemann problem for the 1-D Eulerian droplet model with delta initial condition. The main novelty of this paper lies in that, by providing a global analytical solution, it fills a gap in reference (Keita and Bourgault in J Math Anal Appl 472:1001-1027, 2019) in the literature, in which it was shown that a solution exists for the associated generalized Rankine–Hugoniot conditions with delta initial condition, but no analytical solution was given. By constructing solutions to the perturbed initial value problem, and then passing to the limit to recover the delta initial condition and the limit analytical solution, we successfully develop the global analytical solution to the delta initial value problem. During this process, four different cases of wave interactions are studied and a new type of wave (called delta contact discontinuity) is introduced. The above analysis gives a nice illustration of wave interactions for degenerate hyperbolic conservation laws with source term, which illustrations are relatively rare in the literature. Moreover, with above methods and results, it will give us some new insights into our future study on 1-D or 2-D conservation laws with source term
© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2021
