Delta shock wave for the equations of constant pressure fluid dynamics with a composite source term. (English) Zbl 1489.76035
Summary: In this paper, the Riemann problem for the equations of constant pressure fluid dynamics with composite source term is studied. It is shown that the Riemann solutions involve delta shock wave and vacuum state. Furthermore, the generalized Riemann problem with the initial data containing Dirac delta function is also studied and four kinds of different structures for solutions are exhibited. The effects of composite source term on the Riemann solutions are studied in detail.
MSC:
| 76L05 | Shock waves and blast waves in fluid mechanics |
| 35Q35 | PDEs in connection with fluid mechanics |
References:
| [1] | Hsiao, L., Quasilinear Hyperbolic Systems and Dissipative Mechanisms (1998), Singapore: World Scientific, Singapore · doi:10.1142/3538 |
| [2] | Hsiao, L.; Liu, T., Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Commun. Math. Phys., 143, 599-605 (1992) · Zbl 0763.35058 · doi:10.1007/BF02099268 |
| [3] | Hsiao, L.; Serre, D., Global existence of solutions for the system of compressible adiabatic flow through porous media, SIAM J. Math. Anal., 27, 1, 70-77 (1996) · Zbl 0849.35069 · doi:10.1137/S0036141094267078 |
| [4] | Tan, Z.; Wang, Y., Global solution and large-time behavior of the 3D compressible Euler equations with damping, J. Differ. Equ., 254, 4, 1686-1704 (2013) · Zbl 1259.35162 · doi:10.1016/j.jde.2012.10.026 |
| [5] | Tan, Z.; Wu, Z.; Xie, M., Solution semiflow to the compressible Euler equations with damping, J. Math. Anal. Appl., 503, 1, 125-313 (2021) · Zbl 1473.35428 · doi:10.1016/j.jmaa.2021.125313 |
| [6] | Zhao, H., Convergence to strong nonlinear diffusion waves for solutions of p-system with damping, J. Differ. Equ., 174, 200-236 (2001) · Zbl 0990.35091 · doi:10.1006/jdeq.2000.3936 |
| [7] | Nishihara, K., Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping, J. Differ. Equ., 137, 384-395 (1997) · Zbl 0881.35076 · doi:10.1006/jdeq.1997.3268 |
| [8] | Huang, F.; Pan, R., Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum, J. Differ. Equ., 220, 207-233 (2006) · Zbl 1082.35031 · doi:10.1016/j.jde.2005.03.012 |
| [9] | Zhang, Y.; Zhang, R., The Riemann problem for the equations of constant pressure fluid dynamics with nonlinear damping, Int. J. Non-Linear Mech., 133 (2021) · doi:10.1016/j.ijnonlinmec.2021.103712 |
| [10] | Shen, C., The Riemann problem for the Chaplygin gas equations with a source term, Z. Angew. Math. Mech., 96, 6, 681-695 (2016) · Zbl 1538.35221 · doi:10.1002/zamm.201500015 |
| [11] | Sun, M., The exact Riemann solutions to the generalized Chaplygin gas equations with friction, Commun. Nonlinear Sci. Numer. Simul., 36, 342-353 (2016) · Zbl 1470.82032 · doi:10.1016/j.cnsns.2015.12.013 |
| [12] | Zhang, Y.; Zhang, Y., Riemann problems for a class of coupled hyperbolic systems of conservation laws with a source term, Commun. Pure Appl. Anal., 18, 3, 1523-1545 (2019) · Zbl 1415.35197 · doi:10.3934/cpaa.2019073 |
| [13] | Zhang, Y.; Zhang, Y., The Riemann problem for the Eulerian droplet model with buoyancy and gravity forces, Eur. Phys. J. Plus, 135, 171 (2020) · doi:10.1140/epjp/s13360-020-00145-w |
| [14] | Shen, C., The Riemann problem for the pressureless Euler system with the Coulomb-like friction term, IMA J. Appl. Math., 81, 76-99 (2016) · Zbl 1336.35281 |
| [15] | Hu, J., Two-dimensional Riemann problem for equations of constant pressure fluid dynamics with functional solutions, Q. Appl. Math., LVIII, 251-264 (2000) · Zbl 1157.35417 · doi:10.1090/qam/1753398 |
| [16] | Keita, S.; Bourgault, Y., Eulerian droplet model: delta-shock waves and solution of the Riemann problem, J. Math. Anal. Appl., 472, 1001-1027 (2019) · Zbl 1416.35162 · doi:10.1016/j.jmaa.2018.11.061 |
| [17] | Courant, R.; Friedrichs, K., Supersonic Flow and Shock Waves (1976), New York: Springer, New York · Zbl 0365.76001 · doi:10.1007/978-1-4684-9364-1 |
| [18] | Hu, J., One-dimensional Riemann problem for equations of constant pressure fluid dynamics with measure solutions by viscosity method, Acta Appl. Math., 55, 209-229 (1999) · Zbl 0921.35104 · doi:10.1023/A:1006101529302 |
| [19] | Weinan, E.; Rykov, Yu. G.; Sinai, Y. G., Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Commun. Math. Phys., 177, 349-380 (1996) · Zbl 0852.35097 · doi:10.1007/BF02101897 |
| [20] | Li, J.; Yang, S.; Zhang, T., The Two-Dimensional Riemann Problem in Gas Dynamics (1998), Harlow: Longman, Harlow · Zbl 0935.76002 |
| [21] | Yang, H., Generalized plane delta-shock waves for n-dimensional zero-pressure gas dynamics, J. Math. Anal. Appl., 260, 1, 18-35 (2001) · Zbl 0985.35044 · doi:10.1006/jmaa.2000.7426 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
