Geodesic
The Riemann problem for the main model cases of the Euler---Poisson equations
Contemporary Mathematics. Fundamental Directions, Functional spaces. Differential operators. Problems of mathematics education, Tome 70 (2024) no. 1, pp. 38-52.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we construct a solution to the Riemann problem for an inhomogeneous nonstrictly hyperbolic system of two equations, which is a corollary of the Euler–Poisson equations without pressure [9]. These equations can be considered for the cases of attractive and repulsive forces as well as for the cases of zero and nonzero underlying density background. The solution to the Riemann problem for each case is nonstandard and contains a delta-shaped singularity in the density component. In [16], solutions were constructed for the combination corresponding to the cold plasma model (repulsive force and nonzero background density). In this paper, we consider the three remaining cases. 
@article{CMFD_2024_70_1_a3,
     author = {L. V. Gargyants and O. S. Rozanova and M. K. Turzynsky},
     title = {The {Riemann} problem for the main model cases of the {Euler---Poisson} equations},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {38--52},
     publisher = {mathdoc},
     volume = {70},
     number = {1},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2024_70_1_a3/}
}
TY  - JOUR
AU  - L. V. Gargyants
AU  - O. S. Rozanova
AU  - M. K. Turzynsky
TI  - The Riemann problem for the main model cases of the Euler---Poisson equations
JO  - Contemporary Mathematics. Fundamental Directions
PY  - 2024
SP  - 38
EP  - 52
VL  - 70
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CMFD_2024_70_1_a3/
LA  - ru
ID  - CMFD_2024_70_1_a3
ER  - 
%0 Journal Article
%A L. V. Gargyants
%A O. S. Rozanova
%A M. K. Turzynsky
%T The Riemann problem for the main model cases of the Euler---Poisson equations
%J Contemporary Mathematics. Fundamental Directions
%D 2024
%P 38-52
%V 70
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CMFD_2024_70_1_a3/
%G ru
%F CMFD_2024_70_1_a3
L. V. Gargyants; O. S. Rozanova; M. K. Turzynsky. The Riemann problem for the main model cases of the Euler---Poisson equations. Contemporary Mathematics. Fundamental Directions, Functional spaces. Differential operators. Problems of mathematics education, Tome 70 (2024) no. 1, pp. 38-52. http://geodesic.mathdoc.fr/item/CMFD_2024_70_1_a3/

[1] Gurevich A. V., Zybin K. P., “Nedissipativnaya gravitatsionnaya turbulentnost”, Zh. eksperiment. i teor. fiz., 94:1 (1988), 3–25 | MR

[2] Kochin N. E., Vektornoe ischislenie i nachala tenzornogo analiza, Nauka, M., 1965 | MR

[3] Rozhdestvenskii B. L., Yanenko N. N., Sistemy kvazilineinykh uravnenii i ikh prilozheniya k gazovoi dinamike, Nauka, M., 1978 | MR

[4] Chizhonkov E. V., Matematicheskie aspekty modelirovaniya kolebanii i kilvaternykh voln v plazme, Fizmatlit, M., 2018

[5] Shelkovich V. M., “Singulyarnye resheniya sistem zakonov sokhraneniya tipa $\delta$ i $\delta'$-udarnykh voln i protsessy perenosa i kontsentratsii”, Usp. mat. nauk, 63:3 (2008), 73–146 | DOI | MR | Zbl

[6] Brunelli J. C., Das A., “On an integrable hierarchy derived from the isentropic gas dynamics”, J. Math. Phys., 45:7 (2004), 2633–2645 | DOI | MR | Zbl

[7] Chae D., Tadmor E., “On the finite time blow-up of the Euler–Poisson equations in ${\mathbb R}^n$”, Commun. Math. Sci., 6:3 (2008), 785–789 | DOI | MR | Zbl

[8] Dafermos C. M., Hyperbolic conservation laws in continuum physics, Springer, Berlin–Heidelberg, 2016 | MR | Zbl

[9] Engelberg S., Liu H., Tadmor E., “Critical thresholds in Euler–Poisson equations”, Indiana Univ. Math. J., 50:1 (2001), 109–157 | DOI | MR

[10] Gao B., Tian K., Liu Q. P., Feng L., “Conservation laws of the generalized Riemann equations”, J. Nonlinear Math. Phys., 25:1 (2018), 122–135 | DOI | MR | Zbl

[11] Huang F., Wang Zh., “Well posedness for pressureless flow”, Commun. Math. Phys., 222:1 (2001), 117–146 | DOI | MR | Zbl

[12] Hunter J. K., Saxton R., “Dynamics of director fields”, SIAM J. Appl. Math., 51:6 (1991), 1498–1521 | DOI | MR | Zbl

[13] Pavlov M. V., “The Gurevich–Zybin system”, J. Phys. A: Math. Gen., 38:17 (2005), 3823–3840 | DOI | MR

[14] Popowicz Z., Prykarpatski A. K., “The non-polynomial conservation laws and integrability analysis of generalized Riemann type hydrodynamical equations”, Nonlinearity, 23:10 (2010), 2517–2537 | DOI | MR | Zbl

[15] Rozanova O. S., “On the behavior of multidimensional radially symmetric solutions of the repulsive Euler–Poisson equations”, Phys. D: Nonlinear Phenom., 443 (2022), 133578 | DOI | MR

[16] Rozanova O. S., “The Riemann problem for equations of a cold plasma”, J. Math. Anal. Appl., 527:1 (2023), 127400 | DOI | MR | Zbl

[17] Rozanova O. S., Turzynsky M. K., “On the properties of affine solutions of cold plasma equations”, Commun. Math. Sci., 22:1 (2024), 215–226 | DOI | MR

[18] Schäfer T., Wayne C. E., “Propagation of ultra-short optical pulses in cubic nonlinear media”, Phys. D.: Nonlinear Phenom., 196:1 (2004), 90–105 | DOI | MR | Zbl

[19] Tan C., “Eulerian dynamics in multidimensions with radial symmetry”, SIAM J. Math. Anal., 53:3 (2021), 3040–3071 | DOI | MR | Zbl

[20] Wei D., Tadmor E., Bae H., “Critical thresholds in multi-dimensional Euler–Poisson equations with radial symmetry”, Commun. Math. Sci., 10:1 (2012), 75–86 | DOI | MR | Zbl

[21] Wei L., “Wave breaking, global existence and persistent decay for the Gurevich–Zybin system”, J. Math. Fluid Mech., 22:4 (2020), 1–14 | DOI | MR

[22] Wei L., Wang Y., “The Cauchy problem for a generalized Riemann-type hydrodynamical equation”, J. Math. Phys., 62:4 (2021), 041502 | DOI | MR | Zbl

[23] Xia S., “Existence of a weak solution to a generalized Riemann-type hydrodynamical equation”, Appl. Anal., 102:18 (2023), 4997–5007 | DOI | MR