The Riemann problem for the main model cases of the Euler-Poisson equations
- Authors: Gargyants L.V.1, Rozanova O.S.2, Turzynsky M.K.3,4
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Affiliations:
- Bauman Moscow State Technical University
- Lomonosov Moscow State University
- Russian University of Transport
- National Research University “Higher School of Economics” (HSE University)
- Issue: Vol 70, No 1 (2024): Functional spaces. Differential operators. Problems of mathematics education
- Pages: 38-52
- Section: Articles
- URL: https://journals.rudn.ru/CMFD/article/view/38695
- DOI: https://doi.org/10.22363/2413-3639-2024-70-1-38-52
- EDN: https://elibrary.ru/YYQSXD
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Abstract
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.
About the authors
L. V. Gargyants
Bauman Moscow State Technical University
Author for correspondence.
Email: gargyants@bmstu.ru
Moscow, Russia
O. S. Rozanova
Lomonosov Moscow State University
Email: rozanova@mech.math.msu.su
Moscow, Russia
M. K. Turzynsky
Russian University of Transport; National Research University “Higher School of Economics” (HSE University)
Email: M13041@yandex.ru
Moscow, Russia
References
- Гуревич А.В., Зыбин К.П. Недиссипативная гравитационная турбулентность// Ж. эксперимент. и теор. физ.-1988.- 94, № 1.-С. 3-25.
- Кочин Н.Е. Векторное исчисление и начала тензорного анализа.-М.: Наука, 1965.
- Рождественский Б.Л., Яненко Н.Н. Системы квазилинейных уравнений и их приложения к газовой динамике. -М.: Наука, 1978.
- Чижонков Е.В. Математические аспекты моделирования колебаний и кильватерных волн в плазме. -М.: Физматлит, 2018.
- Шелкович В.М. Сингулярные решения систем законов сохранения типа-ударных волн и процессы переноса и концентрации// Усп. мат. наук.-2008.-63, № 3.- C. 73-146.
- Brunelli J.C., Das A. On an integrable hierarchy derived from the isentropic gas dynamics// J. Math. Phys. -2004.-45, № 7.- С. 2633-2645.
- Chae D., Tadmor E. On the finite time blow-up of the Euler-Poisson equations in Rn// Commun. Math. Sci. -2008.- 6, № 3.- С. 785-789.
- Dafermos C.M. Hyperbolic conservation laws in continuum physics.- Berlin-Heidelberg: Springer, 2016.
- Engelberg S., Liu H., Tadmor E. Critical thresholds in Euler-Poisson equations// Indiana Univ. Math. J.- 2001.-50, № 1.- С. 109-157.
- Gao B., Tian K., Liu Q.P., Feng L. Conservation laws of the generalized Riemann equations// J. Nonlinear Math. Phys.- 2018.- 25, № 1.-С. 122-135.
- Huang F., Wang Zh. Well posedness for pressureless flow// Commun. Math. Phys.- 2001.- 222, № 1.- С. 117-146.
- Hunter J.K., Saxton R. Dynamics of director fields// SIAM J. Appl. Math.- 1991.- 51, № 6.- С. 1498- 1521.
- Pavlov M. V. The Gurevich-Zybin system// J. Phys. A: Math. Gen. -2005.- 38, № 17.- С. 3823-3840.
- Popowicz Z., Prykarpatski A.K. The non-polynomial conservation laws and integrability analysis of generalized Riemann type hydrodynamical equations// Nonlinearity.-2010.- 23, № 10.-С. 2517-2537.
- Rozanova O.S. On the behavior of multidimensional radially symmetric solutions of the repulsive Euler- Poisson equations// Phys. D: Nonlinear Phenom. -2022.-443.- 133578.
- Rozanova O.S. The Riemann problem for equations of a cold plasma// J. Math. Anal. Appl. - 2023.- 527, № 1, Part 1.- 127400.
- Rozanova O.S., Turzynsky M.K. On the properties of affine solutions of cold plasma equations// Commun. Math. Sci.- 2024.- 22, № 1.- С. 215-226.
- Sch¨afer T., Wayne C.E. Propagation of ultra-short optical pulses in cubic nonlinear media// Phys. D.: Nonlinear Phenom. -2004.- 196, № 1.-С. 90-105.
- Tan C. Eulerian dynamics in multidimensions with radial symmetry// SIAM J. Math. Anal. -2021.- 53, № 3. -С. 3040-3071.
- Wei D., Tadmor E., Bae H. Critical thresholds in multi-dimensional Euler-Poisson equations with radial symmetry// Commun. Math. Sci. -2012.- 10, № 1.-С. 75-86.
- Wei L. Wave breaking, global existence and persistent decay for the Gurevich-Zybin system// J. Math. Fluid Mech. -2020.- 22, № 4.- С. 1-14.
- Wei L., Wang Y. The Cauchy problem for a generalized Riemann-type hydrodynamical equation// J. Math. Phys. -2021.-62, № 4.- 041502.
- Xia S. Existence of a weak solution to a generalized Riemann-type hydrodynamical equation// Appl. Anal. -2023.-102, № 18.- С. 4997-5007.
