On the evolution of the hierarchy of shock waves in a~two-dimensional isobaric medium

Yu. G. Rykov

Izvestiya. Mathematics , Tome 88 (2024) no. 2, pp. 284-312.

Résumé

In the proposed paper, the process of propagation of shock waves in two-dimensional media without its own pressure drop is studied. The model of such media is a system of equations of gas dynamics, where formally the pressure is assumed to be zero. From the point of view of the theory of systems of conservation laws, the system of equations under consideration is in some sense degenerate, and, consequently, the corresponding generalized solutions have strong singularities (evolving shock waves with density in the form of delta functions on manifolds of different dimensions). We will denote this property as the evolution of the hierarchy of strong singularities or the evolution of the hierarchy of shock waves. In the paper, in the two-dimensional case, the existence of such an interaction of strong singularities with density delta function along curves in the space $\mathbb{R}^2$ is proved, at which a density concentration occurs at a point, that is, a hierarchy of shock waves arises. The properties of such dynamics of strong singularities are described. The results obtained provide a starting point for moving on to a much more interesting multidimensional case in the future.

Détail

Keywords: conservation laws, pressureless medium, shock waves, shock waves hierarchy, Rankine–Hugoniot conditions, matter concentration.

Bibliographie

[1] G. G. Chernyi, Introduction to hypersonic flow, Academic Press, New York–London, 1961 | MR

[2] L. I. Sedov, Similarity and dimensional methods in mechanics, Academic Press, New York–London, 1959 | MR | Zbl

[3] K. P. Stanyukovich, Unsteady motions of a continuous medium, Nauka, Moscow, 1971 (Russian) | MR | Zbl

[4] Ya. B. Zel'dovich, “Gravitational instability: an approximate theory for large density perturbations”, Astron. Astrophys., 5 (1970), 84–89

[5] S. N. Gurbatov, A. I. Saichev, and S. F. Shandarin, “Large-scale structure of the Universe. The Zeldovich approximation and the adhesion model”, Phys. Usp., 55:3 (2012), 223–249 | DOI

[6] L. V. Ovsyannikov, “Isobaric gas motions”, Differ. Equ., 30:10 (1994), 1656–1662

[7] A. P. Chupakhin, “On barochronic gas motions”, Dokl. Phys., 42:2 (1997), 101–104 | Zbl

[8] A. P. Chupakhin, Baroochronous motions of a gas, Author's abstract diss. doct. fiz.-mat. nauk, Lavrentyev Institute of Hydrodynamics of Siberian Branch of the RAS, Novosibirsk, 1999 (Russaian)

[9] A. N. Kraĭko, “On discontinuity surfaces in a medium devoid of “proper” pressure”, J. Appl. Math. Mech., 43:3 (1979), 539–549 | DOI | Zbl

[10] A. N. Kraiko and S. M. Sulaimanova, “Two-fluid flows of a mixture of a gas and solid particles with “films” and “filaments” appearing in flows past impermeable surfaces”, J. Appl. Math. Mech., 47:4 (1983), 507–516 | DOI | Zbl

[11] A. N. Kraiko, “The Mathematical Models for Description of Flow of Gas and Foreign Particles and for Non-Stationary Filtration of Liquids and Gas in Porous Medium”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 7:1 (2014), 34–48 | DOI | Zbl

[12] M. Y. Nemtsev, I. S. Menshov, and I. V. Semenov, “Numerical simulation of dynamic processes in the medium of fine-grained solid particles”, Math. Models Comput. Simul., 15:2 (2023), 210–226 | DOI

[13] Yu. G. Rykov, “A variational principle for a two-dimensional system of equations of gas dynamics without stress”, Russian Math. Surveys, 51:1 (1996), 162–164 | DOI

[14] A. I. Aptekarev and Yu. G. Rykov, “Emergence of a hierarchy of singularities in zero-pressure media. Two-dimensional case”, Math. Notes, 112:4 (2022), 495–504 | DOI

[15] N. V. Klyushnev and Yu. G. Rykov, “On model two-dimensional pressureless gas flows: variational description and numerical algorithm based on adhesion dynamics”, Comput. Math. Math. Phys., 63:4 (2023), 606–622 | DOI | MR

[16] F. Bouchut, “On zero pressure gas dynamics”, Advances in kinetic theory and computing, Ser. Adv. Math. Appl. Sci., 22, World Sci. Publ., River Edge, NJ, 1994, 171–190 | DOI | MR | Zbl

[17] I. Veinan, Yu. G. Rykov, and Ya. G. Sinai, “The Lax–Oleinik variational principle for some one-dimensional systems of quasilinear equations”, Russian Math. Surveys, 50:1 (1995), 220–222 | DOI

[18] E. Grenier, “Existence globale pour la système des gaz sans pression”, C. R. Acad. Sci. Paris Sér. I Math., 321:2 (1995), 171–174 | MR | Zbl

[19] Weinan E, Yu. G. Rykov, and Ya. G. Sinai, “Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics”, Comm. Math. Phys., 177:2 (1996), 349–380 | DOI | MR | Zbl

[20] Yu. G. Rykov, “Solutions with substance decay in pressureless gas dynamics systems”, Math. Notes, 108:3 (2020), 465–468 | DOI

[21] N. V. Klyushnev and Yu. G. Rykov, “Non-conventional and conventional solutions for one-dimensional pressureless gas”, Lobachevskii J. Math., 42:11 (2021), 2615–2625 | DOI | MR | Zbl

[22] Feimin Huang and Zhen Wang, “Well posedness for pressureless flow”, Comm. Math. Phys., 222:1 (2001), 117–146 | DOI | MR | Zbl

[23] Jiequan Li and G. Warnecke, “Generalized characteristics and the uniqueness of entropy solutions to zero-pressure gas dynamics”, Adv. Differential Equations, 8:8 (2003), 961–1004 | MR | Zbl

[24] R. Hynd, “Sticky particle dynamics on the real line”, Notices Amer. Math. Soc., 66:2 (2019), 162–168 | DOI | MR | Zbl

[25] R. Hynd, “A trajectory map for the pressureless Euler equations”, Trans. Amer. Math. Soc., 373:10 (2020), 6777–6815 | DOI | MR | Zbl

[26] Jiequan Li, Tong Zhang, and Shuli Yang, The two-dimensional Riemann problem in gas dynamics, Pitman Monogr. Surveys Pure Appl. Math., 98, Longman, Harlow, 1998 | DOI | MR | Zbl

[27] J. F. Colombeau, Elementary introduction to new generalized functions, North-Holland Math. Stud., 113, Notes on Pure Math., 103, North-Holland Publishing Co., Amsterdam, 1985 | MR | Zbl

[28] Yu. G. Rykov, “The Singularities of Type of Shock Waves in Pressureless Medium, the Solutions in the Sense of Measures and Kolombo's Sense”, Keldysh Institute preprints, 1998, 030

[29] Yu. G. Rykov, “On the nonhamiltonian character of shocks in 2-D pressureless gas”, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 5:1 (2002), 55–78 | MR | Zbl

[30] A. I. Aptekarev and Yu. G. Rykov, “Variational principle for multidimensional conservation laws and pressureless media”, Russian Math. Surveys, 74:6 (2019), 1117–1119 | DOI

[31] A. I. Aptekarev and Yu. G. Rykov, “Detailed description of the evolution mechanism for singularities in the system of pressureless gas dynamics”, Dokl. Math., 99:1 (2019), 79–82 | DOI

[32] Yicheng Pang, “The Riemann problem for the two-dimensional zero-pressure Euler equations”, J. Math. Anal. Appl., 472:2 (2019), 2034–2074 | DOI | MR | Zbl

[33] Jiequan Li and Hanchun Yang, “Delta-shocks as limits of vanishing viscosity for multidimensional zero-pressure gas dynamics”, Quart. Appl. Math., 59:2 (2001), 315–342 | DOI | MR | Zbl

[34] V. M. Shelkovich, “$\delta$- and $\delta'$-shock wave types of singular solutions of systems of conservation laws and transport and concentration processes”, Russian Math. Surveys, 63:3 (2008), 473–546 | DOI

[35] S. Albeverio, O. S. Rozanova, and V. M. Shelkovich, Transport and concentration processes in the multidimensional zero-pressure gas dynamics model with the energy conservation law, arXiv: 1101.5815

[36] K. Khanin and A. Sobolevski, “Particle dynamics inside shocks in Hamilton–Jacobi equations”, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 368:1916 (2010), 1579–1593 | DOI | MR | Zbl

[37] K. Khanin and A. Sobolevski, “On dynamics of Lagrangian trajectories for Hamilton–Jacobi equations”, Arch. Ration. Mech. Anal., 219:2 (2016), 861–885 | DOI | MR | Zbl

[38] M. Sever, “An existence theorem in the large for zero-pressure gas dynamics”, Differential Integral Equations, 14:9 (2001), 1077–1092 | DOI | MR | Zbl

[39] A. Bressan and Truyen Nguyen, “Non-existence and non-uniqueness for multidimensional sticky particle systems”, Kinet. Relat. Models, 7:2 (2014), 205–218 | DOI | MR | Zbl

[40] S. Bianchini and S. Daneri, On the sticky particle solutions to the multi-dimensioanl pressureless Euler equations, arXiv: 2004.06557

[41] Yu. G. Rykov, “2D pressureless gas dynamics and variational principle”, Keldysh Institute preprints, 2016, 094 | DOI

[42] Yu. G. Rykov, “On the interaction of shock waves in two-dimensional isobaric media”, Russian Math. Surveys, 78:4 (2023), 779–781 | DOI

[43] Yu. G. Rykov, “Concentration processes in a two dimensional system of gas dynamics equations without pressure”, Sb. Tez. 2nd Matem. Konf. Centrov, Russia Nov. 7–11, 2022 (Moscow State University), Moscow State University Press, Moscow, 2022, 192–194

[44] A. Chertock, A. Kurganov, and Yu. Rykov, “A new sticky particle method for pressureless gas dynamics”, SIAM J. Numer. Anal., 45:6 (2007), 2408–2441 | DOI | MR | Zbl