Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-21T18:16:59.343Z Has data issue: false hasContentIssue false
  • English
  • Français

Distributions as initial values in a triangular hyperbolic system of conservation laws

Part of: Hyperbolic equations and systems Distributions, generalized functions, distribution spaces Representations of solutions

Published online by Cambridge University Press:  19 July 2019

C. O. R. Sarrico  and
A. Paiva
C. O. R. Sarrico
Affiliation:
CMAFCIO, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal (corsarrico@gmail.com; paiva@cii.fc.ul.pt)
A. Paiva
Affiliation:
CMAFCIO, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal (corsarrico@gmail.com; paiva@cii.fc.ul.pt)
Get access Rights & Permissions [Opens in a new window]

Abstract

The present paper concerns the system ut + [ϕ(u)]x = 0, vt + [ψ(u)v]x = 0 having distributions as initial conditions. Under certain conditions, and supposing ϕ, ψ: ℝ → ℝ functions, we explicitly solve this Cauchy problem within a convenient space of distributions u,v. For this purpose, a consistent extension of the classical solution concept defined in the setting of a distributional product (not constructed by approximation processes) is used. Shock waves, δ-shock waves, and also waves defined by distributions that are not measures are presented explicitly as examples. This study is carried out without assuming classical results about conservation laws. For reader's convenience, a brief survey of the distributional product is also included.

Keywords

Products of distributionsconservation lawstriangular systemgeneralized pressureless gas dynamics systemδ-shock wavesδ′-shock wavesCauchy problem

MSC classification

Primary: 46F10: Operations with distributions
Secondary: 35D99: None of the above, but in this section 35L67: Shocks and singularities
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 2757 - 2775
Check for updates
Copyright
Copyright © 2019 The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Bressan, A. and Rampazzo, F.. On differential systems with vector valued impulsive controls. Bull. Un. Mat. Ital. 2B (1988), 641–656.Google Scholar
2Chen, G.-Q. and Liu, H.. Formation of δ-shocks and vacuum states in the vanishing pressure limit of solutions to Euler equations for isentropic fluids. SIAM J. Math. Anal. 34 (2003), 925–938.Google Scholar
3Colombeau, J. F. and Le Roux, A.. Multiplication of distributions in elasticity and hydrodynamics. J. Math. Phys. 29 (1988), 315–319.CrossRefGoogle Scholar
4Dal Maso, G., Le Floch, P. and Murat, F.. Definitions and weak stability of nonconservative products. J. Math. Pure Appl. 74 (1995), 483–548.Google Scholar
5Danilov, V. G. and Mitrovic, D.. Delta shock wave formation in the case of triangular hyperbolic system of conservation laws. J. Differ. Equ. 245 (2008), 3704–3734.CrossRefGoogle Scholar
6Hilden, S. T., Nilsen, H. M. and Raynaud, X.. Study of the well-posedness of models for the inaccessible pore volume in polymer flooding. Transp. Porous Media 114 (2016), 65–86.CrossRefGoogle Scholar
7Huang, F.. Existence and uniqueness of discontinuous solutions for a hyperbolic system. Proc. R. Soc. Edinb. Sect. A-Math. 127 (1997), 1193–1205.10.1017/S0308210500027013CrossRefGoogle Scholar
8Kalisch, H. and Mitrovic, D.. Singular solutions of a fully nonlinear 2 × 2 system of conservation laws. Proc. Edinb. Math. Soc. (2) 55 (2012), 711–729.CrossRefGoogle Scholar
9Kalisch, H. and Teyekpiti, V.. A shallow-water system with vanishing buoyancy. Appl. Anal. (2018), 1–15, doi: 10.1080/00036811.2018.1546000.Google Scholar
10Kalisch, H., Mitrovic, D. and Teyekpiti, V.. Delta shock waves in shallow water flow. Phys. Lett. A 381 (2017), 1138–1144.CrossRefGoogle Scholar
11Keyfitz, B. L. and Kranzer, H. C.. A viscosity approximation to a system of conservation laws with no classical Riemann solution. In Nonlinear Hyperbolic Problems (Bordeaux, 1988), Lecture Notes in Math., vol. 1402, pp. 185–197 (Berlin and New York: Springer-Verlag, 1989).Google Scholar
12Keyfitz, B. L. and Kranzer, H. C.. A strictly hyperbolic system of conservation laws admitting singular shocks, In Nonlinear evolution equations that change type, IMA Vol. Math. Appl., 27, pp. 107–125 (New York, NY: Springer, 1990).10.1007/978-1-4613-9049-7CrossRefGoogle Scholar
13Keyfitz, B. L. and Kranzer, H. C.. Spaces of weighted measures for conservation laws with singular shock solutions. J. Differ. Equ. 118 (1995), 420–451.10.1006/jdeq.1995.1080CrossRefGoogle Scholar
14Korchinski, D. J.. Solution of a Riemann problem for a 2 × 2 system of conservation laws possessing no classical weak solution, Ph.D. Thesis, Adelphi University (1977).Google Scholar
15Mazzotti, M., Tarafder, A., Cornel, J., Gritti, F. and Guiochon, G.. Experimental evidence of a δ-shock in nonlinear chromatography. J. Chromatogr. A 1217 (2010), 2002–2012.CrossRefGoogle ScholarPubMed
16Mitrovic, D., Bojkovic, V. and Danilov, V. G.. Linearization of the Riemann problem for a triangular system of conservation laws and delta shock wave formation process. Math. Methods Appl. Sci. 33 (2010), 904–921.Google Scholar
17Nedeljkov, M.. Shadow waves: entropies and interactions for delta and singular shocks. Arch. Ration. Mech. Anal. 197 (2010), 489–537.10.1007/s00205-009-0281-2CrossRefGoogle Scholar
18Panov, E. Yu. and Shelkovich, V. M.. δ′-shock waves as a new type of solutions to systems of conservation laws. J. Differ. Equ. 228 (2006), 49–86.10.1016/j.jde.2006.04.004CrossRefGoogle Scholar
19Sarrico, C. O. R.. About a family of distributional products important in the applications. Port. Math. 45 (1988), 295–316.Google Scholar
20Sarrico, C. O. R.. Global solutions of first order linear systems of ordinary differential equations with distributional coefficients. J. Math. Anal. Appl. 276 (2002), 611–627.10.1016/S0022-247X(02)00368-2CrossRefGoogle Scholar
21Sarrico, C. O. R.. Distributional products and global solutions for nonconservative inviscid Burgers equation. J. Math. Anal. Appl. 281 (2003), 641–656.CrossRefGoogle Scholar
22Sarrico, C. O. R.. New solutions for the one-dimensional nonconservative inviscid Burgers equation. J. Math. Anal. Appl. 317 (2006), 496–509.10.1016/j.jmaa.2005.06.037CrossRefGoogle Scholar
23Sarrico, C. O. R.. Collision of delta waves in a turbolent model studied via a distributional product. Nonlinear Anal-Theor. 73 (2010), 2868–2875.CrossRefGoogle Scholar
24Sarrico, C. O. R.. Products of distributions and singular travelling waves as solutions of advection-reaction equations. Russ. J. Math. Phys. 19 (2012), 244–255.CrossRefGoogle Scholar
25Sarrico, C. O. R.. Products of distributions, conservation laws and the propagation of δ′-shock waves. Chin. Ann. Math. Ser. B 33 (2012), 367–384.10.1007/s11401-012-0713-4CrossRefGoogle Scholar
26Sarrico, C. O. R.. The multiplication of distributions and the Tsodyks model of synapses dynamics. Int. J. Math. Anal. 6 (2012), 999–1014.Google Scholar
27Sarrico, C. O. R.. A distributional product approach to δ-shock wave solutions for a generalized pressureless gas dynamics system. Int. J. Math. 25 (2014), 1450007 (12 pages).10.1142/S0129167X14500074CrossRefGoogle Scholar
28Sarrico, C. O. R.. New distributional global solutions for the Hunter-Saxton equation. Abstr. Appl. Anal. (2014), 9, Art. ID 809095, doi: 10.1155/2014/809095.Google Scholar
29Sarrico, C. O. R.. The Brio system with initial conditions involving Dirac masses: a result afforded by a distributional product. Chin. Ann. Math. 35B (2014), 941–954. doi: 10.1007/s11401-014-0862-8.Google Scholar
30Sarrico, C. O. R.. The Riemann problem for the Brio system: a solution containing a Dirac mass obtained via a distributional product. Russ. J. Math. Phys. 22 (2015), 518–527.10.1134/S1061920815040111CrossRefGoogle Scholar
31Sarrico, C. O. R.. Multiplication of distributions and a nonlinear model in elastodynamics. Pac. J. Math. 294 (2018), 195–212. doi: 10.2140/pjm.2018.294.195.CrossRefGoogle Scholar
32Sarrico, C. O. R.. Multiplication of distributions and travelling wave solutions for the Keyfitz–Kranzer system. Taiwan. J. Math. 22 (2018), 677–693.Google Scholar
33Sarrico, C. O. R. and Paiva, A.. Products of distributions and collision of a δ-wave with a δ′-wave in a turbulent model. J. Nonlinear Math. Phys. 22 (2015), 381–394.CrossRefGoogle Scholar
34Sarrico, C. O. R. and Paiva, A.. Delta shock waves in the shallow water system. J. Dyn. Differ. Equ. 30 (2018), 1187–1198.CrossRefGoogle Scholar
35Sarrico, C. O. R. and Paiva, A.. New distributional travelling waves for the nonlinear Klein–Gordon equation. Differ. Integral Equ. 30 (2017), 853–878.Google Scholar
36Sarrico, C. O. R. and Paiva, A.. The multiplication of distributions in the study of a Riemann problem in fluid dynamics. J. Nonlinear Math. Phys. 24 (2017), 328–345.10.1080/14029251.2017.1341696CrossRefGoogle Scholar
37Sarrico, C. O. R. and Paiva, A.. Newton's second law and the multiplication of distributions. J. Math. Phys. 59 (2018), 013505. doi: 10.1063/1.5021949.CrossRefGoogle Scholar
38Schwartz, L.. Théorie des Distributions (Hermann, Paris, 1966).Google Scholar
39Shelkovich, V. M.. The Riemann problem admitting δ-,δ′-shocks and vacuum states, the vanishing viscosity approach. J. Differ. Equ. 231 (2006), 459–500.CrossRefGoogle Scholar
40Shen, C. and Sun, M.. A distributional product approach to the delta shock wave solution for the one dimensional zero pressure gas dynamic system. Int. J. Non-Linear Mech. 105 (2018), 105–112.CrossRefGoogle Scholar
41Sun, M.. The multiplication of distributions in the study of delta shock wave for the nonlinear chromatography system. Appl. Math. Lett. (2019), 61–68.10.1016/j.aml.2019.04.015CrossRefGoogle Scholar