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 TMF, 1998, Volume 114, Number 1, Pages 3–55 (Mi tmf827)

This article is cited in 38 scientific papers (total in 38 papers)

Algebras of the singularities of singular solutions to first-order quasi-linear strictly hyperbolic systems

V. G. Danilova, V. P. Maslovb, V. M. Shelkovichc

a Moscow State Institute of Electronics and Mathematics (Technical University)
b M. V. Lomonosov Moscow State University
c St. Petersburg State University of Architecture and Civil Engineering

Abstract: An associative commutative algebra of distributions that contains homogeneous and associated homogeneous distributions is constructed. This algebra is used to analyze generalized solutions to strictly hyperbolic partial differential equations. Possible types of singularities are studied and the necessary (analogues of Hugoniót conditions for shock waves) and sufficient conditions for the existence of such solutions are obtained.

DOI: https://doi.org/10.4213/tmf827

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English version:
Theoretical and Mathematical Physics, 1998, 114:1, 1–42

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Received: 29.07.1997

Citation: V. G. Danilov, V. P. Maslov, V. M. Shelkovich, “Algebras of the singularities of singular solutions to first-order quasi-linear strictly hyperbolic systems”, TMF, 114:1 (1998), 3–55; Theoret. and Math. Phys., 114:1 (1998), 1–42

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
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2. S. Yu. Dobrokhotov, “Integrability of truncated Hugoniot–Maslov chains for trajectories of mesoscale vortices on shallow water”, Theoret. and Math. Phys., 125:3 (2000), 1724–1741
3. Dobrokhotov, SY, “On Maslov's conjecture about the structure of weak point singularities of shallow-water equations”, Doklady Mathematics, 64:1 (2001), 127
4. Dobrokhotov, SY, “Proof of Maslov's conjecture about the structure of weak point singular solutions of the shallow water equations”, Russian Journal of Mathematical Physics, 8:1 (2001), 25
5. Danilov V.G., Shelkovich V.M., “Propagation and interaction of nonlinear waves to quasilinear equations”, Hyperbolic Problems: Theory, Numerics, Applications, International Series of Numerical Mathematics, 140, 2001, 267–276
6. Khrennikov, AY, “Locally convex spaces of vector-valued distributions with multiplicative structures”, Infinite Dimensional Analysis Quantum Probability and Related Topics, 5:4 (2002), 483
7. Danilov, VG, “Dynamics of the interface between two immiscible liquids with nearly equal densities under gravity”, European Journal of Applied Mathematics, 13 (2002), 497
8. Smolyanov, OG, “Multiplicative structures in the linear space of vector-valued distributions”, Doklady Mathematics, 65:2 (2002), 169
9. S. Yu. Dobrokhotov, E. S. Semenov, B. Tirozzi, “Hugoniót–Maslov Chains for Singular Vortical Solutions to Quasilinear Hyperbolic Systems and Typhoon Trajectory”, Journal of Mathematical Sciences, 124:5 (2004), 5209–5249
10. Albeverio, S, “Associated homogeneous p-adic distributions”, Doklady Mathematics, 68:3 (2003), 354
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14. S. Yu. Dobrokhotov, E. S. Semenov, B. Tirozzi, “Calculation of Integrals of the Hugoniot–Maslov Chain for Singular Vortical Solutions of the Shallow-Water Equation”, Theoret. and Math. Phys., 139:1 (2004), 500–512
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16. Shvedov, OY, “Approximations for strongly singular evolution equations”, Journal of Functional Analysis, 210:2 (2004), 259
17. Danilov, VG, “Propagation and interaction of delta-shock waves for hyperbolic systems of conservation laws”, Doklady Mathematics, 69:1 (2004), 4
18. Shelkovich V.M., “Delta-shocks, the Rankine-Hugoniot conditions and singular superposition of distributions”, Days on Diffraction 2004, Proceedings, 2004, 175–196
19. Danilov, VG, “Delta-shock wave type solution of hyperbolic systems of conservation laws”, Quarterly of Applied Mathematics, 63:3 (2005), 401
20. Shelkovich, VM, “New versions of the Colombeau algebras”, Mathematische Nachrichten, 278:11 (2005), 1318
21. Danilov, VG, “Dynamics of propagation and interaction of delta-shock waves in conservation law systems”, Journal of Differential Equations, 211:2 (2005), 333
22. Albeverio, S, “p-adic Colombeau-Egorov type theory of generalized functions”, Mathematische Nachrichten, 278:1–2 (2005), 3
23. Dobrokhotov S., Tirozzi B., “A perturbative theory of the evolution of the center of typhoons”, Zeta Functions, Topology and Quantum Physics, Developments in Mathematics, 14, 2005, 31–50
24. V. M. Shelkovich, “The Rankine–Hugoniot conditions and balance laws for $\delta$-shocks”, J. Math. Sci., 151:1 (2008), 2781–2792
25. K. A. Volosov, “Eigenfunctions of structures described by the “shallow water” model in a plane”, J. Math. Sci., 151:1 (2008), 2639–2650
26. Kulagin, DA, “Interaction of kinks for semilinear wave equations with a small parameter”, Nonlinear Analysis-Theory Methods & Applications, 65:2 (2006), 347
27. Albeverio, S, “Associated homogeneous p-adic distributions”, Journal of Mathematical Analysis and Applications, 313:1 (2006), 64
28. Reutskiy, S, “Forecast of the trajectory of the center of typhoons and the Maslov decomposition”, Russian Journal of Mathematical Physics, 14:2 (2007), 232
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33. Sarrico C.O.R., Paiva A., “Products of distributions and collision of a ? -wave with a ?? -wave in a turbulent model”, J. Nonlinear Math. Phys., 22:3 (2015), 381–394
34. Sarrico C.O.R., Paiva A., “New Distributional Travelling Waves For the Nonlinear Klein-Gordon Equation”, Differ. Integral Equ., 30:11-12 (2017), 853–878
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