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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 Hugoniуt 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
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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  mathnet  crossref  crossref  mathscinet  zmath
    3. Dobrokhotov, SY, “On Maslov's conjecture about the structure of weak point singularities of shallow-water equations”, Doklady Mathematics, 64:1 (2001), 127  mathscinet  zmath  isi
    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  mathscinet  zmath  isi
    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  mathscinet  isi
    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus  scopus
    8. Smolyanov, OG, “Multiplicative structures in the linear space of vector-valued distributions”, Doklady Mathematics, 65:2 (2002), 169  mathscinet  zmath  isi
    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  mathnet  crossref  mathscinet  zmath
    10. Albeverio, S, “Associated homogeneous p-adic distributions”, Doklady Mathematics, 68:3 (2003), 354  isi
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    19. Danilov, VG, “Delta-shock wave type solution of hyperbolic systems of conservation laws”, Quarterly of Applied Mathematics, 63:3 (2005), 401  crossref  mathscinet  isi  scopus  scopus
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    22. Albeverio, S, “p-adic Colombeau-Egorov type theory of generalized functions”, Mathematische Nachrichten, 278:1–2 (2005), 3  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    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  crossref  mathscinet  zmath  isi
    24. V. M. Shelkovich, “The Rankine–Hugoniot conditions and balance laws for $\delta$-shocks”, J. Math. Sci., 151:1 (2008), 2781–2792  mathnet  crossref  mathscinet  zmath  elib  elib
    25. K. A. Volosov, “Eigenfunctions of structures described by the “shallow water” model in a plane”, J. Math. Sci., 151:1 (2008), 2639–2650  mathnet  crossref  mathscinet  zmath  elib  elib
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  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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