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Transactions of the Moscow Mathematical Society
Transactions of the Moscow Mathematical Society
ISSN 1547-738X(online) ISSN 0077-1554(print)

 

Stable pencils of hyperbolic polynomials and the Cauchy problem for hyperbolic equations with a small parameter at the highest derivatives


Authors: L. R. Volevich and E. V. Radkevich
Translated by: O. Khleborodova
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 65 (2004).
Journal: Trans. Moscow Math. Soc. 2004, 63-104
MSC (2000): Primary 35B25, 35L25.
Published electronically: November 4, 2004
MathSciNet review: 2193437
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Abstract | References | Similar Articles | Additional Information

Abstract: We study pencils of hyperbolic polynomials of the form ${\mathcal R}(\tau,\xi)= \sum_{j=0}^N(-i)^j\gamma_j P_j(\tau,\xi)$, where $P_j(\tau,\xi)$is a real homogeneous polynomials of degree $m-j$ resolved with respect to the highest power of $\tau$ and $P_j(1,0)=1$; the numbers $\gamma_0,\dots ,\gamma_N$ are positive. In the first part of the paper we find necessary and close to sufficient conditions of stability of the polynomial ${\mathcal R}(\tau,\xi)$ (i.e., the condition that its roots $\tau_j(\xi)$ lie in the open upper half-plane of the complex plane). This problem is closely related to the problem on uniform (with respect to a small parameter) estimates for the solution of the Cauchy problem for hyperbolic equations with a small parameter. The latter problem (both for constant and variable coefficients) is the topic of the second part of the paper.


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  • 1. Harold Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math. 2 (1949), 331–407. MR 0033674 (11,473a)
  • 2. E. V. Radkevich, Well-posedness of mathematical models in continuum mechanics and thermodynamics. Contemporary Mathematics. Fundamental Directions. 2003. Vol. 3. pp. 5-32.
  • 3. M. M. Postnikov, Stable polynomials, URSS, Moscow, 2004. (Russian)
  • 4. L. R. Volevich and M. G. Dzhavadov, Uniform estimates for solutions of hyperbolic equations with a small parameter multiplying the highest derivatives, Differentsial′nye Uravneniya 19 (1983), no. 12, 2082–2090 (Russian). MR 729566 (85j:35020)
  • 5. L. R. Volevich and E. V. Radkevich, Uniform estimates for solutions of the Cauchy problem for hyperbolic equations with a small parameter at the highest derivative, Differentsial'nye Uravneniya 39 (2003), no. 4, 486-499; English transl. in Differential Equations 39 (2003).
  • 6. Ch. Hermite, Oeuvres I. Paris, 1905, pp. 397-414.
  • 7. Lars Hörmander, Linear partial differential operators, Die Grundlehren der mathematischen Wissenschaften, Bd. 116, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0161012 (28 #4221)
  • 8. M. I. Višik and L. A. Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameter, Uspehi Mat. Nauk (N.S.) 12 (1957), no. 5(77), 3–122 (Russian). MR 0096041 (20 #2539)
  • 9. S. G. Gindikin and L. R. Volevich, Mixed problem for partial differential equations with quasihomogeneous principal part, Translations of Mathematical Monographs, vol. 147, American Mathematical Society, Providence, RI, 1996. Translated from the Russian manuscript by V. M. Volosov. MR 1357662 (96j:35145)
  • 10. L. R. Volevich, The Vishik-Lyusternik method in general elliptic boundary value problems with small parameter, Keldysh Inst. Appl. Math. RAS, 2002, Preprint no. 26.
  • 11. S. Gindikin and L. R. Volevich, The method of Newton’s polyhedron in the theory of partial differential equations, Mathematics and its Applications (Soviet Series), vol. 86, Kluwer Academic Publishers Group, Dordrecht, 1992. Translated from the Russian manuscript by V. M. Volosov. MR 1256484 (95a:35001)

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Additional Information

L. R. Volevich
Affiliation: Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow 125047, Russia
Email: volevich@spp.keldysh.ru

E. V. Radkevich
Affiliation: Moscow State University, Mechanics and Mathematics Department, Moscow 119899, Russia
Email: evrad@land.ru

DOI: http://dx.doi.org/10.1090/S0077-1554-04-00147-5
PII: S 0077-1554(04)00147-5
Published electronically: November 4, 2004
Additional Notes: The first author was supported by the Russian Foundation for Basic Research (Grant 03–01–00189) and INTAS (Project no. 899). The second author was supported by the Russian Foundation for Basic Research (Grant 03–01–00189).
Article copyright: © Copyright 2004 American Mathematical Society