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The generalized Cauchy problem with data on two surfaces for a quasilinear analytic system

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Abstract

We consider the generalized Cauchy problem with data on two surfaces for a second-order quasilinear analytic system. The distinction of the generalized Cauchy problem from the traditional statement of the Cauchy problem is that the initial conditions for different unknown functions are given on different surfaces: for each unknown function we pose its own initial condition on its own coordinate axis. Earlier, the generalized Cauchy problem was considered in the works of C. Riquier, N. M. Gyunter, S. L. Sobolev, N. A. Lednev, V. M. Teshukov, and S. P. Bautin. In this article we construct a solution to the generalized Cauchy problem in the case when the system of partial differential equations additionally contains the values of the derivatives of the unknown functions (in particular outer derivatives) given on the coordinate axes. The last circumstance is a principal distinction of the problem in the present article from the generalized Cauchy problems studied earlier.

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References

  1. Riquier C., Les systemes d’equations aux derivees partielles, Gauthier-Villars, Paris (1910).

    Google Scholar 

  2. Finikov S. P., Cartan’s Method of Exterior Forms [in Russian], OGIZ, Moscow; Leningrad (1948).

    Google Scholar 

  3. Gyunter N. M., “On analytic solutions of the equation \(\tfrac{{\partial ^2 u}}{{\partial u\partial y}} = f\left( {x,y,u,\tfrac{{\partial u}}{{\partial x}},\tfrac{{\partial u}}{{\partial y}},\tfrac{{\partial ^2 u}}{{\partial x^2 }},\tfrac{{\partial ^2 u}}{{\partial y^2 }}} \right)\),” Mat. Sb., 32, 26–42 (1925).

    Google Scholar 

  4. Gyunter N. M., “On extension of the Cauchy theorem to an arbitrary system of partial differential equations,” Mat. Sb., 32, 367–447 (1925).

    Google Scholar 

  5. Sobolev S. L., “On analytic solutions of systems of partial differential equations in two independent variables,” Mat. Sb., 38, No. 1–2, 107–147 (1931).

    Google Scholar 

  6. Sobolev S. L., “To the question on analytic solutions of systems of partial differential equations in two independent variables,” Trudy Fiz.-Mat. Inst. V. A. Steklov, 5, 265–282 (1934).

    Google Scholar 

  7. Lednev N. A., “A new method for solving partial differential equations,” Mat. Sb., 22, No. 2, 205–266 (1948).

    MathSciNet  Google Scholar 

  8. Bautin S. P., “The Cauchy problem with initial data on different surfaces,” Dokl. Ross. Akad. Nauk, 345, No. 5, 586–589 (1995).

    MathSciNet  Google Scholar 

  9. Bautin S. P., “The Cauchy problem with initial data on different surfaces for a quasilinear analytic system,” Differentsial’nye Uravneniya, 32, No. 6, 804–813 (1996).

    MathSciNet  Google Scholar 

  10. Bautin S. P. and Kazakov A. L., “A Cauchy problem with initial data on different surfaces for a system with singularity,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 10, 13–23 (1997).

  11. Teshukov V. M., “Destruction of an arbitrary discontinuity on a curvilinear surface,” Prikl. Mekh. i Tekhn. Fizika, No. 2, 126–133 (1980).

  12. Teshukov V. M., “Construction of the shock wave front in a spatial piston problem,” Dinamika Sploshn. Sredy, No. 33, 114–133 (1978).

  13. Teshukov V. M., “Regular reflection of a shock wave from a rigid wall,” Prikl. Mat. i Mekh., 46, No. 2, 225–234 (1982).

    MathSciNet  Google Scholar 

  14. Teshukov V. M., “Spatial interaction of strong discontinuities in a gas,” Prikl. Mat. i Mekh., 50, No. 4, 225–234 (1986).

    MathSciNet  Google Scholar 

  15. Bautin S. P. and Kazakov A. L., “Gas flows with shock waves which diverge from an axis or center of symmetry with finite velocity,” Prikl. Mat. i Mekh., 60, No. 3, 465–474 (1996).

    MathSciNet  Google Scholar 

  16. Kazakov A. L., “Construction of piecewise analytic gas flows join by shock waves near an axis or center of symmetry,” Prikl. Mekh. i Tekhn. Fizika, 39, No. 5, 25–38 (1998).

    MATH  MathSciNet  Google Scholar 

  17. Bautin S. P., Analytical Heat Wave [in Russian], Fizmatlit, Moscow (2003).

    Google Scholar 

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Authors and Affiliations

  1. Ural State University of Railway Transportation, Ekaterinburg, Russia

    A. L. Kazakov

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  1. A. L. Kazakov
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Correspondence to A. L. Kazakov.

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__________

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 5, pp. 1041–1055, September–October, 2007.

Original Russian Text Copyright © 2007 Kazakov A. L.

The author was supported by the Russian Foundation for Basic Research (Grants 02-01-011-22 and 04-01-00-205).

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Kazakov, A.L. The generalized Cauchy problem with data on two surfaces for a quasilinear analytic system. Sib Math J 48, 837–848 (2007). https://doi.org/10.1007/s11202-007-0085-2

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  • DOI: https://doi.org/10.1007/s11202-007-0085-2

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