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Notion of blowup of the solution set of differential equations and averaging of random semigroups

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Abstract

We propose a unique approach to studying the violation of the well-posedness of initial boundary-value problems for differential equations. The blowup of the set of solutions of a problem for a differential equation is defined as a discontinuity of a multivalued map associating an initial boundary-value problem with the set of solutions of this problem. We show that such a definition not only describes effects of the solution destruction or its nonuniqueness but also permits prescribing a procedure for extending the solution through the singularity origination instant by using an appropriate random process. Considering the initial boundary-value problems whose solution sets admit singularities of the blowup type and a neighborhood of these problems in the space of problems permits associating the initial problem with the set of limit points of a sequence of solutions of the approximating problems. Endowing the space of problems with the structure of a space with measure, we obtain a random semigroup generated by the initial problem. We study the properties of the mathematical expectations (means) of a random semigroup and their equivalence in the sense of Chernoff to semigroups with averaged generators.

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References

  1. M. W. Hirsch and C. Pugh, “Stable manifolds and hyperbolic sets,” in: Global Analysis (Proc. Symp. Pure Math., Vol. 14, S.-S. Chern and S. Smale, eds.), Amer. Math. Soc., Providence, R. I. (1970), pp. 133–164.

    Article  MathSciNet  Google Scholar 

  2. J. Palis, Proc. Am. Math. Soc., 27, 85–90 (1971).

    MathSciNet  MATH  Google Scholar 

  3. M. Shub and S. Smail, Ann. Math., 96, 587–591 (1972).

    Article  MATH  Google Scholar 

  4. I. U. Bronshtein, Nonautonomous Dynamical Systems [in Russian], Shtiintsa, Kishinev (1984).

    Google Scholar 

  5. D. V. Anosov, S. Kh. Aranson, V. Z. Grines, R. V. Plykin, E. A. Sataev, A. V. Safonov, V. V. Solodov, A. N. Starkov, A. M. Stepin, and S. V. Shlyachkov, “Dynamical systems with hyperbolic behavior [in Russian],” in: Dynamical Systems–9 (Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., Vol. 66), VINITI, Moscow (1991), pp. 5–242.

    Google Scholar 

  6. L. S. Efremova, Vestn. NNGU im. N. I. Lobachevskogo, 3(1), 130–136 (2012).

    Google Scholar 

  7. L. S. Efremova, J. Math. Sci., 202, 794–808 (2014).

    Article  MATH  Google Scholar 

  8. V. A. Galaktionov, S. P. Kurdyumov, and A. G. Mikhailov, Peaking Regimes [in Russian], Nauka, Moscow (1986).

    Google Scholar 

  9. S. I. Pokhozhaev, Soviet Math. Dokl., 6, 1408–1411 (1965).

    MATH  Google Scholar 

  10. S. N. Kruzhkov, Lectures on Partial Differential Equations [in Russian], Moscow State Univ., Moscow (1970).

    Google Scholar 

  11. V. Zh. Sakbaev, “Cauchy problem for a linear differential equation and averaging of its approximating regularizations [in Russian],” in: Partial Differential Equations (SMFN, Vol. 43), RUDN, Moscow (2012), pp. 3–172.

    Google Scholar 

  12. O. A. Oleinik, Uspekhi Mat. Nauk, 12, No. 3(75), 3–73 (1957).

    MathSciNet  Google Scholar 

  13. V. Zh. Sakbaev, Dokl. Math., 77, 208–211 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  14. S. K. Godunov and E. I. Romensky, Elements of Continuum Mechanics and Conservation Laws [in Russian], Universitetskaya Seriya, Novosibirsk (1998)

    MATH  Google Scholar 

  15. S. K. Godunov and E. I. Romensky, English transl., Kluwer, New York (2003).

    Google Scholar 

  16. I. V. Volovich and V. Zh. Sakbaev, Proc. Steklov Inst. Math., 285, 56–80 (2014); arXiv:1312.4302v1 [math.AP] (2013).

    Article  MATH  Google Scholar 

  17. V. P. Burskii, Methods of Investigation of Boundary-Value Problems for General Differential Equations [in Russian], Naukova Dumka, Kiev (2002).

    Google Scholar 

  18. V. I. Oseledets, Theory Probab. Appl., 10, 499–504 (1965).

    Article  Google Scholar 

  19. L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York (1974).

    MATH  Google Scholar 

  20. M. Blank, Stability and Localization in Chaotic Dynamics [in Russian], MTsNMO, Moscow (2001).

    Google Scholar 

  21. Yu. N. Orlov, V. Zh. Sakbaev, and O. G. Smolyanov, Proc. Steklov Inst. Math., 285, 222–232 (2014).

    Article  MATH  Google Scholar 

  22. N. N. Bogoliubov, On Some Statistical Methods in Mathematical Physics [in Russian], Izdat. Akad. Nauk SSSR, Kiev (1945).

    Google Scholar 

  23. L. Accardi, Y. G. Lu, and I. V. Volovich, Quantum Theory and Its Stochastic Limit, Springer, Berlin (2002).

    Book  MATH  Google Scholar 

  24. L. Accardi, A. N. Pechen, and I. V. Volovich, J. Phys. A: Math. Gen., 35, 4889–4902 (2002); arXiv:quant-ph/0108112v2 (2001).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. O. G. Smolyanov and E. T. Shavgulidze, Path Integrals [in Russian], Moscow State Univ., Moscow (2015).

    Google Scholar 

  26. V. K. Ivanov, I. V. Mel’nikova, and A. I. Filinkov, Operator-Differential Equations and Ill-Posed Problems [in Russian], Fizmatlit, Moscow (1995).

    MATH  Google Scholar 

  27. A. B. Bakushinskii and M. Yu. Kokurin, Iteration Methods for Solving Ill-Posed Operator Equations with Smooth Operators [in Russian], URSS, Moscow (2002).

    Google Scholar 

  28. A. N. Tikhonov and V. Ya. Arsenin, Methods for the Solution of Ill-Posed Problems [in Russian], Nauka, Moscow (1986).

    Google Scholar 

  29. M. I. Vishik and V. V. Chepyzhov, Sb. Math., 200, 471–497 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  30. E. Mitidieri and S. I. Pokhozhaev, Proc. Steklov Inst. Math., 234, 1–362 (2001).

    MathSciNet  Google Scholar 

  31. V. Zh. Sakbaev, Proc. Steklov Inst. Math., 283, 165–180 (2013).

    Article  Google Scholar 

  32. V. A. Galaktionov and J. L. Vazquez, Arch. Rational Mech. Anal., 129, 225–244 (1995).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. V. Zh. Sakbaev, J. Math. Sci., 151, 2741–2753 (2008).

    Article  MathSciNet  Google Scholar 

  34. V. I. Bogachev, N. V. Krylov, and M. Röckner, Russ. Math. Surveys, 64, 973–1078 (2009).

    Article  ADS  MATH  Google Scholar 

  35. V. V. Zhikov, Funct. Anal. Appl., 35, 19–33 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  36. E. Yu. Panov, Matem. Mod., 14, No. 3, 17–26 (2002).

    MathSciNet  Google Scholar 

  37. R. J. Di Perna and P. Lions, Invent. Math., 98, 511–547 (1989).

    Article  ADS  MathSciNet  Google Scholar 

  38. N. Mizoguchi, F. Quiros, and J. L. Vazquez, Math. Ann., 350, 811–827 (2011).

    Article  MathSciNet  Google Scholar 

  39. T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin (1995).

    MATH  Google Scholar 

  40. A. A. Arsen’ev, “On the existence of invariant measures on the space of solutions of the evolution equation,” Preprint No. 73, Keldysh Inst. Appl. Mech., Russ. Acad. Sci., Moscow (1976).

    Google Scholar 

  41. A. M. Vershik and O. A. Ladyzhenskaya, Soviet Math. Dokl., 17, 18–22 (1976).

    MATH  Google Scholar 

  42. O. G. Smolyanov, H. von Weizsäcker, and O. Wittich, Potential Anal., 26, 1–29 (2007).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia

    L. S. Efremova

  2. Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow Oblast, Russia

    V. Zh. Sakbaev

Authors
  1. L. S. Efremova
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  2. V. Zh. Sakbaev
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Correspondence to L. S. Efremova.

Additional information

The results in Secs. 1 and 2 were obtained by L. S. Efremova, and the results in Secs. 3 and 4 were obtained by V. Zh. Sakbaev.

The research of L. S. Efremova was supported by the Russian Federation Ministry of Education and Science (Grant No. 10-14).

The research of V. Zh. Sakbaev was funded by a grant from the Russian Science Foundation (Project No. 14-11-00687) and was performed at the Steklov Mathematical Institute of the Russian Academy of Sciences.

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 185, No. 2, pp. 252–271, November, 2015.

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Efremova, L.S., Sakbaev, V.Z. Notion of blowup of the solution set of differential equations and averaging of random semigroups. Theor Math Phys 185, 1582–1598 (2015). https://doi.org/10.1007/s11232-015-0366-z

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  • DOI: https://doi.org/10.1007/s11232-015-0366-z

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