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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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.
J. Palis, Proc. Am. Math. Soc., 27, 85–90 (1971).
M. Shub and S. Smail, Ann. Math., 96, 587–591 (1972).
I. U. Bronshtein, Nonautonomous Dynamical Systems [in Russian], Shtiintsa, Kishinev (1984).
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.
L. S. Efremova, Vestn. NNGU im. N. I. Lobachevskogo, 3(1), 130–136 (2012).
L. S. Efremova, J. Math. Sci., 202, 794–808 (2014).
V. A. Galaktionov, S. P. Kurdyumov, and A. G. Mikhailov, Peaking Regimes [in Russian], Nauka, Moscow (1986).
S. I. Pokhozhaev, Soviet Math. Dokl., 6, 1408–1411 (1965).
S. N. Kruzhkov, Lectures on Partial Differential Equations [in Russian], Moscow State Univ., Moscow (1970).
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.
O. A. Oleinik, Uspekhi Mat. Nauk, 12, No. 3(75), 3–73 (1957).
V. Zh. Sakbaev, Dokl. Math., 77, 208–211 (2008).
S. K. Godunov and E. I. Romensky, Elements of Continuum Mechanics and Conservation Laws [in Russian], Universitetskaya Seriya, Novosibirsk (1998)
S. K. Godunov and E. I. Romensky, English transl., Kluwer, New York (2003).
I. V. Volovich and V. Zh. Sakbaev, Proc. Steklov Inst. Math., 285, 56–80 (2014); arXiv:1312.4302v1 [math.AP] (2013).
V. P. Burskii, Methods of Investigation of Boundary-Value Problems for General Differential Equations [in Russian], Naukova Dumka, Kiev (2002).
V. I. Oseledets, Theory Probab. Appl., 10, 499–504 (1965).
L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York (1974).
M. Blank, Stability and Localization in Chaotic Dynamics [in Russian], MTsNMO, Moscow (2001).
Yu. N. Orlov, V. Zh. Sakbaev, and O. G. Smolyanov, Proc. Steklov Inst. Math., 285, 222–232 (2014).
N. N. Bogoliubov, On Some Statistical Methods in Mathematical Physics [in Russian], Izdat. Akad. Nauk SSSR, Kiev (1945).
L. Accardi, Y. G. Lu, and I. V. Volovich, Quantum Theory and Its Stochastic Limit, Springer, Berlin (2002).
L. Accardi, A. N. Pechen, and I. V. Volovich, J. Phys. A: Math. Gen., 35, 4889–4902 (2002); arXiv:quant-ph/0108112v2 (2001).
O. G. Smolyanov and E. T. Shavgulidze, Path Integrals [in Russian], Moscow State Univ., Moscow (2015).
V. K. Ivanov, I. V. Mel’nikova, and A. I. Filinkov, Operator-Differential Equations and Ill-Posed Problems [in Russian], Fizmatlit, Moscow (1995).
A. B. Bakushinskii and M. Yu. Kokurin, Iteration Methods for Solving Ill-Posed Operator Equations with Smooth Operators [in Russian], URSS, Moscow (2002).
A. N. Tikhonov and V. Ya. Arsenin, Methods for the Solution of Ill-Posed Problems [in Russian], Nauka, Moscow (1986).
M. I. Vishik and V. V. Chepyzhov, Sb. Math., 200, 471–497 (2009).
E. Mitidieri and S. I. Pokhozhaev, Proc. Steklov Inst. Math., 234, 1–362 (2001).
V. Zh. Sakbaev, Proc. Steklov Inst. Math., 283, 165–180 (2013).
V. A. Galaktionov and J. L. Vazquez, Arch. Rational Mech. Anal., 129, 225–244 (1995).
V. Zh. Sakbaev, J. Math. Sci., 151, 2741–2753 (2008).
V. I. Bogachev, N. V. Krylov, and M. Röckner, Russ. Math. Surveys, 64, 973–1078 (2009).
V. V. Zhikov, Funct. Anal. Appl., 35, 19–33 (2001).
E. Yu. Panov, Matem. Mod., 14, No. 3, 17–26 (2002).
R. J. Di Perna and P. Lions, Invent. Math., 98, 511–547 (1989).
N. Mizoguchi, F. Quiros, and J. L. Vazquez, Math. Ann., 350, 811–827 (2011).
T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin (1995).
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).
A. M. Vershik and O. A. Ladyzhenskaya, Soviet Math. Dokl., 17, 18–22 (1976).
O. G. Smolyanov, H. von Weizsäcker, and O. Wittich, Potential Anal., 26, 1–29 (2007).
Author information
Authors and Affiliations
Corresponding author
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.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11232-015-0366-z