Skeleton decomposition of linear operators in the theory of nonregular systems of partial differential equations

N. A. Sidorov   ; D. N. Sidorov

The Bulletin of Irkutsk State University. Series Mathematics, Tome 20 (2017), pp. 75-95

Résumé

The linear system of partial differential equations is considered. It is assumed that there is the irreversible linear operator in the main part of the system, which enjoy the skeletal decomposition. The differential operators is such system are assumed to have a sufficiently smooth coefficients. In the concrete situations the domains of such differential operators are linear manifolds of smooth enough functions with ranges in Banach space. Such functions are assumed to satisfy an additional boundary conditions. The concept of a skeleton chain of linear operator is introduced. It is assumed that the operator generates a skeleton chain of the finite length. In this case, the problem of solution of given system is reduced to a regular split system of equations. The system is resolved with respect to the highest differential expressions taking into account the certain initial and boundary conditions. The possible generalization of the approach and the application to the formulation of boundary value problems in the nonlinear case. Presented results develop the theory of degenerate differential equations in the monographs N. A. Sidorov [General regularization questions in problems of branching theory. (1982; MR 87a:58036)]; N. A. Sidorov, B. V. Loginov, A. V. Sinitsyn and M. V. Falaleev [Lyapunov–Schmidt methods in nonlinear analysis and applications (Math. Appl. 550, Kluwer Acad. Publ., Dordrecht) (2002; Zbl 1027.47001)].

Détail

Keywords: ill-posed problems, Cauchy problems, irreversible operator, skeleton decomposition, skeleton chain, boundary value problems.

Bibliographie

[1] Vainberg M. M., Theory of branching of solutions of non-linear equations, Leyden, 1974 | MR | Zbl

[2] Gantmacher F. R., The theory of matrices, F. Trans. from Russian by K. A. Hirsch, v. I, II, Chelsea, New York, 1959 | MR

[3] Korpusov M.O., Blow-up in nonclassic wave equations, Libercom Publ., M., 2010, 240 pp.

[4] Loginov B.V., Rousak Yu.B., Kim-Tyan L.R., “Differential equations with degenerate, depending on the unknown function operator at the derivative”, Proceedings of the Seventh International Conference on Differential and Functional-Differential Equations (Moscow, August 22-29, 2014), v. 2, CMFD, 59, PFUR, M., 2016, 119–147

[5] Petrowsky I.G., Selected works, part 1, v. I, Classics of Soviet Mathematics, 5, Systems of partial differential equations and algebraic geometry, ed. Oleinik O.A., Gordon and Breach Publishers, Amsterdam, 1996

[6] Sveshnikov A.G., Alshin A.B., Korpusov M.O., Pletner Yu. D., Linear and Nonlinear Sobolev Equations, Fizmatlit, M., 2007

[7] Sobolev S.L., “The Cauchy problem for a special case of system that are not of Kovalevskaya type”, Dokl. Akad. Nauk SSSR, 82:2 (1952), 1007–1009 | Zbl

[8] Sviridyuk G.A., “On the general theory of operator semigroups”, Russian Mathematical Surveys, 49:4 (1994), 47–74 | Zbl

[9] Sidorov N. A., General Issues of Regularisation in the Problems of the Theory of Branching, Irkutsk State University Publ., Irkutsk, 1982, 312 pp.

[10] Sidorov N.A., Blagodatskaya E.B., Differential equations with a Fredholm operator at the higher order derivative, Preprint No 1, ICC AS USSR, Irkutsk, 1986, 34 pp. (in Russian)

[11] Sidorov N.A., Romanova O.A., Blagodatskaya E.B., PDE with finie index operator in the main part, Preprint No 3, ICC AS USSR, Irkutsk, 1992, 29 pp. (in Russian)

[12] Sidorov N. A., Blagodatskaya E.B., “Differential Equations with Fredholm Operator in the Leading Differential Expression”, Soviet Math. Dokl., 44:1 (1992), 302–305 | MR | Zbl

[13] Tikhonov A.N., Samarskii A.A., Equations of mathematical physics, Courier Corporation, 2013 | MR

[14] N. Sidorov, D. Sidorov, Y. Li, Skeleton decomposition of linear operators in the theory of degenerate differential equations, 2015, arXiv: 1511.08976

[15] D. N. Sidorov, N. A. Sidorov, “Solution of irregular systems of partial differential equations using skeleton decomposition of linear operators”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 10:2 (2017), 63–73

[16] G.A. Sviridyuk, V.E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, Inverse and Ill-Posed Problems Series, De Gruyter, 2003, 224 pp. | MR

[17] D. Sidorov, Integral Dynamical Models: Singularities, Signals and Control, World Scientific Series on Nonlinear Science, Series A, 87, ed. L. O. Chua, World Scientific Publ., Singapore–London, 2015, 258 pp. | MR | Zbl

[18] N. Sidorov, B. Loginov, A. Sinitsyn, M. Falaleev, Lyapunov-Schmidt Methods in Nonlinear Analysis and Applications, Mathematics and Its Applications, Springer Netherlands, 2013 | MR