Solvability of the Showalter-Sidorov problem for Sobolev type equations with operators in the form of first-order polynomials from the Laplace-Beltrami operator on differential forms. (English) Zbl 1458.76106

Shafranov, Dmitriĭ Evgen’evich; Adukova, Natal’ya Viktorovna

J. Comput. Eng. Math. 4, No. 3, 27-34 (2017).

Summary: We consider solvability of the Showalter-Sidorov problem for the Barenblatt-Zheltov-Kochina equations and the Hoff linear equation. The equations are linear representatives of the class of linear Sobolev type equations with an irreversible operator under derivative. We search for a solution to the problem in the space of differential \(k\)-forms defined on a Riemannian manifold without boundary. Both equations are the special cases of an equation with operators in the form of polynomials of the first degree from the Laplace-Beltrami operator, generalizing the Laplace operator in spaces of differential \(k\)-forms up to a sign. Applying the Sviridyuk theory and the Hodge-Kodaira theorem, we prove an existence of the subspace in which there exists a unique solution to the problem.

MSC:

76S05	Flows in porous media; filtration; seepage
35Q35	PDEs in connection with fluid mechanics
53Z05	Applications of differential geometry to physics

Keywords:

Riemannian manifold without boundary; Barenblatt-Zheltov-Kochina problem

References:

[1]	G. I. Barenblatt, Yu. P. Zheltov, I. N. Kochina, “Basic Concepts in the Theory of Seepage of Homogeneous Fluids in Fissurized Rocks [Strata]”, Journal of Applied Mathematics and Mechanics, 24:5 (1960), 1286-1303 · Zbl 0104.21702 · doi:10.1016/0021-8928(60)90107-6
[2]	N. A. Hoff, “Greep buckling”, Journal of the Aeronautical Sciences, 7:1 (1965), 1-20
[3]	G. A. Sviridyuk, N. A. Manakova, “Dinamicheskie modeli sobolevskogo tipa s usloviem Shouoltera-Sidorova i additivnymi «shumami»”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 7:1 (2014), 90-103 · Zbl 1303.60060 · doi:10.14529/mmp140108
[4]	G. A. Sviridyuk, S. A. Zagrebina, “Zadacha Shouoltera-Sidorova kak fenomen uravnenii sobolevskogo tipa”, Izv. Irkutskogo gos. un-ta. Ser. Matematika, 3:1 (2010), 104-125 · Zbl 1260.35074
[5]	F. Warner, Foundations of Differentiable Manifolds and Lie Groups, v. 94, Graduate Texts in Mathematics, Springer-Verlag, 1983 · Zbl 0516.58001 · doi:10.1007/978-1-4757-1799-0
[6]	D. E. Shafranov, A. I. Shvedchikova, “Uravnenie Khoffa kak model uprugoi obolochki”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 2012, no. 12, 77-81 · Zbl 1413.35135
[7]	A.A. Zamyshlyaeva, Dzh.K.T. Al-Isavi, “O nekotorykh svoistvakh reshenii odnogo klassa evolyutsionnykh matematicheskikh modelei sobolevskogo tipa v kvazisobolevykh prostranstvakh”, Vestnik YuUrGU. Seriya: Matematicheskoe modelirovanie i programmirovanie, 8:4 (2015), 113-119 (in English) · Zbl 1344.47026
[8]	M. A. Sagadeeva, F. L. Hasan, “Bounded solutions of Barenblatt-Zheltov-Kochina model in quasi-Sobolev spaces”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 8:4 (2015), 138-144 · Zbl 1344.47028 · doi:10.14529/mmp150414
[9]	G. A. Sviridyuk, V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, v. 42, Inverse and Ill-Posed Problems Series, de Gruyer, 2012, 216+viii pp. · doi:10.1515/9783110915501
[10]	A. A. Dezin, Mnogomernyi analiz i diskretnye modeli, Nauka, Novosibirsk, 1990 · Zbl 0704.39005
[11]	K. Maurin, Metody Przestrzeni Hilberta, Państwowe Wydawnictwo Naukowe, Warsawa, 1959 · Zbl 0086.30901
[12]	R. Palais, Seminar on the Atiyah-Singer Index Theorem, Princeton University Press, Princeton, 1965 · Zbl 0137.17002