Sobolev Type Equations in Spaces of Differential Forms on Riemannian Manifolds without Boundary

The article contains a review of the results obtained by the author both independently and in collaboration with other members of the Chelyabinsk scientific school founded by
G.A. Sviridyuk and devoted to Sobolev-type equations in specific spaces, namely the spaces of differential forms defined on some Riemannian manifold without boundary. Sobolev type equations are nonclassical equations of mathematical physics and are characterized by an irreversible operator at the highest derivative. In our spaces, we need to use special generalizations of operators to the space of differential forms, in particular, the Laplace operator is replaced by its generalization, the Laplace-Beltrami operator. We consider specific interpretations of equations with the relatively bounded operators: linear Barenblatt-Zheltov-Kochina, linear and semilinear Hoff, linear Oskolkov ones. For these equations, we investigate the solvability of the Cauchy, Showalter-Sidorov and initial-final value problems in different cases. Depending on the choice of the type of equation (linear or semi-linear), we use the corresponding modification of the phase space method. In the spaces of differential forms, in order to use this method based on domain splitting and the actions of the corresponding operators, the basis is the Hodge-Kodaira theorem on the splitting of the domain of the Laplace-Beltrami operator.

Keywords

Ключевые слова: Sobolev-type equations; phase space method; differential forms; Riemannian manifold without boundary.

References

1. Barenblatt G.I., Zheltov Iu.P., Kochina I.N. Basic Concepts in the Theory of Seepage of Homogeneous Liquids in Fissured Rocks. Journal of Applied Mathematics and Mechanics, 1960, vol. 24, iss. 5, pp. 852-864. DOI: 10.1016/0021-8928(60)90107-6
2. Favini A., Sviridyuk G.A., Manakova N.A. Linear Sobolev Type Equations with Relatively p-Sectorial Operators in Space of ``Noises''. Abstract and Applied Analysis, 2015, article ID: 69741, 8 p. DOI: 10.1155/2015/697410
3. Favini A., Sviridyuk G.A., Sagadeeva M.A. Linear Sobolev Type Equations with Relatively p-Radial Operators in Space of ``Noises''. Mediterranean Journal of Mathematics, 2016, vol. 13, no. 6, pp. 4607-4621. DOI: 10.1007/s00009-016-0765-x
4. Favini A., Zagrebina S.A., Sviridyuk G.A. Multipoint Initial-Final Value Problems for Dynamical Sobolev-Type Equations in the Space of Noises. Electronic Journal of Differential Equations, 2018, vol. 2018, no. 128, pp. 1-10.
5. Hoff N.J. Creep Buckling. Journal of Aeronautical Sciences, 1956, no. 1, pp. 1-20.
6. Kitaeva O.G. Exponential Dichotomies of a Non-Classical Equations of Differential Forms on a Two-Dimensional Torus with ``Noises''. Journal of Computational and Engineering Mathematics, 2019, vol. 6, no. 3, pp. 26-38. DOI: 10.14529/jcem190303
7. Kitaeva O.G. Dichotomies of Solutions to the Stochastic Ginzburg-Landau Equation on a Torus. Journal of Computational and Engineering Mathematics, 2020, vol. 7, no. 4, pp. 17-25. DOI: 10.14529/jcem200402
8. Kitaeva O.G. Exponential Dichotomies of a Stochastic Non-Classical Equation on a Two-Dimensional Sphere. Journal of Computational and Engineering Mathematics, 2021, vol. 8, no. 1, pp. 60-67. DOI: 10.14529/jcem210105
9. Kitaeva O.G. Invariant Spaces of Oskolkov Stochastic Linear Equations on the Manifold. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2021, vol. 13, no. 2, pp. 5-10. DOI: 10.14529/mmph210201
10. Kitaeva O.G., Shafranov D.E., Sviridyuk G.A. Exponential Dichotomies in Barenblatt-Zheltov-Kochina Model in Spaces of Differential Forms with ``Noise''. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2019, vol. 12, no. 2, pp. 47-57. DOI: 10.14529/mmp190204
11. Kitaeva O.G., Shafranov D.E., Sviridyuk G.A. Degenerate Holomorphic Semigroups of Operators in Spaces of K-``Noises'' on Riemannian Manifolds. Springer Proceedings in Mathematics and Statistics, Springer, Cham, 2020, vol. 325, pp. 279-292. DOI: 10.1134/S0012266121040078
12. Sagadeeva M.A., Zagrebina S.A., Manakova N.A. Optimal Control of Solutions of a Multipoint Initial-Final Problem for Non-Autonomous Evolutionary Sobolev Type Equation. Evolution Equations and Control Theory, 2019, vol. 8, no. 3, pp. 473-488. DOI: 10.3934/eect.2019023
13. Shafranov D.E. Numeral Solution of the Barenblatt-Zheltov-Kochina Equation with Additive ``White Noise'' in Spaces of Differential Forms on a Torus. Journal of Computational and Engineering Mathematics, 2019, vol. 6, no. 4, pp. 31-43. DOI: 10.14529/jcem190403
14. Shafranov D.E. Numerical solution of the Dzektser Equation with ``White Noise'' in the Space of Smooth Differential Forms Defined on a Torus. Journal of Computational and Engineering Mathematics, 2020, vol. 7, no. 2, pp. 58-65. DOI: 10.14529/jcem200206
15. Shafranov D.E., Adukova N.V. 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. Journal of Computation and Engineering Mathematics, 2017, vol. 4, no. 3, pp. 27-34. DOI: 10.14529/jcem170304
16. Shafranov D.E., Kitaeva O.G. The Barenblatt-Zheltov-Kochina Model with the Showalter-Sidorov Condition and Аdditive ``White Noise'' in Spaces of Differential Forms on Riemannian Manifolds without Boundary. Global and Stochastic Analysis, 2018, vol. 5, no. 2, pp. 145-159.
17. Shafranov D.E., Kitaeva O.G., Sviridyuk G.A. Stochastic Equations of Sobolev Type with Relatively p-Radial Operators in Spaces of Differential Forms. Differential Equations, 2021, vol. 57, no. 4, pp. 507-516. DOI: 10.1134/S0012266121040078
18. Shafranov D.E., Shvedchikova A.I. The Hoff Equation as a Model of Elastic Shell. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2012, no. 18 (277), pp. 77-81. (in Russian)
19. Shestakov A.L., Sviridyuk G.A. On the Measurement of the ``White Noise''. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2012, no. 27 (286), pp. 99-108. (in Russian)
20. Sviridyuk G.A. A Problem of Generalized Boussinesq Filtration Equation. Soviet Mathematics, 1989, vol. 33, no. 2, pp. 62-73.
21. Sviridyuk G.A. Solvability of a Problem of the Termoconvection of a Viscoelastic Incompressible Fluid. Soviet Mathematics, 1990, vol. 34, no. 12, pp. 80-86.
22. Sviridyuk G.A. On the General Theory of Operator Semigroups. Russian Mathematical Surveys, 1994, vol. 49, no. 4, pp. 45-74. DOI: 10.1070/RM1994v049n04ABEH002390
23. Sviridyuk G.A., Efremov A.A. Optimal Control Problem for a Class of Linear Equations of Sobolev Type. Russian Mathematics, 1996, vol. 40, no. 12, pp. 60-71.
24. Sviridyuk G.A., Kazak V.O. The Phase Space of a Generalized Model of Oskolkov. Siberian Mathematical Journal, 2003, vol. 44, iss. 5, pp. 877-882. DOI: 10.1023/A:1026080506657
25. Sviridyuk G.A., Manakova N.A. The Dynamical Models of Sobolv Type with Showalter-Sidorov Condition and Additive ``Noise''. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 1, pp. 90-103. DOI: 10.14529/mmp140108 (in Russian)
26. Sviridyuk G.A., Shafranov D.E. The Cauchy Problem for the Barenblatt-Zheltov-Kochina Equation on a Smooth Manifold. Vestnik Chelyabinskogo gosudarstvennogo universiteta, 2003, vol. 9, pp. 171-177. (in Russian)
27. Sviridyuk G.A., Shemetova V.V. Hoff Equations on Graphs. Differential Equations, 2006, vol. 42, no. 1, pp. 139-145. DOI: 10.1134/S0012266106010125
28. Sviridyuk G.A., Sukacheva T.G. Cauchy Problem for a Class of Semilinear Equations of Sobolev Type. Siberian Mathematical Journal, 1990, vol. 31, iss. 5, pp. 794-802. DOI: 10.1007/BF00974493
29. Sviridyuk G.A., Yakupov M.M. The Phase Space of the Initial-boundary Value Problem for the Oskolkov System. Differential Equations, 1996, vol. 232, no. 11, pp. 1535-1540.
30. Sviridyuk G.A., Zagrebina S.A. The Showalter-Sidorov Problem as a Phenomena of the Sobolev Type Equations. The Bulletin of Irkutsk State University. Series Mathematics, 2010, vol. 3, no. 1, pp. 104-125.
31. Warner F.W. Foundations of Differentiable Manifolds and Lie Groups. \ New York, Springer Science and Business Media, 1983.
32. Zagrebina S.A. The Initial-Finite Problems for Nonclassical Models of Mathematical Physics. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2013, vol. 6, no. 2, pp. 5-24.
33. Zagrebina S.A., Sviridyuk G.A., Shafranov D.E. The Initial-Final Problem for Measuring the Bending of a Beam, Which Is an Elastic Shell. Proceedings of the 24th National Scientific Symposium with International Participation Metrology and Metrology Assurance, Sozopol, 2014, pp. 144-147. (in Russian)
34. Zamyshlyaeva A.A., Bychkov E.V. The Cauchy Problem for the Sobolev Type Equation of Higher Order. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2018, vol. 11, no. 1, pp. 5-14. DOI: 10.14529/mmp180101