A Multipoint Initial-Final Value Problem for a Linear Model of Plane-Parallel Thermal Convection in Viscoelastic Incompressible Fluid

The linear model of plane-parallel thermal convection in a viscoelastic incompressible Kelvin-Voigt material amounts to a hybrid of the Oskolkov equations and the heat equations in the Oberbeck-Boussinesq approximation on a two-dimensional region with B'enard's conditions. We study the solvability of this model with the so-called multipoint initial-final conditions. We use these conditions to reconstruct the parameters of the processes in question from the results of multiple observations at various points and times. This enables us, for instance, to predict emergency situations, including the violation of continuity of thermal convection processes as a result of breaching technology, and so forth.
For thermal convection models, the solvability of Cauchy problems and initial-final value problems has been studied previously. In addition, the stability of solutions to the Cauchy problem has been discussed. We study a multipoint initial-final value problem for this model for the first time. In addition, in this article we prove a generalized decomposition theorem in the case of a relatively sectorial operator. The main result is a theorem on the unique solvability of the multipoint initial-final value problem for the linear model of plane-parallel thermal convection in a viscoelastic incompressible fluid.

Keywords

multipoint initial-final value problem; Sobolev-type equation; generalized splitting theorem; linear model of plane-parallel thermal convection in viscoelastic incompressible fluid.

References

1. Dudko L.L. Issledovanie polugrupp operatorov s yadrami: dis. ... kand. fiz.-mat. nauk [Investigation of Semigroups of Operators with Kernels]. Novgorod, 1996. (in Russian)
2. Zagrebina S.A. On the Showalter - Sidorov problem. Russian Mathematics (Izvestiya VUZ. Matematika), 2007, vol. 51, no. 3, pp. 19-24.
3. Zagrebina S.A. The Showalter - Sidorov - Verigin Problem for the Linear Sobolev-type Equations. Neklassicheskie uravneniya matematicheskoy fiziki [Nonclassical Mathematical Physics equations], Novosibirsk, 2007, pp. 150-157. (in Russian)
4. Zagrebina S.A., Yakupov M.M. Existence and Stability of Solutions of one Class of Semilinear Sobolev Type Equations. Bulletin of the South Ural State University. Series 'Mathematical Modelling, Programming & Computer Software', 2008, no. 27 (127), issue 2, pp. 10-18. (in Russian)
5. Zagrebina S.A. The Multipoint Initial-Finish Problem for the Stochastic Barenblatt - Zheltov - Kochina Model. Bulletin of the South Ural State University. Series 'Computer Technologies, Automatic Control, Radio Electronics', 2013, vol. 13, no. 4, pp. 103-111. (in Russian)
6. Zamyshlyaeva A.A., Tsyplenkova O.N. The Optimal Control over Solutions of the Initial-finish Value Problem for the Boussinesque-Love Equation. Bulletin of the South Ural State University. Series 'Mathematical Modelling, Programming & Computer Software', 2012, vol. 5 (264), issue 11, pp. 13-24. (in Russian)
7. Keller A.V. The Algorithm for Solution of the Showalter - Sidorov Problem for Leontief Type Models. Bulletin of the South Ural State University. Series 'Mathematical Modelling, Programming & Computer Software', 2012, no. 4 (241), issue 7, pp. 40-46. (in Russian)
8. Landau L.D., Lifshitz E.M. Fluid Mechanics. Oxford, Pergamon Press, 1959.
9. Manakova N.A., Dylkov A. G. Optimal Control of the Solutions of the Initial-Finish Problem for the Linear Hoff Model. Mathematical Notes, 2013, vol. 94, issue 2, pp. 220-230. DOI: 10.1134/S0001434613070225
10. Matveeva O.P., Sukacheva T.G. Matematicheskie modeli vyazkouprugikh neszhimaemykh zhidkostey nenulevogo poryadka [The mathematical model of a viscoelastic incompressible fluid of nonzero order]. Chelyabinsk, Publishing center of South Ural State University, 2014. (in Russian)
11. Oskolkov A. P. Nonlocal Problems for Some Class Nonlinear Operator Equations Arising in the Theory Sobolev Type Equations. Zap. Nauchn. Sem. LOMI, 1991, vol. 198, pp. 31-48. (in Russian)
12. Pankov A.A., Pankova T.E. Nonlinear Evolution Equations with Irreversible Operator Coefficient for the Derivative. Doklady Akademii Nauk Ukraine, 1993, no. 9, pp. 18-20. (in Russian)
13. Sviridyuk G.A. Solvability of a Problem of the Thermoconvection of a Viscoelastic Incompressible Fluid. Soviet Mathematics (Izvestiya VUZ. Matematika), 1990, vol. 34, no. 12, pp. 80-86.
14. Sviridyuk G.A. Semilinear Equations of Sobolev Type with a Relatively Sectorial Operator. Doklady Mathematics, 1993, vol. 329, no. 3, pp. 274-277.
15. Sviridyuk G.A. Phase Portraits of Sobolev-Type Semilinear Equations with a Relatively Strongly Sectorial Operator. St. Petersburg Mathematical Journal, 1995, vol. 6, no. 5, pp. 1109-1126.
16. Sviridyuk G.A., Bokareva T.A. The Number of Deborah and One Class Semilinear Equations of Sobolev Type, Doklady Mathematics , 1991, vol. 319, no. 5, pp. 1082-1086.
17. Sviridyuk G.A., Efremov A.A. Optimal Control Of Sobolev-Type Linear Equations With Relatively $p$-Sectorial Operators. Differential Equations, 1995, vol. 31, no. 11, pp. 1882-1890.
18. Sviridyuk G.A., Zagrebina S.A. On the Verigin Problem for the Generalized Boussinesq Filtration Equation. Russian Mathematics (Izvestiya VUZ. Matematika), 2003, vol. 47, no. 7, pp. 55-59.
19. Sviridyuk G.A., Zagrebina S.A. Verigin's Problem for Linear Equations of the Sobolev Type with Relatively p-Sectorial Operators. Differential Equations, 2002, vol. 38, no. 12, pp. 1745-1752. DOI: 10.1023/A:1023812213901
20. 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. 51-72. (in Russian)
21. Sviridyuk G.A., Keller A.V. Invariant Spaces and Dichotomies of Solutions of a Class of Linear Equations of the Sobolev Type. Russian Mathematics (Izvestiya VUZ. Matematika), 1997, vol. 41, no. 5, pp. 57-65.
22. Sviridyuk G.A., Kuznetsov G.A. Relatively Strongly p-Sectorial Linear Operators. Doklady Mathematics, 1999, vol. 59, no. 2, pp. 298-300.
23. Sviridyuk G.A., Sukacheva T.G. Some Mathematical Problems of Dynamics Viscoelastic Incompressible Media. Vestnik Magnitogorskogo gosudarstvennogo universiteta. Seria 'Matematika', [Bulletin of Magnitogorsk State University. Series ' Mathematics'], 2005, issue 8, pp. 5-33. (in Russian)
24. Sviridyuk G.A., Fedorov V. E. Analytic Semigroups with Kernel and Linear Equations of Sobolev Type. Siberian Mathematical Journal, 1995, vol. 36, no. 5, pp. 973-987. DOI: 10.1007/BF02112539
25. Sviridyuk G.A., Fedorov V.E. On Units of Analytic Semigroups of Operators with Kernels. Siberian Mathematical Journal, 1998, vol. 39, no. 3, pp. 522-533. DOI: 10.1007/BF02673910
26. Sviridyuk G.A., Yakupov M.M. The Phase Space Of The Initial-Boundary Value Problem For The Oskolkov System. Differential Equations, 1996, vol. 32, no. 11, pp. 1535-1540.
27. Sukacheva T.G. , Matveeva O.P. Spline Approximations of the Solution of a Singular Integro-Differential Equation. Russian Mathematics (Izvestiya VUZ. Matematika), 2001, vol. 45, no. 11, pp. 44-51.
28. Henry, D. Geometric Theory of Semilinear Parabolic Equations. Berlin, Heidelberg, N.-Y., Springer Verlag, 1981.
29. Shestakov A.L., Keller A.V., Nazarova E.I. The Numerical Solution of the Optimal Demension Problem. Automation and Remote Control, 2011, vol. 73, no. 1, pp. 97-104. DOI: 10.1134/S0005117912010079
30. Al'shin, A.B., Korpusov, M.O., Sveshnikov, A.G. Blow-up in Nonlinear Sobolev Type Equations. Berlin, Walter de Gruyter GmbH& Co.KG, 2011.
31. Demidenko G.V., Uspenskii S.V. Partial Differential Equations and Systems not Solvable with Respect to the Highest Order Derivative. N.-Y., Basel, Hong Kong, Marcel Dekker, Inc., 2003.
32. Pyatkov S.G. Operator Theory. Nonclassical Problems. Utrecht, Boston, Koln, Tokyo, VSP, 2002. DOI: 10.1515/9783110900163
33. Showalter R.E. The Sobolev Type Equations. I (II). Appl. Anal., 1975, vol. 5, no. 1, pp. 15-22 (no. 2, pp. 81-89).
34. Zagrebina S.A., Sagadeeva M.A. The Generalized Splitting Theorem for Linear Sobolev type Equations in Relatively Radial Case. The Bulletin of Irkutsk State University. Series 'Mathematics', 2014, vol. 7, pp. 19-33.