On boundary value problems for the Boussinesq-type equation with dynamic and non-dynamic boundary conditions

Abstract

The work studies boundary value problems with non-dynamic and dynamic boundary conditions for one- and two-dimensional Boussinesq-type equations in domains representing a trapezoid, triangle, "curvilinear" trapezoid, "curvilinear" triangle, truncated cone, cone, truncated "curvilinear" cone, and "curvilinear" cone. Combining the methods of the theory of monotone operators and a priori estimates, in Sobolev classes, we have established theorems on the unique weak solvability of the boundary value problems under study.

Keywords

• Boussinesq equation
• \; dynamical boundary condition
• \; non-cylindrical domains
• \; non-dynamical boundary condition

Kaynakça

1. [1] H. P. McKean, Boussinesq’s Equation on the Circle, Commun. Pure. Appl. Math. 27 (1981), 599–691.
2. [2] Z. Y. Yan, F.D. Xie, H.Q. Zhang, Symmetry Reductions, Integrability and Solitary Wave Solutions to Higher-Order Modified Boussinesq Equations with Damping Term, Commun. Theor. Phys. 36 (2001), 1–6.
3. [3] V. F. Baklanovskaya, A. N. Gaipova, On a two-dimensional problem of nonlinear filtration, Zh. Vychisl. Math. i Math. Phiz., 6 (1966), 237–241 (in Russian).
4. [4] J. L. Vazquez, The Porous Medium Equation. Mathematical Theory, Oxford University Press, Oxford (2007). XXII+625p.
5. [5] P. Ya. Polubarinova-Kochina, On a nonlinear differential equation encountered in the theory of infiltration, Dokl. Akad. Nauk SSSR, 63 (1948), 623–627.
6. [6] P.Ya. Polubarinova-Kochina, Theory of Groundwater Movement, Princeton Univ. Press, Princeton (1962).
7. [7] Ya. B. Zel’dovich, A. S. Kompaneets, Towards a theory of heat conduction with thermal conductivity depending on the temperature, In Collection of Papers Dedicated to 70th Anniversary of A. F. Ioffe. Izd. Akad. Nauk SSSR, Moscow (1950), 61–72.
8. [8] Ya. B. Zel’dovich, G. I. Barenblatt, On the dipole-type solution in the problems of a polytropic gas flow in porous medium, Appl. Math. Mech. 21 (1957), 718–720.

9. [9] Ya. B. Zel’dovich, G. I. Barenblatt, The asymptotic properties of self-modelling solutions of the nonstationary gas filtration equations, Sov. Phys. Doklady. 3 (1958), 44–47..
10. [10] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Amer. Math. Soc., Providence (1997). XIII+270=283p.
11. [11] M. M. Vainberg, Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations, Wiley, New York (1973).
12. [12] X. Zhong, Strong solutions to the nonhomogeneous Boussinesq equations for magnetohydrodynamics convection without thermal diffusion, Electron. J. Qual. Theory Differ. Equ. 24 (2020), 1–23.
13. [13] H. Zhang, Q. Hu, G. Liu, Global existence, asymptotic stability and blow-up of solutions for the generalized Boussinesq equation with nonlinear boundary condition Mathematische Nachrichten, 293: 2 (2020), 386–404.
14. [14] G. Oruc, G. M. Muslu, Existence and uniqueness of solutions to initial boundary value problem for the higher order Boussinesq equation, Nonlinear Anal. Real. World. Appl. 47 (2019), 436–445.
15. [15] W. Ding, Zh.-A. Wang, Global existence and asymptotic behavior of the BoussinesqBurgers system, J. Math. Anal. Appl. 424 (2015), 584–597.
16. [16] N. Zhu, Zh. Liu, K. Zhao, On the Boussinesq-Burgers equations driven by dynamic boundary conditions, J. Differ. Equ. 264 (2018), 2287–2309.
17. [17] J. Crank, Free and Moving Boundary Problems, Oxford University Press, 1984.
18. [18] M.T. Jenaliyev, A.S. Kassymbekova, M.G. Yergaliyev, A.A. Assetov, An initial boundary value problem for the Boussinesq equation in a Trapezoid, Bull. of the KarU, Math. Series, 106 (2022), 117–127.
19. [19] M.T. Jenaliyev, A.S. Kassymbekova, M.G. Yergaliyev, On a boundary value problem for a Boussinesq-type equation in a triangle, Jour. Math. Mech. Comp. Scie. 115 (2022), 36–48.
20. 20] M.T. Jenaliyev, M.G. Yergaliyev, A.A. Assetov, A.M. Ayazbayeva, On a Neumanntype problem for the Burgers equation in a degenerate corner domain, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz. 206 (2022), 46–62.
21. [21] J.-L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod Gauthier-Villars, Paris (1969).
22. [22] I. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Acad. Scient. Hung. 7 (1956), 81–94.
23. [23] V. Lakshmikantham, S. Leela, A.A. Martynyuk, Stability Analysis of Nonlinear Systems, Marcell Dekker, Inc., New York-Basel (1989). IX+315 p.
24. [24] J.-L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. V. 1, Springer Verlag, Berlin (1972).