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

Year 2023, , 377 - 386, 23.07.2023

Muvasharkhan Jenaliyev , Arnay Kassymbekova Madi Yergaliyev , Bekzat Orynbasar

https://doi.org/10.31197/atnaa.1215178

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

Supporting Institution

the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan

Project Number

AP09258892

References

• [1] H. P. McKean, Boussinesq’s Equation on the Circle, Commun. Pure. Appl. Math. 27 (1981), 599–691.
• [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] 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] J. L. Vazquez, The Porous Medium Equation. Mathematical Theory, Oxford University Press, Oxford (2007). XXII+625p.
• [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] P.Ya. Polubarinova-Kochina, Theory of Groundwater Movement, Princeton Univ. Press, Princeton (1962).
• [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] 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] 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] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Amer. Math. Soc., Providence (1997). XIII+270=283p.
• [11] M. M. Vainberg, Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations, Wiley, New York (1973).
• [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] 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] 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] W. Ding, Zh.-A. Wang, Global existence and asymptotic behavior of the BoussinesqBurgers system, J. Math. Anal. Appl. 424 (2015), 584–597.
• [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] J. Crank, Free and Moving Boundary Problems, Oxford University Press, 1984.
• [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] 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] 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] J.-L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod Gauthier-Villars, Paris (1969).
• [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] V. Lakshmikantham, S. Leela, A.A. Martynyuk, Stability Analysis of Nonlinear Systems, Marcell Dekker, Inc., New York-Basel (1989). IX+315 p.
• [24] J.-L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. V. 1, Springer Verlag, Berlin (1972).