On a boundary value problem for a Boussinesq-type equation in a triangle
DOI:
https://doi.org/10.26577/JMMCS.2022.v115.i3.04Keywords:
Boussinesq equation, degenerating domain, a priori estimates, Sobolev spaceAbstract
Earlier, we considered an initial-boundary value problem for a one-dimensional Boussinesq-type equation in a domain that is a trapezoid, in which the theorems on its unique weak solvability in Sobolev classes were established by the methods of the theory of monotone operators. In this article, we continue research in this direction and study the issues of correct formulation of the boundary value problem for a one-dimensional Boussinesq-type equation in a degenerate domain, which is a triangle. A scalar product is proposed with the help of which the monotonicity of the main operators is shown, and uniform a priori estimates are obtained. Further, using the methods of the theory of monotone operators and a priori estimates, theorems on its unique weak solvability in Sobolev classes are established. A theorem on increasing the smoothness of a weak solution is established. In proving the smoothness enhancement theorem, we use a generalization of the classical result on compactness in Banach spaces proved by Yu.I. Dubinsky ("Weak convergence in nonlinear elliptic and parabolic equations Sbornik: Mathematics, 67 (109): 4 (1965)) in the presence of a bounded set from a semi-normed space instead of a normed one. It is also shown that the solution may have a singularity at the point of degeneracy of the domain. The order of this feature is determined, and the corresponding theorem is proved.
References
[2] M.T. Jenaliyev, A.S. Kasymbekova, M.G. Yergaliyev, A.A. Assetov, "An Initial Boundary Value Problem for the
Boussinesq Equation in a Trapezoid" , Bulletin of the Karaganda University. Mathematics, 106: 2 (2022), 11p.
[3] H. P. McKean, "Boussinesq’s Equation on the Circle" , Communications on Pure and Applied Mathematics, XXXIV (1981): 599–691
[4] 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" , Communications in Theoretical Physics, 36: 1 (2001): 1–6.
[5] V. F. Baklanovskaya, A. N. Gaipova, :On a two-dimensional problem of nonlinear filtration" , Zh. vychisl. math. i math. phiz., 6: 4 (1966): 237–241 (in Russian).
[6] J. L. Vazquez, The Porous Medium Equation. Mathematical Theory, (Oxford University Press, Oxford (2007).
XXII+625p).
[7] P. Ya. Polubarinova-Kochina, "On a nonlinear differential equation encountered in the theory of infiltration" , Dokl. Akad. Nauk SSSR, 63: 6 (1948): 623–627.
[8] P.Ya. Polubarinova-Kochina, Theory of Groundwater Movement, (Princeton Univ. Press, Princeton (1962)).
[9] Ya. B. Zel’dovich and 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).
[10] Ya. B. Zel’dovich and G. I. Barenblatt, "On the dipole-type solution in the problems of a polytropic gas flow in porous medium" , Appl. Math. Mech., 21: 5 (1957): 718–720.
[11] Ya. B. Zel’dovich and G. I. Barenblatt, "The asymptotic properties of self-modelling solutions of the nonstationary gas filtration equations: , Sov. Phys. Doklady, 3 (1958): 44–47.
[12] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, (Amer. Math. Soc., Providence (1997). XIII+270=283p).
[13] M. M. Vainberg, Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations, (Wiley, New York (1973)).
[14] X. Zhong, "Strong solutions to the nonhomogeneous Boussinesq equations for magnetohydrodynamics convection without thermal diffusion" , Electronic Journal of Qualitative Theory Differential Equations, 2020: 24: 1–23.
[15] 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.
[16] G. Oruc, G. M. Muslu, "Existence and uniqueness of solutions to initial boundary value problem for the higher order Boussinesq equation" , Nonlinear Analysis – Real World Applications, 47 (2019): 436–445.
[17] W. Ding, Zh.-A. Wang, "Global existence and asymptotic behavier of the Boussinesq-Burgers system" , Journal of Mathematical Analysis and Applications, 424: 1 (2015): 584–597.
[18] N. Zhu, Zh. Liu, K. Zhao, "On the Boussinesq-Burgers equations driven by dynamic boundary conditions" , Journal of Differential Equations, 264: 3 (2018): 2287–2309.
[19] J. Crank, Free and Moving Boundary Problems, (Oxford University Press, 1984).
[20] J.-L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, (Dunod Gauthier-Villars, Paris (1969)).
[21] Yu.A. Dubinsky, "Weak convergence in nonlinear elliptic and parabolic equations" , Sbornik:Math., 67(109): 4 (1965): 609–642.
[22] P.A. Raviart, "Sur la resolution et l’approximation de certaines equations paraboliques non lineaires degenerees" , Archive Rat. Mech. Anal., 25 (1967): 64–80
