An initial boundary value problem for the Boussinesq equation in a Trapezoid

Jenaliyev, M.T.; Kassymbekova, A.S.; Yergaliyev, M.G.; Assetov, A.A.

doi:10.31489/2022m2/117-127

Cited by 2 publications

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“…Now we are ready to show the uniqueness of the solution in problem ( 1)-( 2). Let u 1 (t) and u 2 (t) be two solutions to problem (1)- (2). Then their difference u(t) = u 1 (t) − u 2 (t) satisfies the homogeneous problem:…”

Section: Thus We Getmentioning

confidence: 99%

On a boundary value problem for a Boussinesq-type equation in a triangle

Jenaliyev

Kassymbekova²,

Yergaliyev³

2022

JMMCS

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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.

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Section: Thus We Getmentioning

confidence: 99%

On a boundary value problem for a Boussinesq-type equation in a triangle

Jenaliyev

Kassymbekova²,

Yergaliyev³

2022

JMMCS

View full text Add to dashboard Cite

show abstract

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

Jenaliyev

KASSYMBEKOVA

Yergaliyev

et al. 2023

Advances in the Theory of Nonlinear Analysis and Its Application

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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.

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An initial boundary value problem for the Boussinesq equation in a Trapezoid

Abstract: This paper considers an initial boundary value problem for a one-dimensional Boussinesq-type equation in a domain, that is, a trapezoid. Using the methods of the theory of monotone operators, we establish theorems on their unique weak solvability in Sobolev classes.

Cited by 2 publications

References 0 publications

On a boundary value problem for a Boussinesq-type equation in a triangle

On a boundary value problem for a Boussinesq-type equation in a triangle

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

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