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

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.


Introduction
The theory of Boussinesq equations and its modications always attracts the attention of mathematicians and applied scientists.The Boussinesq equation, as well as its modications, take an important place in the description of the motion of liquids and gases, including in the theory of non-stationary ltration in porous media [1] [11].The works [12] [17] can also be noted here.In recent years, boundary value problems for these equations have been actively studied, since they simulate processes in porous media.Processes occurring in porous media acquire special importance for deep understanding and comprehension in the problems of exploration and ecient development of oil and gas elds.
The papers [18][20] previously studied the solvability of the boundary value problems for one-dimensional Boussinesq-type and Burgers equations with Dirichlet boundary conditions in a domain, which is represented by a trapezoid or a triangle, respectively.
In this paper, we study the questions of the correctness of the formulation of boundary value problems for one-and two-dimensional Boussinesq-type equations in a domain on the moving part of the boundary of which dynamic nonlinear conditions are set.Domains are represented by a trapezoid, triangle, truncated cone, cone, truncated "curvilinear" cone, and "curvilinear" cone.We establish theorems on the unique weak solvability of the considered boundary problems.

Statements of the problems and main results
Problem 1.Let Ω t = {0 < x < t} and ∂Ω t be the boundary of Ω t , 0 < t 0 < T < ∞.In the domain , which is a trapezoid, we consider an initial boundary value problem for Boussinesq-type equation with boundary conditions where where f (x, t), g(t), u 0 (x) are given functions, u 00 is a given number.
Problem 2. Let Ω t = {0 < x < t} and ∂Ω t be the boundary of Ω t , T < ∞.In the domain Q xt = Ω t × (0, T ), representing a triangle, we consider a boundary value problem for the Boussinesq-type equation with boundary conditions where du(t,t) dt = [∂ t u(x, t) + ∂ x u(x, t)] |x=t , and the functions f (x, t), g(t) are given.
Problem 3. Let Ω t = {0 < x < φ(t)} and ∂Ω t be the boundary of In the domain Q xt = Ω t × (t 0 , T ), representing a "curvilinear" trapezoid, we consider the initial-boundary problem for the Boussinesq-type equation with boundary conditions where du(t,t) dt , and with initial condition where f (x, t), g(t), u 0 (x) are given functions, u 00 is a given number.
Problem 4. Let Ω t = {0 < x < φ(t)} and ∂Ω t be the boundary of which is a "curvilinear" triangle, we consider a boundary value problem for Boussinesq-type equation with boundary conditions where , and the functions f (x, t), g(t) are given.
Remark 2.1.We consider problems with a dynamic boundary condition only on the moving part of the boundary.The latter in no way detracts from the generality; this is done only for the sake of simplicity of presentation.It would be possible to put a dynamic condition on the xed part of the boundary as well.
Problem 5. Let x = (x 1 , x 2 ), Ω t = {|x| < t} and ∂Ω t be the boundary of , which is a truncated cone, we consider an initial-boundary problem for a two-dimensional Boussinesq-type equation with boundary conditions where where f (x, t), g(x, t), u 0 (x), u 00 (x) are given functions.Problem 6.Let x = (x 1 , x 2 ), Ω t = {|x| < t} and ∂Ω t be the boundary of Ω t , T < ∞.In the , which is a cone, we consider a boundary value problem for a two-dimensional Boussinesq-type equation with boundary conditions where |x|=t , ⃗ n is a unit outward normal to the circle |x| = t, and the functions f (x, t), g(x, t) are given.
Problem 7. Let x = (x 1 , x 2 ), Ω t = {|x| < φ(t)} and ∂Ω t be the boundary of a truncated cone (with a curvilinear generatrix determined by the function φ(t)), we consider an initial boundary value problem for a two-dimensional Boussinesq-type equation with boundary conditions where , ⃗ n is a unit outward normal to the circle |x| = φ(t), and with initial condition where f (x, t), g(x, t), u 0 (x), u 00 (x) are given functions.
Problem 8. Let x = (x 1 , x 2 ), Ω t = {|x| < φ(t)} and ∂Ω t be the boundary of , representing a cone (with a curvilinear generatrix determined by the function φ(t)), we consider a boundary value problem for a two-dimensional Boussinesq-type equation with boundary conditions where D t u(x, t) |x|=φ(t) ≜ [∂ t u(x, t) + φ ′ (t)∂ ⃗ n u(x, t)] |x|=φ(t) , ⃗ n is a unit outward normal to the circle |x| = φ(t), and the functions f (x, t), g(x, t) are assumed to be given.

Main results
Using and developing the results of [18][19], we have established the validity of the following theorems.

Theorem 3.1 (Trapezoid). Let
u 00 is a given number.(21) Then the initial boundary value problem (1)(3) has a unique solution where t ∈ (t 0 , T ), s ∈ (0, Then the boundary value problem (4)(5) has a unique solution where t ∈ (0, T ), s ∈ (0, (25) Theorem 3.3 ("Curvilinear" trapezoid).Let Then the initial boundary value problem (6)( 8) has a unique solution where v(s) s=s(t) = u(φ(t), t), s = s(t) Theorem 3.4 ("Curvilinear" triangle).Let Then the boundary value problem (9) (10) has a unique solution Thus, we get the following initial boundary value problem Problem 1.2.Find a solution to the initial boundary value problem for the Boussinesq equation with boundary conditions and with initial condition where f (x, t), h(t) = v( √ 2 (t − t 0 )), u 0 (x) are given functions.The solvability of the problem (43)(45) was previously established by us in [18].Thus, the solvability of Problems 1.1 and 1.2 allows us to obtain the assertion of Theorem 3.1.This is a brief outline of the proof of this theorem.Now about the proof of Theorem 3.2.First of all, let us formulate an analog of Problem 1.1.
Problem 2.1.Find a solution to the Cauchy problem for an (ordinary) dierential equation where where g(t) is a given function.
Under the conditions of Theorem 3.2 in the Cauchy problem (46)(47) the operator 1 2 |u|u has the monotonicity condition.This allows us to establish the validity of the assertion that Problem 2.1 has a unique solution {v(s), s ∈ (0, √ with boundary conditions where f (x, t), h(t) = v( √ 2 t) are given functions.The solvability of the problem (48)(49) was established by us earlier in [19].Remark 4.1.Let us show that the solution u(x, t) of the boundary value problem (23) (24) having a singularity of the order specied in (25) will belong to the space L 3 (Q t 0 xt ), where Q t 0 xt = {x, t| 0 < x < t, 0 < t < t 0 ≪ T }.For this purpose, it suces to show that the following integral is bounded for t 0 → 0+: We have where g(x, t) is a given function.
Under the conditions of Theorem 3.6 in the Cauchy problem (56)(57) the operator 1 2 |u|u has the monotonicity condition.This allows us to establish the validity of the assertion that Problem 6.1 has a unique solution {v(x, s), s ∈ (0, √ t), where u(x, 0) = v(x, 0) = u 00 (x).Thus, we get the following boundary value problem Problem 6.2.Find a solution to the boundary value problem for the Boussinesq equation with boundary conditions u(x, t) = h(x, t) at {x = t, t ∈ (0, T )}, u(x, t) = 0 at {x = 0, t ∈ (0, T )}, where f (x, t), h(x, t) = v(x, √ 2 t) are given functions.The solvability of the problem (58)(59) is set in the same way as in [19], following [21][24].Thus, the solvability of Problems 6.1 and 6.2 allows us to obtain the assertion of Theorem 2.6.This is a brief outline of the proof of this theorem.
Theorems 3.7 and 3.8 are proved similarly to Theorems 3.5 and 3.6.

Conclusion
In the work boundary value problems for one-and two-dimensional Boussinesq-type equations in domains representing a trapezoid, a triangle, a "curvilinear" trapezoid, a "curvilinear" triangle, a truncated cone, a cone, a truncated "curvilinear" cone, and " curvilinear" cone are studied.Using the methods of the theory of monotone operators and a priori estimates, we prove theorems on their unique weak solvability in Sobolev classes.
T ), which allows us to obtain the Dirichlet boundary condition u(t, t) = v( √ 2 t), where u(0, 0) = v(0) = 0. Thus, we get the following boundary value problem Problem 2.2.Find a solution to the boundary value problem for the Boussinesq equation