Solvability of the Boussinesq Approximation for Water Polymer Solutions

: We consider nonlinear Boussinesq-type equations that model the heat transfer and steady viscous ﬂows of weakly concentrated water solutions of polymers in a bounded three-dimensional domain with a heat source. On the boundary of the ﬂow domain, the impermeability condition and a slip condition are provided. For the temperature ﬁeld, we use a Robin boundary condition corresponding to the classical Newton law of cooling. By using the Galerkin method with special total sequences in suitable function spaces, we prove the existence of a weak solution to this boundary-value problem, assuming that the heat source intensity is bounded. Moreover, some estimates are established for weak solutions.


Introduction and Problem Formulation
In this work, we examine the solvability of a boundary-value problem for the Boussinesq approximation describing steady-state flows of weakly concentrated water polymer solutions [1,2] in a sufficiently regular bounded domain Ω ⊂ R 3 with a heat source, under the assumption that on solid walls of Ω the impermeability condition, a vorticity-slip condition and a Robin-type temperature boundary condition are valid: Obviously, in the case when α = 0, we formally recover the stationary Navier-Stokes-Boussinesq system with a heat source, but, in what follows, we always assume that α > 0. As shown in [5][6][7], the parameter α must be non-negative, in view of thermodynamic restrictions.
It is worth pointing out that there is a vast amount of literature on the model of weakly concentrated water polymer solutions and its various modifications, including the so-called Kelvin-Voigt equations that describe viscous fluid flows in which, after the instantaneous removal of stresses, the velocity does not vanish instantaneously but decays exponentially. Starting with the pioneering series of works by A.P. Oskolkov [8][9][10][11][12][13][14], there is an ever growing list of contributions. An interested reader can see the papers [15][16][17][18][19][20][21][22][23][24][25][26][27][28]; this list is by no means exhaustive, but gives a number of current mathematical results obtained for these types of viscoelastic fluids.
In the majority of works, isothermal flows are investigated, although from the point of view of applications in engineering, the analysis of heat transfer in polymer flows is no less important than the studying of hydrodynamic fields. Motivated by this, we consider problem (1)-(4). The present paper continues the investigations initiated in the articles [10,29], where the thermal convection is studied for a simplified version of the model of water polymer solutions. Namely, the authors of these papers focus on the Kelvin-Voigt-Boussinesq equations, which have a lower order (for partial derivatives with respect to the space variables) compared with the equations considered herein.
Under the assumption that the intensity of the heat source is bounded, by the Galerkin method with special total sequences in suitable function spaces, we show the existence of a weak solution to boundary-value problem (1)-(4) and derive some estimates for the norms of the velocity and temperature fields.
Finally, by Y β (Ω) denote the space consisting of functions from H 1 (Ω) with the following scalar product and norm: It can easily be checked that the scalar product (·, ·) Y β (Ω) is well defined and the norm · Y β (Ω) is equivalent to the standard H 1 -norm.

Weak Formulation of Problem (1)-(4) and Main Results
We propose the problem of determining the velocity field u ∈ X α (Ω) and the temperature T ∈ Y β (Ω) that satisfy the system of equations: for any pair ( v, S) from the space J 1 n (Ω) × H 1 (Ω). (5) and (6) is called a weak solution of boundary value problem (1)-(4).
The following theorem gives the main result of this paper.
Then, boundary value problem (1)-(4) has at least one weak solution ( u, T) such that with

Proof of Theorem 1
To construct a weak solution of problem (1)-(4), we shall use the Galerkin method. Let us take a where δ ij is the Kronecker symbol. In addition, fix a sequence {S j } ∞ j=1 that is an orthonormal basis of Y β (Ω).
For an arbitrary fixed number N ∈ {1, 2, . . . }, we consider the 2N-dimensional auxiliary problem: where λ is a parameter, λ ∈ [0, 1]. Our immediate goal is to obtain a priori estimates for solutions to problem (9)-(11). Let a vector h N λ = (a N1 , . . . , a NN , b N1 , . . . , b NN ) be a solution of problem (9)-(11) under a fixed parameter λ ∈ [0, 1]. It is easy to see that Therefore, we wish to find estimates for the norms u N X α (Ω) and T N Y β (Ω) . Let us multiply Equation (9) by a Nj and add the results for j = 1, . . . , N. Then, we get This equality can be rewritten as follows: Note that the term Q 1 vanishes. Indeed, using integration by parts, we find that Taking into account the relation we can rewrite (13) as It follows from this equality that whence, using the Cauchy-Bunyakovsky-Schwarz inequality, the estimates 0 ≤ λ ≤ 1 and we derive and hence Next, we multiply Equation (10) by b Nj and add the results for j = 1, . . . , N; this gives Applying integration by parts, it can easily be checked that the term Q 2 vanishes. Indeed,
Let { u N * } ∞ N=1 and {T N * } ∞ N=1 be sequences of functions that satisfy (9) and (10) with the parameter λ = 1, i.e., It follows from estimates (16) and (17) that the set { u N * } ∞ N=1 is bounded in the space X α (Ω) and the set {T N * } ∞ N=1 is bounded in the space Y β (Ω). Hence, there exists a vector function u 0 from the space X α (Ω) and a function T 0 from the space Y β (Ω) such that u N * converges to u 0 weakly in X α (Ω) and T N * converges to T 0 weakly in Y β (Ω), for some subsequence N → ∞. Without loss of generality, we can assume that lim Using standard compactness results for the Sobolev spaces (see, e.g., [30] Chap. 6), we derive that the space X α (Ω) is compactly imbedded into C(Ω) and the space Y β (Ω) is compactly imbedded into L q (Ω) when 1 ≤ q < 6. Therefore, it follows from (22) and (23) that lim N→∞ u N * = u 0 strongly in C(Ω), lim N→∞ T N * = T 0 strongly in L q (Ω), where 1 ≤ q < 6.
Furthermore, by using the theorem of M. A. Krasnoselskii on a superposition operator (see the Appendix A, Proposition A2), we conclude from conditions (i)-(iii) and (25) that lim N→∞ ψ ·, T N * (·) = ψ ·, T 0 (·) strongly in L 2 (Ω). (26) Fix an arbitrary number j ∈ {1, 2, . . . }. Then, letting m → ∞ in (20) and (21), we get Of course, in this passage-to-limit procedure, we used all the convergence results (22) is an isomorphism. Therefore, the sequence { w j − α∆ w j } ∞ j=1 is total in the space J 1 n (Ω). This fact is a key tool in our proof. Indeed, due to this property, equality (27) remains valid if we replace w j − α∆ w j with an arbitrary vector function v from J 1 n (Ω). In addition, since the set {S j } ∞ j=1 is total in H 1 (Ω), we see that (28) is true with an arbitrary function S from H 1 (Ω) instead of S j . Thus, we have established that the pair ( u 0 , T 0 ) is a weak solution of problem (1)-(4).
Moreover, in view of estimates (16) and (17), we obviously have the inequalities (7) and (8) with u = u 0 and T = T 0 . The proof of Theorem 1 is complete.

Conclusions
In this paper, we considered nonlinear Boussinesq-type equations describing the heat transfer and steady viscous flows of weakly concentrated water solutions of polymers in a bounded three-dimensional domain with sufficiently smooth boundary. We proved the existence of weak solutions in suitable function classes. Besides, some estimates for weak solutions are obtained in terms of the data of this model.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
For the reader's convenience, let us state an important generalization of the Brouwer fixed-point theorem, which is used in our proof.
Then, for any λ ∈ [0, 1], the equation F( x, λ) = 0 has at least one solution x λ , which belong to the ball B r .