A large number of works are devoted to the issues concerning mathematical modeling the processes of fluid flows taking into account convective effects. These papers explore both applied questions, concerning the derivation and justification of basic convection models, and purely mathematical questions. These include methods for constructing exact and approximate solutions, theoretic-group analysis (or Li-Ovsyannikov symmetry) method of investigating the qualitative properties of solutions to differential equations underlying the convection models under study, and theoretical analysis of the solvability and uniqueness of solutions of boundary value problems for basic convection models.

Among the various convection models for binary and/or thermally conducting liquids, an impotant role is played by so-called Oberbeck–Boussinesq models [1,2], which are derived from the exact Navier-Stokes equations of fluid dynamics taking into account the following assumptions (see [1], Section 54):

1. Convective motion is similar to that of an incompressible fluid with constant density ${\rho }_0$, but the possible deviation of the true density $\rho$ from ${\rho }_0$ is taken into account in the momentum conservation equation in the form of a term describing the additional volumetric force – the buoyancy (Archimedes) force.

2. The change in density is caused by changes in temperature and concentration of dissolved substance, but not changes in the pressure.

3. The velocity gradients are small enough so that the process of transition of work to heat during movement does not lead to a change in the temperature of the medium.

The model that results from these simplifications is called the Oberbeck-Boussinesq model. In turn, the Oberbeck-Boussinesq model allows for further simplification, often called the simplified or classical Boussinesq model.

When modeling heat transfer processes in a viscous heat-conducting liquid, by (classical) Boussinesq model one usually understands the model in which the bulk buoyancy force included in the momentum conservation equation linearly depends on temperature, and, besides, the main parameters of the fluid, namely: the viscosity coefficient and the thermal conductivity coefficient are positive constants. When modeling mass transfer processes in a binary fluid, by (classical) Boussinesq model one usually understands the model in which the bulk buoyancy force (included in the momentum conservation equation) linearly depends on the concentration of dissolved substance while the main parameters, namely: the viscosity coefficient, diffusion coefficient, as well as another parameter, called the reaction coefficient, are the positive constants. The latter parameter is responsible for the possible decay of the dissolved substance in the main medium due to the chemical reaction. When all or some of the above conditions are not fulfilled, the corresponding model is often referred to as a generalized Boussinesq model.

Let us emphasize that for the classical Boussinesq model, many theoretical issues are quite fully studied. This, in particular, holds for the study of the correctness of boundary or initial-value problems for stationary or non-stationary models of heat and mass transfer. Among many works in this area, we note the cycle [3,4,5,6,7,8] of works by the first author and his coauthors on the study of the correctness of boundary value problems for stationary equations of heat and/or mass transfer. We emphasize that in these works, in addition to the study of correcteess a boundary value problems, a theoretical analysis of control problems for the models of heat and mass transfer was performed. The analysis of the results obtained in [3,4,5,6,7,8] made it possible to identify interesting regularities related to the interaction of hydrodynamic and thermal fields in binary and/or heat-conducting media and, in particular, to establish the most effective mechanisms for controlling thermohydrodynamic processes in viscous liquids. The close problems of boundary or distributed control for the heat transfer equations in the Boussinesq approximation have also been investigated in the works [9,10,11,12,13,14].

Theoretical questions for the generalized Boussinesq model have been studied to a much lesser extent. However, significant progress has been made in recent years in this area as well. Over the past decades, a large number of papers have been published regarding the study of heat and mass transfer equations with variable transfer coefficients and with variable buoyancy force depending on temperature and/or concentration of dissolved substance. These works can be divided into several groups. The first group contains papers that develop methods for finding exact solutions to these equations (see, for example, [15,16,17,18,19] and review [20]). The second group contains works devoted to application of Li-Ovsyannikov symmetry method to study qualitative properties of solutions of equations of heat and mass transfer in viscous binary and/or heat-conducting liquids. This group includes very large quantily of works (see e.g. [21,22,23,24,25], monograph [26] and reviews [27,28]). Another group of works is that in which mathematical modeling of fluid motion processes takes into account thermodiffusion effects (or Sorét effects) and/or concentration diffusion effects (or Dufort effects). A detailed list and analysis of these works can be found in the reviews [27,28].

At the same time, the authors know only a few papers in which the solvability of boundary value problems for equations of heat and mass transfer with variable coefficients is investigated. The mentioned works can be divided into several groups. The first group includes works [29,30,31] or [32], devoted to the study of the solvability of boundary value problems for stationary equations of heat or mass transfer. The second group is joined by works [33,34,35,36,37,38], devoted to the study of the solvability of boundary value and control problems for non-stationary Boussinesq equations of heat (or mass) transfer. The works [39,40,41] form once more group in which the solvability of boundary value and control problems is studied for the stationary mass transfer model in the case where the reaction coefficient can depend on the concentration of matter and spatial variables.

Close questions on the study of correctness of boundary value or control problems for stationary equations of magnetic hydrodynamics of viscous incompressible or heat-conducting liquid in the Boussinesq approximation were investigated in [42,43,44,45,46]. In [47], the solvability of the initial-boundary problem for the non-stationary MHD-Boussinesq system, considered under mixed boundary conditions for velocity, magnetic field, and temperature, in the case when the viscosity coefficient, magnetic permeability, electrical conductivity, thermal conductivity and specific heat of the fluid depend on the temperature.

Finally, we mention papers [48,49,50,51,52,53,54,55] that touch upon issues close to the subject of this paper from nonlinear diffusion, viscoelasticity, engineering mechanics, complex heat exchange, acoustics and oceanology.

The purpose of this work is to analyze the global solvability and local uniqueness of solutions of the boundary value problem for a generalized Boussinesq mass transfer model describing the flow of binary fluid in which the diffusion, viscosity and reaction coefficients and the buoyancy force depend on the substance concentration.

The paper is organized as follows. In Section 2, we will formulate the main boundary value problem, to which we will refer below as Problem 1. Besides, we introduce functional spaces and formulate a number of auxiliary results in the form of Lemmas 1, 2 and 3 which will be used when studying the solvability and uniqueness of Problem 1. In Section 3, we will formulate and prove the theorem on the global existence of a weak solution to Problem 1 and establish the maximum principle for substance concentration $\phi$. In Section 4, we will establish sufficient conditions on the data of Problem 1 that provide conditional uniqueness of the weak solution having an additional property of smoothness for concentration. The last Section 5 (Conclusion) contains a brief summary of the results obtained in our paper.

Let $\Omega$ be a bounded domain in the space ${\mathbb{R}}^3$ with a Lipschitz boundary $\Gamma$. Below, we will consider the following boundary value problem describing the motion of binary fluid within the framework the generalized Boussinesq model of mass transfer: Here $\mathbf{u}$ is the velocity vector, $\phi$ is the concentration of dissolved substance, $p=P/{\rho }_0$, where P is the pressure, $\rho =\mathrm{const}$ is the fluid density, $\nu =\nu \left(\phi \right)>0$ is the kinematic (molecular) viscosity coefficiemt, $\lambda =\lambda \left(\phi \right)>0$ is the diffusion coefficient, $b\equiv b\left(\phi \right)$ is the mass expansion factor, $k=k\left(\phi \right)$ is the reaction coefficient, $\mathbf{G}=-(0,0,G)$ is the gravitational acceleration, $\mathbf{f}$ or f is the bulk density of external forces or of the external sources of matter, respectively. Below the problem (1) – (3) for the given functions $\nu \left(\phi \right)$, $\lambda \left(\phi \right)$, $b\left(\phi \right)$, $k\left(\phi \right)$, $\mathbf{f}$, and f will be referred to as Problem 1.

When studying Problem 1, we will use the Sobolev functional spaces $H^s\left(D\right)$, $s\in \mathbb{R}$. Here, D means either domain $\Omega$ or some subset $Q\subset \Omega$, or the boundary $\Gamma$. By ${\parallel ·\parallel }_{s,Q}$, ${|·|}_{s,Q}$ and ${(·,·)}_{s,Q}$ the norm, half-norm and in inner product in $H^s\left(Q\right)$ will be defended, respectively. The norms and scalar product in $L^2\left(Q\right)$ or in $L^2(\Omega )$ are denoted by ${\parallel ·\parallel }_Q$, ${(·,·)}_Q$ or by ${\parallel ·\parallel }_{\Omega }$, and $(·,·)$, respectively. $X^{*}$ denotes the dual space of Hilbert space X, while the duality relation for the pair of dual spaces X and $X^{*}$ is written as ${〈·,·〉}_{X^{*}\times X}$, or simply as $〈·,·〉$.

An important role in our analysis will be played by the following functional spaces:

Note that each of the spaces $H^1(\Omega )$ and $H_0^1(\Omega )$ is Hilbert at norm ${\parallel ·\parallel }_{1,\Omega }$ which is equivalent to ${|·|}_{1,\Omega }$ for $\phi \in H_0^1(\Omega )$ due to the Friedrichs-Poincaré inequality It is well known (see, for example, [56,57]) that for the domain $\Omega$ with the Lipschitz boundary the spaces H and V are characterized as follows:

Let us define the products of spaces $X=H_0^1{(\Omega )}^3\times H_0^1(\Omega )$ and $W=V\times H_0^1(\Omega )\subset X$ with the norm and denote by $X^{*}$ the space $(H^{-1}{(\Omega )}^3\times H^{-1}(\Omega )$ which is dual of X.

Let the following conditions be met:

2.1.$\Omega$ is a bounded domain in ${\mathbb{R}}^3$ with a boundary $\Gamma \in C^{0,1}$ consisting of N component ${\Gamma }^{\left(i\right)}$, $i=1,2,...,N$.

2.2.$\mathbf{f}\in H^{-1}{(\Omega )}^3$, $f\in H^{-1}(\Omega )$, $\psi \in H^{1/2}(\Gamma )$.

2.3.$\mathbf{g}\in H^{1/2}{(\Gamma )}^3$, ${(\mathbf{g},\mathbf{n})}_{{\Gamma }^{\left(i\right)}}=0$, $i=1,2,...,N$.

2.4. For any function $\phi \in H^1(\Omega )$, the embedding $b\left(\phi \right)\in L^p(\Omega )$ is valid where $p\ge 3/2$ is a fixed number independent of $\phi$, and the vector-function $\mathbf{b}\left(\phi \right)\equiv b\left(\phi \right)\mathbf{G}$ satisfy Here ${\widehat{\beta }}_p$ is a positive constant dependent on p. In addition, for any pair of functions ${\phi }_1,{\phi }_2\in H^1(\Omega )$ belonging to sphere $B_r=\{\phi \in H^1{(\Omega ):\parallel \phi \parallel }_{1,\Omega }\le r\}$ of radius r, the following inequality is true: Here $L_b$ is a constant that depends on b and on r, but does not depend on ${\phi }_1,{\phi }_2\in B_r$.

2.5. For any function $\phi \in H^1(\Omega )$, the embedding $k\left(\phi \right)\in L_{+}^p(\Omega )$ is valid, where $p\ge 3/2$ is a fixed number independent of $\phi$, and the following estimate for ${\parallel k\left(\phi \right)\parallel }_{L^p(\Omega )}$ is valid Here ${\widehat{\gamma }}_p$ is a positive constant dependent on p. Also for any sphere $B_r=\{\phi \in H^1{(\Omega ):\parallel \phi \parallel }_{1,\Omega }\le r\}$ the following inequality: holds. Here $L_k$ is a constant that depends on k and on r, but does not depend on ${\phi }_1,{\phi }_2\in B_r$.

2.6.$\nu \in C^0\left(\mathbb{R}\right)$, $\lambda \in C^0\left(\mathbb{R}\right)$ and there are positive constants ${\nu }_{min}$, ${\nu }_{max}$, ${\lambda }_{min}$ and ${\lambda }_{max}$ such that

Consider the function $\mu \in C^0\left(\mathbb{R}\right)$, satisfying the following condition: Clearly $\mu \left(h\right)\in L^{\infty }(\Omega )$ for any $h\in H^1(\Omega )$ and the following estimates: hold. In addition, $\mu \left(h_n\right)\to \mu \left(h\right)$ a.e. in $\Omega$ if $h_n\to h$ a.e. in $\Omega$ as $n\to \infty$.

Let ${\mu }_n=\mu \left(h_n\right)$. Since $|\mu \left(h_n\right)|\le {\mu }_{max}$ a.e. in $\Omega$, the Lebesgue’s theorem on majorant convergence implies that The property (11) will play an important role when proving the solvability of Problem 1 which contains the variable leading coefficients $\nu \left(\phi \right)$ and $\lambda \left(\phi \right)$.

Below, we will often use the following inequalities: Here $C_s$ is a constant dependent on $\Omega$ and $s\in [1,6]$, $C^p$ is a constant dependent on $\Omega$ and p at $p\ge 3/2$. The inequality (12) is a consequence of Sobolev’s embedding theorem, according to which the space $H^1(\Omega )$ is imbedded into $L^s(\Omega )$ continuously at $s\le 6$ and compactly at $s<6$. The inequality (14) is a consequence of the H$\ddot{o}$lder inequality for the three functions. In turn, the following inequality: is the consequence of inequalities (12) and (14) where A similar inequality holds for the scalar functions $q,\phi$, and h. It has the form

Along with inequalities (12)–(16), we will use a number of other important inequalities and properties of bilinear and trilinear forms, which we will write as the following Lemma.

We prove, for example, (27) and establish a relationship between the constant ${\beta }_p$ and the constant ${\widehat{\beta }}_p$ defined in (5). To do this we use the inequality (15) at $\mathbf{q}=\mathbf{b}\left(\phi \right)$ and property (5). Using these relations successively we have: Proof of the remaining statements constituting Lemma 1 we leave to the reader. They can also be found in [56,57,58,59].

Similarly, the following estimate for the difference $k\left({\phi }_1\right)-k\left({\phi }_2\right)$ follows from the estimate (25) and the property (8) of the function $k\left(\phi \right)$:

The following lemmas about the existence of liftings for the velocity and concentration will play an important role below.

Our nearest goal is to derive the weak formulation of Problem 1 and to prove the existence of its solution. To this end, we multiply the first equation in (1) by the function $\mathbf{v}\in H_0^1{(\Omega )}^3$, equation (2) by the function $h\in H_0^1(\Omega )$ and integrate the result over $\Omega$ using Green’s formulas. As a result, we will get a weak formulation of Problem 1. It consists in finding a trio of functions $(\mathbf{u},\phi ,p)\in H^1{(\Omega )}^3\times H^1(\Omega )\times L_0^2(\Omega )$ that satisfy the relations: The specified three functions $(\mathbf{u},\phi ,p)$ satisfying (31) – (33), will be called below a weak solution to Problem 1.

Consider the restriction of the identity (31) by the space V which, taking into account the condition $\mathrm{div}\mathbf{v}=0$ for $\mathbf{v}\in V$, becomes: It is well known that for the proof of existence of a weak solution of Problem 1 it is enough to prove existence of the solution $(\mathbf{u},\phi )\in H^1{(\Omega )}^3\times H^1(\Omega )$ of problem (32)–(34), and then, using the standard scheme, to restore the pressure $p\in L_0^2(\Omega )$ so that the identity (31) is fulfilled. One can read more about pressure recovery in [57], p. 134, [58], Section 3.2.

We will look for the solution $(\mathbf{u},\phi )\in H^1{(\Omega )}^3\times H^1(\Omega )$ of problem (32)–(34) in a form Here ${\mathbf{u}}_0\equiv {\mathbf{u}}_{{\epsilon }_0}$ and ${\phi }_0={\phi }_{{ϵ}_0}$ are the velocity and concentration liftings defined above, and $\tilde{\mathbf{u}}\in V$ and $\tilde{\phi }\in H_0^1(\Omega )$ are the new unknown functions which we are looking for. The value ${\epsilon }_0$ of the parameter $\epsilon$ will be choosen so that the following conditions are fulfilled: The corresponding value ${ϵ}_0$ of the parameter $ϵ$ will be selected below.

Substituting (35) in (32), (34), we come to the following relations with respect to the pair $(\tilde{\mathbf{u}},\tilde{\phi })$:

To prove the existence of the solution $(\tilde{\mathbf{u}},\tilde{\phi })\in V\times H_0^1(\Omega )$ of problem (37), (38), we apply Schauder’s fixed point theorem according to the scheme proposed in [58], Section 4.2. To this end we define the pairs $\mathbf{z}=(\mathbf{w},\eta )\in W$, $\mathbf{y}=(\tilde{\mathbf{u}},\tilde{\phi })\in W$ and construet the operator $F:W\to W$ acting by: $F\left(\mathbf{z}\right)=\mathbf{y}$, where $\mathbf{y}=(\tilde{\mathbf{u}},\tilde{\phi })\in W$ is the solution of the linear problem Here $(\mathbf{w},\eta )\in V\times H_0^1(\Omega )$ is a given pair of functions, functionals ${\mathbf{f}}_1:V\to \mathbb{R}$ and $f_1:H_0^1(\Omega )\to \mathbb{R}$ are defined by formulas

Using (17), (18), (19), (22), (23), (24), (25), (28), (27), (35), (36) and Lemmas 2, 3, we deduce that

From (43), (46), (48), (49), (51), (52) follows that ${\mathbf{f}}_1\in V^{*}$, $f_1\in H^{-1}(\Omega )$ and the following estimates are valid: In addition, for each fixed pair $(\mathbf{w},\eta )\in V\times H_0^1(\Omega )$ bilinear form $a_2^{\mathbf{w},\eta }:H_0^1(\Omega )\times H_0^1(\Omega )$ defined in (40) is continuous and coercive with the constant ${\lambda }_{*}$ defined in (23), while the right-hand side of the identity (40) defines a linear continuous functional over $H_0^1(\Omega )$, and the following estimates hold:

In this case, it follows from the Lax-Milgram theorem that for any pair $(\mathbf{w},\eta )\in V\times H_0^1(\Omega )$ a solution $\tilde{\phi }={\tilde{\phi }}_{\mathbf{w},\eta }\in H_0^1(\Omega )$ of problem (40) exists and is unique. Besides, by (55), (56) and Lemma 3 the following estimates are performed for $\tilde{\phi }$ and $\phi \equiv {\phi }_0+\tilde{\phi }$:

Let us turn to the problem (39) where we put $\tilde{\phi }={\tilde{\phi }}_{\mathbf{w},\eta }$. From (51), (53) and (58) it follows that the right-hand side in (39) is a value on the element $\mathbf{v}\in V$ of the linear continuous functional of $V^{*}$ and we have

In turn, from the estimates (43), (44), (45) and from the second identity in (20) it follows that the bilinear form $a_1^{\mathbf{w},\eta }:V\times V\to \mathbb{R}$ defined in (39), is continuous and coercive with constant $({\nu }_{*}/2)$, where ${\nu }_{*}$ is defined in (18). From Lax–Milgram’s theorem, it then follows that for any pair $(\mathbf{w},\eta )\in V\times H_0^1(\Omega )$ there is a single solution $\tilde{\mathbf{u}}\in V$ of the problem (39) and the following estimate is performed: