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Compressible Navier-Stokes-Fourier flows at steady-state

The heat conducting compressible viscous flows are governed by the Navier-Stokes-Fourier (NSF) system. In this paper, we study the NSF system accomplished by the Newton law of cooling for the heat transfer at the boundary. On one part of the boundary, we consider the Navier slip boundary condition, while in the remaining part the inlet and outlet occur. The existence of a weak solution is proved via a new fixed point argument. With this new approach, the weak solvability is possible in Lipschitz domains, by making recourse to \(L^q\)-Neumann problems with \(q>n\).Thus, standard existence results can be applied to auxiliary problems and the claim follows by compactness techniques. Quantitative estimates are established.


Introduction
The heat conductive flows are described by a coupled system consisting of the equations of continuity, motion and energy. The study of compressible flows depends on the knowledge of solving the continuity equation, because this equation has its shortcomings. We refer to [2] for the existence of stationary solutions if the transport coefficients are, at least, of class W 2,p (Ω) with p > n.
Several works deal with barotropic flows, where the pressure is a function of the density only. To cover the physical point of view, namely, the adiabatic exponent γ = 5/3 for the monoatomic gases or γ = 7/5 for the diatomic gases at ordinary temperature 150 K to 600 K, the imposed assumption on the pressure has being studied in function of the adiabatic exponent γ. To deal with this, the renormalized bounded energy weak solutions, in the context of the theory introduced by P.L. Lions [20], are proved for γ ≥ 5/3 if n = 3. Since then the adiabatic exponent is becoming realistic. In [12], the renormalized bounded energy weak solutions are proved under the assumption that the adiabatic exponent satisfies γ > 4/3. We refer the existence of renormalized weak solutions for γ > (3 + √ 41)/8 to [3], for the flows powered by volume potential forces in a rectangular domain with periodic boundary conditions, and recently, for γ > 1 to [26], in a bounded domain with no-slip boundary condition. For a general case, the existence of a fixed point to the Navier-Stokes system is applied in [29] by using the Schauder theorem under smallness of the H 3 -norm for the velocity field if providing the system by smooth coefficients. The higher order derivatives are essential in establishing the estimate of div u. We remind that a fluid that flows at low velocity is described by the Stokes equations and not by the Navier-Stokes equations.
Nonisothermal steady state studies are well known and there exists a vast literature under the Dirichlet condition, for instance, on optimal control of low Mach number [17] and on uniqueness [24] and the literature cited therein. The better regularity of solutions by introducing the effective viscous flux G = p−(2µ+λ)div u is only possible under constant viscosities µ and λ (see [12], and the references therein). With this assumption, the authors in [23] prove the existence of weak solutions by replacing, in NSF system, the energy equation by the total energy equation. This new system has the particularity of adding the equations, the pressure and the dissipation disappear, in the establishment of the crucial estimates.
Here, we consider the transport coefficients as temperature and spatial dependent. The behavior of the transport coefficients do not allow standard techniques [9] as, for instance, the use of either the above G or the inverse of the Stokes operator.
The inhomogeneous boundary value problems are, in contrast, less common. We refer to [25] to the existence of continuous strong solutions to NSF problem under the assumptions that the Reynolds number and the inverse viscosity ratio are small and the Mach number Ma 1.
The study of the NSF system that the source/sink is the heat transfer at the boundary, which is given by the Newton law of cooling, can be applicable to the physical situations such that come from biomedical engineering (as, for instance, thermal ablation for the treatment of thyroid nodules [4,27]) as well as geological engineering (as, for instance, the natural gas flow in wells at the region that a single phase occurs).
A priori estimates are the core in a fixed point argument. However, they are usually deduced from the boundedness propriety of the operators. Then, there exist a universal constant that is abstract, that is, it does not reflect the data dependency. To fill this gap, additional attention is payed in the determination of quantitative estimates in which the dependence on the data is explicit.
The outline of this paper is as follows. Next section is concerned for modeling of the problem under study and the description of the model itself. Section 3 is devoted to the mathematical framework, the establishment of the data assumptions, and the statement of the main theorems. In Section 4, we delineate the fixed point argument. The following sections (Sections 5, 6 and 7) concentrate on the wellposedness of three auxiliary problems, namely a Dirichlet-Navier problem for the velocity field, a inlet/outlet problem for the density scalar and a Dirichlet-Robin problem for the temperature. The remaining sections (Sections 8 and 9) are devoted to the proofs of the main theorems, respectively, Theorems 3.1 and 3.2.

Statement of the problem
Let Ω be a bounded domain (connected open set) of R n , (n = 2, 3), with Lipschitz boundary. The boundary ∂Ω consists of three pairwise disjoint relatively open (n − 1)dimensional submanifolds, Γ in , Γ out and Γ, with positive Lebesgue measures, whose verify where cl stands for the set closure. The heat conducting fluid at steady-state is governed by the Navier-Stokes-Fourier equations Here, the unknown functions are the density ρ, the velocity field u, and the specific internal energy e. We denote ζ : ς = ζ ij ς ij taking into account the convention on implicit summation over repeated indices. The gravitational force g and the dissipation σ : Du are negligible. Notice that the neglecting the external force fields does not imply that the fluid is at rest. Indeed, the fluid flow is driven both by inlet and outlet flows and by heat transfer on the boundary. In the case of ideal gases, the specific internal energy e is related with the absolute temperature θ by the linear relationship e = c v θ, where c v denotes the specific heat capacity of the fluid at constant volume. Thus, the energy equation (3) can be written in terms of the temperature. Assuming that the thermal conductivity k is a function dependent on both temperature and space variable, the smoothness of the temperature depends on this coefficient.
The pressure p in the case of ideal gases obeys to the Boyle-Marriotte law where R specific = R/M is the specific gas constant, with R = 8.314 J mol −1 K −1 being the gas constant and M denoting the molar mass.
To understand the range of values we are talking to about, we exemplify some well known values for the dry air. For the air (assumed to be at the atmospheric pressure p = 101.325 kPa), the molar mass of dry air is M = 28.96 kg kmol −1 at temperature θ = 298.15 K (= 25°C), then the density ρ = 1.184 kg m −3 . Thus, we have R specific = 287 J kg −1 K −1 . The dry air can be assumed as diatomic, then c v = 5R/2. The dynamic viscosity µ/2 = 0.018 mPa s and the bulk viscosity ν = 0.8µ/2 [16]. Similar values are known for O 2 (see Table 1).
The triple point of the air is reached at temperature of 59.75 K (= −213.4°C) and a correlated pressure (which value varies from author to author because how it is assumed the air composition). Thus, a minimum temperature θ 0 is admissible. The values for the velocity, however, range from that the flow has Reynolds number Re 1, in which case is described by the Stokes equation, until the flow behaves in the turbulent regime of Re ≥ 10 6 . This means Re > 6.5 × 10 4 vL, with v standing for an average velocity and L the maximum length of the cross-section of the domain, in the above conditions. We notice that, in this work, we only assume as constant the specific heat capacity. This assumption is essential to leave the thermal conductivity as space variable dependent, by replacing the specific internal energy by the temperature as an unknown to seek. We leave all the remaining coefficients dependent on the temperature (see, for instance, Table 1) and on the space variable.
This represents both the inflow (u in := u D · n < 0) and outflow (u out := u D · n > 0). On the remaining boundary Γ, the fluid do not penetrate the solid wall, and it obeys the Navier slip boundary condition where n stands for the unit outward vector to Γ, u N , u T are the normal and tangential components of the velocity vector, respectively, τ T = τ · n − τ N n and τ N = (τ · n) · n are the tangential and normal components of the deviator stress tensor τ = σ + pI, respectively, and γ denotes the friction coefficient. For the heat transfer conditions, it is admissible to assume prescribed temperature in the inlet, that is, we consider the Dirichlet condition For the sake of simplicity, we assume θ in as a positive constant. Alternatively, we might assume that θ in may be extended to a functionθ in ∈ H 1 (Ω). On the boundary Γ N = ∂Ω \ cl(Γ in ), we assume the Newton law of cooling where h c denotes the heat transfer coefficient and θ e represents a given (eventually nonconstant) external temperature. This condition is mathematically known as the Robin condition. The heat source/sink is completely driven from the boundary and we denote

Main Results
We assume that Ω ⊂ R n is a bounded domain with its boundary ∂Ω ∈ C 0,1 . The standard notation of Lebesgue and Sobolev spaces is used. Let us define the Hilbert spaces endowed with the norms, respectively, The meaning of the condition v · n = 0 on Γ should be understood as subject to (6), (7) and (9). Here, q stands for the conjugate exponent of q, i.e. 1/q + 1/q = 1, and h = h c θ e .
Remark 3.1. The variational formulations (11)-(13) are standardly derived from the NSF system (1)-(3) by the Green formula. We point out that the general formula The following assertions on the physical parameters appearing in the equations are assumed: (H1): The viscosities µ and λ are Carathéodory functions from Ω × R into R such that for a.e. x ∈ Ω and for all e ∈ R.
for a.e. x ∈ Ω and for all e ∈ R. (H3): The friction coefficient γ is a continuous function from R into R such that for all e ∈ R. Moreover, h = θ e h c with the function θ e ∈ L ∞ (Γ N ). (H5): The boundary term ρ ∞ u D · n ∈ L q (Γ D ), for some q > n, satisfy the compatibility condition There exists u D ∈ H 1 (Ω) such that its trace u D = u D on Γ D and the normal component of trace vanishes on Γ. Indeed, the trace operator has a continuous right inverse operator, and in particular it is surjective from W 1,q (Ω) onto Remark 3.2. We denote by p * = pn/(n − p) the critical Sobolev exponent related to the embedding W 1,p (Ω) → L p * (Ω), if p < n. For the sake of simplicity, we also denote by p * any real value greater than one, if p = n. The Rellich-Kondrachov embedding stands for any exponent between 1 and the critical Sobolev exponent p * . Notice that the Morrey embedding W 1,q (Ω) → C 0,1−n/q (Ω) holds for q > n.
Let us state our first main theorem, where the density function is only defined a.e. in Ω.
Theorem 3.1. Let the assumptions (H1)-(H5) be fulfilled. For any M ∈ N, there exists a triplet (ρ, u, θ) such that • ρ is a measurable function satisfying ρu ∈ L q (Ω), with n < q < n + ε, for some ε depending on Ω; , which is a weak solution to the NSF problem, with (6) replaced by Here, T M stands for the truncation, i.e. T M (z) = z for 0 ≤ z ≤ M and T M ≡ M otherwise.
Let us state our second main theorem, where the density function is assumed to have L r -regularity, for some r > 2n/(4 − n) (n = 2, 3).
for some positive constant R independent on M and r verifying (23). Moreover, the following quantitative estimates hold, where R 1 , R 2 and R 4 are defined in (50), (51) and (58), respectively.
Remark 3.4. The quantitative estimate (26) may be simplified if, for instance, in the assumption (H5) we assume the existence of u D ∈ H 1 (Ω) having the trace

Strategy
Our strategy is based on that the velocity field is not admissible to use for the fixed point argument, because the velocity field is not directly measurable and the linear momentum is easier to be physically determined.
For fixed q > n and r > 2, we define the closed set Here, we consider three auxiliary problems.
(Dirichlet-Navier problem): The auxiliary velocity w ∈ V is the unique solution to the Dirichlet-Navier problem defined by with (Inlet/outlet problem): The auxiliary density ρ is a unique solution to the inlet/outlet problem defined by (Dirichlet-Robin problem): The auxiliary temperature θ − θ in ∈ H 1 in (Ω) is the unique weak solution to the Dirichlet-Robin problem defined by Finally, the auxiliary pressure is given by (24). Let us establish some properties of the linearized convective and advective terms, which are the key points of this paper. Let Ω ⊂ R n be a bounded Lipschitz domain. For each m ∈ L q (Ω), q > n, which verifies (29), the following functionals are well defined and continuous: Moreover, B is skew-symmetric in the sense and, in particular, holds for any e, v ∈ H 1 (Ω).

Wellposedness of the Dirichlet-Navier problem
The following properties are well known in the fluid mechanics theory. However, the quantitative estimate is essential in the fixed point argument and we will fix it.
Proof. This proof is standard, but we sketch it because its quantitative expression.
The following proposition asserts the existence and uniqueness of auxiliary velocity field.
Proof. Let a be the (non symmetric) bilinear form on V × V, which is associated to the energy functional J : V → R, defined by The existence of J is well-defined [28, Appendix C] as the Fréchet derivative of a Nemytskii operator F : Ω × R n × M n×n sym → R, and the form a is sum of the non symmetric and symmetric parts Then, the existence and uniqueness of solution are consequence of the Lax-Milgram Lemma [19].
We finalize this section by proving the continuous dependence.

Proposition 5.3 (Continuous dependence).
Let {(m m , ξ m , π m )} m∈N be a sequence weakly convergent in K q,r , for some q > n and r > 2. Then, the corresponding solutions w m = w(m m , ξ m , π m ) to the problem (30) m , for each m ∈ N, weakly converge to w = w(m, ξ, π) in V, which is the solution to the problem (30) corresponding to the weak limit (m, ξ, π).
Proof. Let us take the sequences m m m in L q (Ω) for some q > n; ξ m ξ in H 1 (Ω); π m π in L r (Ω) for some r > 2.
The above convergences allow to pass to the limit as m tends to infinity in (30) m , concluding that w satisfies the system (30).

Existence and uniqueness of density solution
In this section, our objective is not to apply the artificial viscosity technique that approximates the continuity equation by an elliptic equation through a vanishing viscosity (also known as elliptic approximation) as it has being usual. Our argument goes out in the spirit of the Helmholtz decomposition a = a ω + a ψ , where • a ω = ∇ × ω stands for the solenoidal (divergence-free) component, i.e. it satisfies ∇ · a ω = 0 in Ω. • a ψ = ∇ψ stands for the irrotational component (curl-free), i.e. it satisfies ∇ × a ψ = 0 in Ω.
We refer to [13] for the weak L q -solution to the Dirichlet-Laplace problem being motivated by the Weyl decomposition.
On the other hand, we find the corresponding vector that makes possible the decomposition.
By the potential theory [1,22], it suffices to seek for a unique non-negative density function that satisfies ρu = ∇ψ + z, (41) with z belonging to Taking in (41) the inner product with u, we obtain otherwise, we define ρ = ρ 0 in Ω[|u| = 0] (cf. Remark 6.1). Hereafter, the set A[S] means {x ∈ A : S(x)}, with S denoting a sentence to be pointwisely (a.e.) satisfied in A, which may represent either Ω or Γ. Taking in (41) the inner product with u ⊥ = (−u 2 , u 1 ) ∈ L p (Ω), for any 1 < p < ∞, we find the relation taking ψ = (0, 0, ψ) into account. We emphasize that the absolute value of the real number ∇ × ψ · u/|u| = ∇ψ · u ⊥ /|u| ∈ L q (Ω) is the magnitude of the vector rejection of ∇ψ from u, which is defined as where the right hand side stands for the vector projection of ∇ψ onto u. In the following, for the sake of simplicity, we assume that ∇ψ · u ⊥ ≥ 0, otherwise we may similarly argue by redefining u ⊥ = (u 2 , −u 1 ). Let us consider the following two cases. Case 1. If ∇ × ψ · u = 0, it means that ∇ψ = ±|∇ψ| u |u| .
In accordance with the latter case, we define the vector a ∈ L q (Ω) as follows This definition captures both cases (a) ∇ψ · u > 0 and (b) ∇ψ · u ≤ 0 (see Fig. 1).
The existence and uniqueness of this scalar potential in the quotient space W 1,q (Ω)/R is guaranteed by the range values of q for Lipchitz domains. Setting it is unique and it belongs to H q . Moreover, we have max{ ∇ϕ q,Ω , F q,Ω } ≤ C q a q,Ω , where C q depends only on q and the Lipschitz character of Ω. Taking (44) and the decomposition (43) for ϕ, we have It remains to evaluate the last term of the above relation, namely Or, equivalently f t (t, n) = 0 f n (t, n) = ∇ϕ · e n , by taking the change of coordinates e t e n = u 1 /|u| u 2 /|u| −u 2 /|u| u 1 /|u| into account. Hence, we define z t and z n such that z n = f n and z t = − ∂ n z n dt + ρ 3 (n), with ρ 3 being such that z t < 0. By returning to the Cartesian coordinate system, the boundary condition z · n = 0 guarantees the uniqueness of ρ 3 . Finally, we choose the vector z ∈ H q such that z = F + z t e t + z n e n , by recalling the vector F from (47). From (40), the function ρ verifies the variational formulation (31), which concludes the proof of Proposition 6.1.
Remark 6.1. We call by ρ 0 the constant density at STP (standard temperature and pressure). Notice that the velocity may be zero, the so called stagnation. The no upper boundedness of the density is related to that |u| → 0 means ρ → ∞. However, neither the velocity function nor the density function are continuous. It suggests that some upper boundedness will be possible, but it is still an open problem.
In the three dimensional space, arguing as in the two dimensional Proposition 6.1 we seek for ρ = (∇ψ + ∇ × ω) · u/|u| 2 in Ω[|u| = 0] (49) otherwise, we define ρ = ρ 0 if Ω[|u| = 0] (cf. Remark 6.1). As the objective is to find a scalar function, the argument of the proof of Proposition 6.1 may be repeated in the plane formed by the vectors u and ∇ψ, i.e. we consider the local coordinate system (e t , e n , 0), where e t = u/|u| and e n = ∇ψ × u/|∇ψ × u|.
Finally, we are in conditions to determine the estimate for the linear momentum (cf. (39)). Corollary 6.1. Let Ω be Lipschitz. For n = 2, 3, let ρ be the unique function given at Propositions 6.1 and 6.2. Then, the estimate ρu q,Ω ≤ 3C q ρ ∞ u D · n q,∂Ω := R 1 (50) holds, for any n < q < n + ε and n = 2, 3, with ε depending on Ω and C q depending on n, q, and the Lipschitz character of Ω.

Wellposedness of the Dirichlet-Robin problem
The existence of the solution θ ∈ H 1 (Ω), which satisfies (9), to the problem (32) is stated in the following proposition.
Proof. The existence and uniqueness of θ = u + θ in , with u ∈ H 1 in (Ω), solving (32) is standard by the Lax-Milgram Lemma. The problem (32) reads where the continuous bilinear form a from H 1 in (Ω) × H 1 in (Ω) into R, is defined by Moreover, using the assumptions (17) and (20)- (21), the form a is coercive: taking u 2 ∈ W 1,q (Ω) into account, that is, (29) reads The estimate (51) follows by choosing v = θ − θ in as a test function in (32), arguing as above, considering that ∇u = ∇θ and 2,Γ N after routine computations.
The following minimum-maximum principle is standard, its proof argument differs on the advective and boundary terms. For reader convenience, we provide the proof. Since the advective term verifies taking (29) and next (20)-(21) into account, we deduce Then, we conclude that φ(θ) = 0 in Ω, which means that the lower bound is proved. The upper bound is analogously proved, by defining T max = ess sup{θ 0 (x) : x ∈ ∂Ω} and choosing φ(θ) = (θ − T max ) + = max{θ − T max , 0} ∈ H 1 in (Ω) as a test function in (32).
By the other hand, from ξ m → ξ in L 1 (∂Ω) and a.e. on ∂Ω, and the assumptions (19)- (21), the continuity property of the Nemytskii operator associated to the boundary coefficient h c implies that For each m ∈ N, let θ m = θ(m m , ξ m ) be the corresponding solution to the problem (32) m . The uniform estimate (51) allows to extract at least one subsequence, still denoted by θ m , of the solutions θ m = θ(m m , ξ m ) weakly convergent for some θ ∈ H 1 (Ω).
The above convergences do not be sufficient to the passage to the limit, as m tends to infinity, in (32) m . It remains to pass the advective term to the limit. To this aim, we prove the following strong convergence ∇θ m → ∇θ in L 2 (Ω). Arguing as in [5], we apply the assumption (17) and we decompose to obtain Next, to prove that I 1 also tends to zero, we take v = θ m − θ as a test function in (32) m . Hence, we obtain taking the Rellich-Kondrachov embeddings H 1 (Ω) → → L p (Ω), with p < 2 * , and H 1 (Ω) → → L 2 (∂Ω) into account for n = 2, 3. Then, applying the relation (29) into the left hand side of the above equality, we find the claim, i.e. the strong convergence. Then, the passage to the limit yields that θ satisfies (32), concluding Proposition 7.3.

Existence of a fixed point to the problem (Proof of Theorem 3.1)
We will apply the following Tychonoff extension to weak topology of the Schauder fixed point theorem [10, pp. 453-456 and 470].
Theorem 8.1. Let K be a nonempty weakly sequential compact convex subset of a locally convex linear topological vector space V . Let T : K → K be a weakly sequential continuous operator. Then T has at least one fixed point.
Let V = L q (Ω) × H 1 (Ω) × L r (Ω) and K q,r be the nonempty convex set defined in (28). We define K = K q,r ∩ B, where B is the closed (bounded) ball, with radius R 1 , R 2 , R 3 > 0 defined in (50), (51) and (54), respectively. In the reflexive Banach space V , the closed, convex and bounded set K is compact for the weak topology σ(V, V ), i.e. it is weakly sequential compact.
Let T be the operator defined in Section 4. The fixed point argument (cf. Theorem 8.1) guarantees the existence of the required solution, by proving the following two propositions, namely, Propositions 8.1 and 8.2.
Proposition 8.1. Let the assumptions (H1)-(H5) be fulfilled. Then, the operator T is well defined and it maps K into itself.
Thanks to Corollary 6.1, the quantitative estimate (50) guarantees the existence of R 1 , for q depending on the smoothness of the domain Ω. Thanks to Proposition 7.1, the quantitative estimate (51) guarantees the existence of R 2 .
The existence of R 3 is due to the definition (24), we concretely have p M r,Ω ≤ M |Ω| 1/r R specific ess sup by considering the estimate (52). π m π in L r (Ω).
Thanks to Proposition 5.3, the corresponding solutions w m = w(m m , ξ m , π m ) ∈ V to the problem (30) m , for each m ∈ N, weakly converge to the solution w = w(m, ξ, π) to the problem (30). Thus, we get u m u in H 1 (Ω).
Consequently, we get u m → u a.e. in Ω. Notice that u satisfies − Ω m × u : ∇v dx + Ω µ(ξ)Du : Dv dx + Ω λ(ξ)∇ · u∇ · v dx for all v ∈ V, and the convective term verifies (33). Let ρ m be the unique solution given at Propositions 6.1 and 6.2, for n = 2, 3, respectively. Then, it follows that ρ m a.e. converges to ρ in Ω. Thanks to Corollary 6.1, we have ρ m u m ρu in L q (Ω), which limit satisfies (31). Thanks to Proposition 7.3, the corresponding solutions θ m = θ(m m , ξ m ) to the problem (32) m , for each m ∈ N, weakly converge to the solution θ = θ(m, ξ) in H 1 (Ω). Thus, θ m strongly converges to θ in L p (Ω), for 1 < p < 2 * . Thanks to (54) and the Lebesgue dominated convergence theorem, we have Then, the operator T is weakly sequential continuous, which finishes the proof of Proposition 8.1.
Therefore, we are in condition to obtain the fixed point (m, ξ, π) = (ρu, θ, p M ), which is the required solution. Finally, the argument of Proposition 7.2, with the auxiliary problem (32) being replaced by the variational problem (13), can be applied