On weak (measure valued)–strong uniqueness for Navier–Stokes–Fourier system with Dirichlet boundary condition

In this article, our goal is to define a measure valued solution of compressible Navier–Stokes–Fourier system for a heat conducting fluid with Dirichlet boundary condition for temperature in a bounded domain. The definition will be based on the weak formulation of entropy inequality and ballistic energy inequality. Moreover, we obtain the weak (measure valued)–strong uniqueness property of this solution with the help of relative energy.


Introduction
We consider the Navier-Stokes-Fourier system in the time-space cylinder Q = (0, T )×Ω, where T > 0 and Ω is a bounded domain of R d with d = 2 or 3.The time evolution of the density ̺ = ̺(t, x), velocity u = u(t, x) and the absolute temperature ϑ = ϑ(t, x) of a compressible, viscous and heat conducting fluid is given by the following system of field equations which describes conservation of mass, momentum and internal energy: ∂ t (̺e(̺, ϑ)) + div x (̺e(̺, ϑ)u) + div x q = S : where D x u = 1 2 (∇ x u + ∇ t x u), p = p(̺, ϑ) is the pressure and e = e(̺, ϑ) is the internal energy and g is an external force.The pressure and the internal energy is interrelated by means of Gibbs' equation where s = s(̺, ϑ) is the entropy and D = ∂ ∂̺ ∂ ∂ϑ . We assume the fluid is Newtonian, hence the viscous stress tensor (S) is given by where µ is the shear viscosity coefficient and λ is the bulk viscosity coefficient with µ > 0 and λ ≥ 0. The heat flux q is given by the Fourier law, where κ is the heat conductivity coefficient.Since we consider the bounded domain Ω, our goal is to discuss the solvability of the initial-boundary value problem for the system (1.1)-(1.3)endowed with the constitutive relations (1.5) and (1.6) where we consider homogeneous Dirichlet boundary condition in velocity and in-homogeneous Dirichlet boundary condition in the temperature ϑ| ∂Ω = ϑ B > 0. (1.8) The consideration of (1.8) is closely related to the celebrated Rayleigh-Benard problem.For the above system (1.1)-(1.3)with the consitutive relations (1.4)-(1.8)and the pressure following Boyle's law, the existence of a local in time strong solution was proved by Valli and Zajaczkowski [28].For the global in time weak solutions there are many articles and monographs by P.L. lions, [23] Bresch and Desardin [2], Feireisl and Novotný [17], Bresch and Jabin [3] which focus mainly on space-periodic domains or energy-conserving boundary conditions for the temperature.Also for thermodynamically open systems, Feireisl and Novotný [19] prove the existence of weak solutions, but also require the control of the internal (heat) energy flux q in ∂Ω.Recently, the concept of weak solution was introduced for the particular boundary condition (1.8) in [7] and it also considers very general in flowoutflow boundary data.This weak solution approach is slightly different from the earlier considerations that we will discuss in Section 2. This similar idea is adapted by Pokorný in [27] for steady flows.
The concept of measure valued solutions in the context of inviscid and incompressible fluids has been used by several authors, beginning with the seminal work of DiPerna [9] and DiPerna and Majda [10], see also related results of Kröner and Zajaczkowski [21], Necasova and Novotný [25] and Neustupa [26].Březina, Feireisl and Novotný [5] also introduce the measure valued solution for compressible viscous heat conducting fluids for the "no flux" boundary condition.Their definition follows the approach of considering measure valued solution as a general object as described by Brenier, De Lellis and Szekelyhidi [1], in contrast to the approach described by Malek et.al. [24], where they identify a measure valued solution as a weak limit of weak solutions.We follow here the approach proposed by [5].
For a generalized (weak or measure valued) solution, it is quite important to prove the generalized-strong uniqueness principle.The principle asserts that given the same initial data a weak solution will coincide with the strong or classical solution if the latter exists.For such properties in the context of weak solutions, see Feireisl and Novotný [18] and Feireisl [11].For measure valued solution for the system was avalaible in Březina et al [5].The main idea here is to use the relative energy inequality.It was first introduced by Daefermos [8] for scalar conservation law and the for compressible fluid a suitable adaptation is available in Feireisl and Novotný [17].
An important application of measure valued solutions is their identification as limits of numerical schemes.Interesting results in numerical analysis have been obtained by Fjordholm et al. [20], Feireisl and Lukáčová-Medvid'ová [13] and Feireisl, Lukáčová-Medvid'ová and Mizerová [15].Together with the existing weak(measure valued)-strong uniqueness principle in the class of measure valued solutions, one can show that numerical solutions converge strongly to a strong solution of the system as long as the latter exists.In the context of compressible Navier-Stokes-Fourier system with no-flux boundary condition see Feireisl et al [16] and a detailed discussion is available in the monograph by Feireisl et al. [14].
The plan for the article is as follows: • At first in Section 2, we devote ourselves to the weak formulation of the problem along with a proper definition of the measure valued solution.
• In Section 3, adapting the relative energy inequality suitably for the measure valued solution, we derive the relative energy inequality for the measure valued solution.
• We present our main results in Section 4. We conclude the weak(measure valued)strong uniqueness property for the system.Our first two theorems, Theorem 4.1 and Theorem 4.2, requires some additional hypothesis on measure valued solutions, while Theorem 4.3 is does not need any of such extra assumption on solutions, although we impose some physically relevant structural assumptions on transport coefficients µ, λ and κ.
• Finally, in the Section 5, we discuss briefly the limitation of our results, some comments on existence and validation of the definition.
2 Measure valued solution

Weak formulation: Revisit
At first we quickly recall the weak formulation described in Feireisl and Novotný [17,Chapter 2].Although it deals with a no-flux boundary condition for temperature, i.e. q • n = 0 on ∂Ω. (2.1) The weak formulation of the problem depends on the weak formulation of continuity equation, momentum equation and replaces the internal energy equation (1.3) by the entropy inequality along with the weak form of total energy balance : Instead of boundary condition (2.1), if we consider inhomogeneous Dirichlet boundary condition for the temperature (1.8), we need to proceed as prescribed [7,Section 2.4].In this case, assuming all quantities in consideration are smooth, we have and consequently d dt where σ x is the surface measure on the boundary ∂Ω.Therefore, the total energy balance is unavailable.But, considering a smooth function θ such that θ > 0 in (0, T ) × Ω with θ = ϑ B on ∂Ω, and multiplying the entropy inequality (2.2) by θ along with performing the integrating by parts formula, we obtain Adding (2.4) and (2.5) we obtain the ballistic energy inequality We recall that for some smooth function Θ, the ballistic energy is denoted by H Θ (̺, ϑ) and it reads as Therefore we consider the weak formulation with ballistic energy inequality instead of energy balance.In this paper our goal is to provide a suitable definition of a measure valued solution based on the above discussion for boundary condition (1.8).To define it, we take motivation from Březina, Feireisl and Novotný [5] that covers the the boundary condition (2.1).This definition will give in terms of Young measure and suitable defect measures.For a simpler consideration, from now on we consider g = 0.

Phase space and Young measure
A natural candidate for the phase space is given by the state variables [̺, u, ϑ].Since we are looking for a more general class of solutions and ∇ x u and ∇ x ϑ are present in the system (1.1)-(1.3),thus gradient of velocity and temperature have been included along with the natural choices.Hence a proper phase space is We consider a Young measure V such that V ≡ {V t,x } (t,x)∈(0,T )×Ω and )

Compatibility relations for the Young measure
We consider a very general phase space for the Young measure.It satisfies the following compatibility conditions.
• Velocity compatibility: The identity holds for any T ∈ C 1 (Q T ; R d×d sym ).
• Temperature compatibility: The identity

Compatibility of defect measures
For any where ξ ∈ L 1 (0, T ).

Generalized Korn-Poincaré inequality
The following version of Korn-Poincaré inequality is true: for any U ∈ L 2 (0, T ; W 1,2 0 (Ω; R d )).Remark 2.1.We must note that the ballistic energy inequality is given for a large class of functions Θ and in general the dissipation defect D Θ is dependent of Θ.On the other hand the defect measure (r M ) in (2.12) is independent of Θ.Therefore, the (2.15) is very strong assumption.Although for some physical equation state we will able to conclude that D Θ is independent of Θ, we will discuss it in Section 5.
Remark 2.2.Analogously, on can think of to have a defect measure in the right hand side of inequality (2.14).We are avoiding it.An explanation is available in 5.

Definition of a measure valued solution
Here now we provide the definition of the measure valued solution Moreover, the initial condition V 0 satisfies (2.8).Then {V, D Θ } is a measure valued solution for the system (1.1)-(1.3)with (1.4), (1.5), (1.6)

Relative energy inequality
Following Feireisl and Novotný in [17,Chapter 9], we consider the relative energy with the help of Ballistic energy and it is given by where (̺, θ, ũ) are smooth functions such that they satisfy As observed in [19, Section 1.2], if the relative energy is interpreted in terms of the conservative entropy variables (̺, S = ̺s, m = ̺u), represents a Bregman distance/divergence associated with the energy functional To obtain that the reltive enrgy functional is non-negative, Indeed we need the hypothesis of thermodynamic stability, i.e.
which in turn yields the convexity of the internal energy ̺e(̺, S) with respect to the variables (̺, S).In addition, Thus the relative energy expressed in the conservative entropy variable may be interpreted as The time evolution of the relative energy is given by Given a measure valued solution {V t,x } (t,x)∈Q T of Navier-Stokes-Fourier system, we adapt the relative energy as At first, using the standard expansion, we have In the identity (3.5), the term L 1 is associated with the ballistic energy inequality (2.14), while for the terms L 2 and L 3 we use the momentum equation (2.12) and the continuity equation (2.11), respectively.For the term L 4 we use the identity .
A suitable application of the Gibbs' relation (1.4) yields Our main goal is to establish the weak (measure valued)-strong uniqueness property.To obtain this result we choose (̺, ũ, θ) as a strong solution of the problem emanating from the same initial data V 0,x and they share same boundary condition.Thus, suppose that (̺, ũ, θ) is smooth and ̺, ϑ > 0 in (0, T ) × Ω, then the inequality(3.6)reduces to where Moreover, adjusting a few terms suitably, the above inequality becomes where More precisely, we have Now we use the compatibility of the Young measure (2.9) and (2.10) to deduce where the term R 2 is given by (3.7) and it consists of quadratic error terms.Let us now fix some notation to write (3.8) more precisely.For A = (a ij ) d i,j=1 ∈ R d×d , we consider the symmetric part and the traceless part of A as respectively, where Tr(A) = d i=1 a ii .We write the Newtonian stress tensor as We rewrite the inequality (3.8) as The inequality (3.9) is called the relative energy inequality associated with the problem.Therefore we summarize the above discussion in the following lemma: Lemma 3.1.Let the transport coefficients κ(̺, ϑ), µ(̺, ϑ) and λ(̺, ϑ) be continuously differentiable and positive for ̺ > 0, ϑ > 0. Let the thermodynamic functions satisfy Gibbs equation(1.4) and the thermodynamic stability assumption(3.2).Let {V, D θ} be a measure valued solution of the system (1.1)-(1.3)with initial data V 0 and {̺, ũ, θ} be a strong solution with sufficient regularity and initial data.Then we have the inequality (3.9) holds, where the remainder term R 2 is given by (3.7).

A suitable reduction of relative energy inequality
In this subsection we will try to reduce the (3.9).Since we already notice that the relative energy is a non-negative functional and the remainder term R 2 contains certain quadratic terms.Therefore we introduce the next part to have a close look on relative energy.

Essential and residual part of a function
At first, we introduce a cut-off function χ δ such that For a function H = H(̺, ϑ, u, D u , D ϑ ), we set If ̺, θ is strictly positive, bounded above and bounded below, |ũ| is also bounded, then with the help of the above notation, we have (3.10) Remark 3.2.The inequality (3.10) implies that the relative energy functional is coercive.
At this point, our goal is either to control the remainder term R 2 of (3.9), by the integral of the relative energy or to absorb it into a non-negative term on the left-hand side of (3.9).Lemma 3.3.Let the hypothesis in Lemma 3.1 remains true.Then with the help of (3.10), the inequality (3.9) reduces to Proof.The proof is quite straight forward, since R 2 contains quadratic terms, therefore the essential parts of R 2 will be controlled by C(δ, ̺, θ, ũ) ´τ 0 E mv (t) dt .For the residual parts of R 2 , we control some of the terms that are compatible with the right hand side of (3.10) and we keep the other terms we in the right hand side of the (3.11).

Conditional Weak(measure valued)-strong uniqueness
By the term 'conditional', we mean that, along with the measure valued solution if we assume some additional hypothesis, them we will able to achieve the desired Weak(measure valued)-strong uniqueness property.

Second conditional result
Our next result is for a more physical situation with where pressure law is boyle's law.But the weak-strong uniqueness we prove is imposing certain condition on entropy and additional hypothesis on transport coefficients.

Unconditional weak(measure valued)-strong uniqueness
Here we consider a general pressure law and some structural assumption on the transport coefficients.We call it unconditional result because, here we will not assume any further information on measure valued solution.Let us first consider the pressure law in the following way: where p M stands for the molecular pressure and p R is the radiation pressure.The relation between the molecular pressure p M and the associated internal energy e M is is given by From the Gibb's relation (1.4), we have , for some function P .Moreover, following [17], we assume that P ∈ C 1 [0, ∞) ∩ C 5 (0, ∞), P (0) = 0, P ′ (q) > 0, for all q > 0, 0 < 5 3 P (q) − P ′ (q)q q < c for all q < 0, lim q→∞ P (q) We rewrite the internal energy (e M ) associated with molecular pressure as , and, again using Gibbs relation, we have for some function S with the property S ′ (q) = − 3 2 5 3 P (q) − P ′ (q)q q 2 < 0. (4.12) Finally, we impose the third law of thermodynamics in the form lim q→∞ S(q) = 0. (4.13) the structural assumptions on transport coefficient reads as Assume that [̺, ũ, θ] is a classical solution to the system in [0, T ] × Ω emanating from initial data [̺ 0 , u 0 , ϑ 0 ] with ̺ 0 , ϑ 0 > 0 in Ω. Assume further that {V t,x } (t,x)∈Q T is a measure valued solution of the same problem following the Definition 2.15 such that V 0,x = δ [̺ 0 ,u 0 ,ϑ 0 ] for a.e.x ∈ Ω. Proof.At first, using the structural assumption (4.14), we have Using Young's inequality, we obtain Since β ≤ 2, the second term s controlled by radiation pressure p R .Therefore, our choice of p yields Furthermore, for any ǫ > 0, we get the following inequality On the other hand, the structural assumption on p M and S M gives us Finally, we conclude that Now proper choice of ǫ, generalized Korn-Poincaré inequality and Grönwall's argument gives us the desired result.

Comments on measure valued solution
We try to obtain a priori bounds for the system in which the pressure law follows (4.8) with (4.9) and (4.15) and the transport coefficients µ, λ and κ follow (5.1) We also invoke the third law of thermodynamics (4.13).

A priori estimate
Let us first recall the ballistic energy inequality d dt where θ is a smooth function such that θ > 0 in (0, T ) × Ω with θ = ϑ B on ∂Ω.
In order to control the last integral in (5.2), we proceed analogously as in [7, Section 4.1].Hence, we consider the extension θ to be the unique solution of the Laplace equation The maximum principle for the Laplace equation gives Let us denote this particular extension by ϑ B .At first we note that where K ′ (ϑ) = κ(ϑ) ϑ .Next, as ϑ B is continuously differentiable in time, we obtain Therefore, for the term using the inequality in [18, Section 4, formula (4.6)], we have Consequently, it yields Now we can use Grönwall's argument to deduce a priori estimate for 1 2 ̺|u| 2 + ̺e − θ̺s .Moreover for this particular pressure law (4.8) with (4.9) and (4.15) along with the third law of thermodynamics (4.13), following Feireisl and Březina [4], we are able to conclude that ess sup (0,T ) ̺s L q (Ω) + ess sup (0,T ) ̺su L q (Ω) ≤ C for some q > 1.Although they consider only for the molecular pressure, the radiation pressure case is quite straight forward.Therefore we obtain the following a priori estimate : for some q > 1.

Comments on compatibility conditions
Now suppose we assume a young measure V is generated by a family of sequences (̺ ǫ , u ǫ , D x u ǫ , ϑ ǫ , ∇ x ϑ ǫ ) {ǫ>0} that satisfies the bound ess sup (0,T ) uniformly with respect to ǫ and µ, λ and κ follows (5.1) and q > 1. Clearly the bound (5.3) is motivated from the discussion we have in Section 5.1.Then following [5, Section 2.4], we are able to deduce the velocity and temperature compatibility.Also, Generalized Korn-Poincaré inequality can be derived in the similar way.and κ ǫ (ϑ) = 1 + ϑ β + ǫϑ γ , with β ≤ 2 and γ > 6.
• Navier-slip boundary condition for velocity: Instead of boundary condition (1.7), we consider the Navier-Slip boundary condition We can provide a similar definition by modifying the Korn-Poincare inequality suitably, see [6,Section 1.3.6].A similar weak(measure valued)-strong uniqueness result is expected to be true.
• Limitation with radiation pressure following Stefan-Boltzman law: Unfortunately, we are not able to prove our results for the radiation pressure following Stefan-Boltzman law p R (ϑ) = aϑ 4 with a > 0. The main difficulty is to deal the term [̺s R |u|] res .Following Feireisl [11], we notice that, the estimation of the term needs certain Sobolev embedding which is missing in our definition.Our choice of p R in (4.15) is motivated from the models of Neutron star, see Lattimer et al. [22].