Vanishing viscosity limit for the compressible Navier-Stokes system via measure-valued solutions

We identify a class of measure-valued solutions of the barotropic Euler system on a general (un-bounded) spatial domain as a vanishing viscosity limit for the compressible Navier-Stokes system. Then we establish the weak (measure-valued)-strong uniqueness principle, and, as a corollary, we obtain strong convergence to the Euler system on the lifespan of the strong solution.

We consider the compressible Euler system with damping here ̺ = ̺(t, x) denotes the density, m = m(t, x) the momentum -with the convection that the convective term is equal to zero whenever ̺ = 0 -and p = p(̺) the pressure. The term am, with a ≥ 0, represents "friction". We will study the system on the set (t, x) ∈ (0, T ) × Ω, where T > 0 is a fixed time, Ω ⊆ R N with N = 2, 3, can be a bounded or unbounded domain, along with the boundary condition m · n| ∂Ω = 0, for all t ∈ [0, T ]; if Ω is unbounded, we impose the condition at infinity with ̺ ≥ 0. We also consider the following initial data with ̺ 0 > 0. We finally assume that the pressure p is given by the isentropic state equation where γ > 1 is the adiabatic exponent and A > 0 is a constant. Our goal is to identify a class of generalizeddissipative measure valued (DMV) solutions -for the Euler system (1), (2) as a vanishing viscosity limit of the Navier-Stokes equations. More specifically, we start considering the set Ω R = Ω ∩ B R , B R = {x ∈ R N : |x| < R}, where we assume Ω R to be at least a Lipschitz domain, and we consider the Navier-Stokes system: ∂ t (̺u) + div x (̺u ⊗ u) + ∇ x p(̺) = 1 R div x S(∇ x u) − a̺u; (8) now u = u(t, x) is the velocity and S = S(∇ x u) is the viscous stress, which we assume to be a linear function of the velocity gradient, more specifically to satisfy the Newton's rheological law where µ > 0, η ≥ 0 are constants. Introducing λ = η − 2 N µ we also have As our goal is to perform the vanishing viscosity limit for the Navier-Stokes system, we impose the complete slip boundary conditions on ∂Ω: u · n| ∂Ω = 0, (S · n) × n| ∂Ω = 0, (11) and the no-slip boundary conditions on ∂B R : for all t ∈ [0, T ]. Of course, (11) and (12) are compatible only if ∂B R ∩ ∂Ω = ∅ for R large enough meaning that ∂Ω is a compact set. That is Ω is either (i) bounded, or (ii) exterior domain, or (iii) Ω = R N . For the sake of simplicity, we restrict ourselves to these three cases. Finally, we impose the initial conditions: Our goal will be first to show that the solutions of the Navier-Stokes system converge to the measurevalued solution of the Euler system with damping in the zero viscosity limit, then we will prove the weakstrong uniqueness principle for the Euler system, see Theorem 2.3. Then we conclude that solutions of the Navier-Stokes system converge to smooth solution of the Euler system as long as the latter exists, see Theorem 2.5. Note that the vanishing viscosity limit for the compressible Navier-Stokes system on a bounded domain was studied by Sueur [14]. Our goal is to propose an alternative approach based on the concept of dissipative measure-valued solutions and extend the result to a more general class of domains. The concept of dissipative measure-valued solution is of independent interest and has been use recently in the analysis of convergence of certain numerical schemes, see [5].
1 From the Navier-Stokes to the Euler system 1

.1 Weak formulation
To get the weak formulation of the Navier-Stokes system, we simply multiply both equations (7), (8) by test functions, and, supposing also that the density ̺ and the momentum ̺u are weakly continuous in time, we get ˆΩ for any τ ∈ [0, T ) and all ϕ ∈ C 1 c ([0, T ] × Ω R ), and for any τ ∈ [0, T ) and all ϕ ∈ C 1 c ([0, T ] × Ω ∩ B R ; R N ) with ϕ · n| ∂Ω = 0. Multiplying (8) by u and introducing the pressure potential P as the solution of the equation which, for instance, in our case can be taken as (notice in particular that P (̺) = 0; this will be used later) we get the energy equality from which the energy inequality followŝ for a.e. τ ∈ [0, T ]. For more details see [3].

Existence of weak solutions
Now, we have the following result (for the proof see [3]).

Reformulation of the problem in terms of a background density ̺
Choosing a background density ̺ ≥ 0, we can slightly change the energy inequality; indeed the Navier-Stokes system can be rewritten as Again, multiplying both equations by test functions and using the weak continuity in time we obtain Also, integrating the first equation over Ω R along with condition (12), we get Since P (̺) = 0, we can rewrite (16) as and (17) becomeŝ for a.e. τ ∈ [0, T ].

Weak sequential stability
We can then consider the family {̺ R − ̺, m R = ̺ R u R } R>0 of dissipative weak solutions to the previous Navier-Stokes system with initial data {̺ R,0 − ̺, m R,0 } R>0 defined in (0, T ) × Ω; in particular, they will satisfy the following conditions:ˆT We have replaced Ω R by Ω in the previous integrals. Note that this is correct for R large enough as the test functions are compactly supported in Ω R .
More precisely, thanks to the weak continuity of the densities and momenta, we have for any τ ∈ [0, T ) and all ϕ ∈ C 1 c ([0, T ] × Ω), for any τ ∈ [0, T ) and all ϕ ∈ C 1 c ([0, T ] × Ω; R N ), ϕ · n| ∂Ω = 0. Finally, we have the energy inequalitŷ for a.e. τ ∈ [0, T ]. In (21), we suppose that the initial data have been chosen on Ω in such a way that where the constant E 0 is independent of R. Then, extending u R to be zero and ̺ R as ̺ outside B R , we easily deduce from the energy inequality that ess sup ess sup where the bounds are independent of R. Next, from (25), we can deduce that Now, we can use the following relation for a positive constant c(̺) (see [4]). Following [6], we introduce the decomposition of an integrable function h R : Then we have ess sup and ess sup where means modulo a multiplication constant. In particular this implies that passing to suitable subsequences as the case may be; defining ̺ − ̺ := f ̺R−̺ + g ̺R−̺ , we have that We can repeat the same procedure for the momenta; indeed, using (23) we have ess sup we also have ess sup which, together with (23) and Hölder's inequality with p = γ + 1, gives ess sup Then we obtain passing to suitable subsequences as the case may be. In a similar way we have ess sup and ess sup Also, noticing that from (23) we deduce that also the convective terms are uniformly bounded in the non-reflexive space There are two disturbing phenomena that may occur to bounded sequences in L 1 : oscillations and concentrations. The idea is then to see L 1 ((0, T ) × Ω) as embedded in the space of bounded Radon measures Accordingly, we may assume passing to suitable subsequences as the case may be. This means, × Ω); the same holds for the other convergences.
We can now let R → ∞ in (19), (20); notice that the R-dependent viscous stress tensor vanishes. Indeed, using (26) and Hölder's inequality we get Then we getˆT . As a matter of fact, the limit for ̺ R − ̺ can be strengthened to the same holds for the limit of ̺ R u R : We can then rewrite the last two integral equations aŝ for any τ ∈ [0, T ) and any ϕ ∈ C 1 c ([0, T ] × Ω) and for any τ ∈ [0, T ) and any ϕ ∈ C 1 c ([0, T ] × Ω; R N ), ϕ · n| ∂Ω = 0. Finally, using the generalization of the concept of Lebesgue point to Radon measures, we can deduce from the energy inequality (21) for a.e. τ ∈ (0, T ), whereˆΩ Equations (31), (32), and (33) form a suitable platform for introducing the measure-valued solutions of the Euler system. To state the exact definition, we make a short excursion in the theory of Young measures.

Young measures
We will introduce some useful notations.
is measurable; here and in the sequel we use the standard notation ν x = ν(x), as if measures ν x were parametrized by x. In this case, since is also measurable and we can define Finally, let Then, the following theorem holds. and . Proof. See [10], Chapter 3, Theorem 2.11. From now on we will consider Q = (0, T ) × Ω and the sequence z R = (̺ R − ̺, m R = ̺ R u R ) of solutions to the Navier-Stokes system; we can now construct the Young measure associated to the sequence {z R } R>0 . First, for every R we define the mapping If we define it is weakly- * measurable and we also have that , which by Theorem 1.3 is the dual space of the separable space L 1 (Q; C 0 (R 4 )); we can apply the Banach-Alaoglu theorem to find a subsequence, not relabeled, and ν ∈ L ∞ weak (Q; M(R 4 )) such that Then, for every ϕ ∈ C 0 (R 4 ), knowing that From the weak- * lower semi-continuity of the norm we also have that What we proved is the first statement in the following theorem.
Theorem 1.4. Let Q ⊂ R d be a measurable set and let z R : Q → R s , R > 0, be a sequence of measurable functions. Then there exists a subsequence, still denoted by z R , and a measure-valued function ν with the following properties: for a.e. y ∈ Q and we have for every ϕ ∈ C 0 (R s ), as R → ∞, Proof. See [10], Chapter 4, Theorem 2.1.
Remark 1.5. If z R are uniformly bounded in L p (Q; R s ) for some p ∈ [1, ∞), the condition (35) is satisfied.
Since c is independent of both R and k, we obtain and hence at least one of the terms on the last line must be ≥ k 4 so that For k large enough (k ≥ 4), we have where in particular the constant c(E 0 ) is independent of k and R so that which implies (35). Then we obtain that the Young measure in our case is a parametrized family of probability measures supported on the set [0, ∞) × R N , since the densities are supposed to be non-negative: ). It is also easy to check that Ψ(t) = t p with p > 1 are Young functions that satisfy the ∆ 2 -condition with the constant 2 p , and in that case L Ψ (Q) = L p (Q). Thus, 1. first, we can take Ψ(t) = t 2 and τ 1 (z) = z 1 χ(z 1 + ̺), where z = (z 1 , z 2 , z 3 , z 4 ) in our case, to notice that condition (36) is equivalent in requiring that [̺ R − ̺] ess are uniformly bounded in L 2 ((0, T ) × Ω) which is true from (27). Then we obtain also, taking Ψ(t) = t γ and τ 2 (z) = z 1 (1 − χ(z 1 + ̺)), condition (36) is equivalent in requiring that [̺ R − ̺] res are uniformly bounded in L γ ((0, T ) × Ω) which is true from (28). Then we obtain Unifying the two results we get We will write ν t,x ; ̺ − ̺ = (̺ − ̺)(t, x) for almost every (t, x) ∈ (0, T ) × Ω just to make the notation readable; 2. secondly, we can take Ψ(t) = t 2 and τ 1 (z) = z i χ(z 1 + ̺) with i = 2, 3, 4 to see that condition (36) which we will write ν t,x ; m = m(t, x) for almost every (t, x) ∈ (0, T ) × Ω.

Concentration measures and dissipation defect
In the previous subsection we showed that the Young measure, applied to proper continuous functions, coincides almost everywhere with the density ̺ − ̺ and the momentum m. Now, we examine what happens for those functions H for which we only know that Without loss of generality, we can consider |H| or, equivalently, assume that H ≥ 0. We take a family of cut-off functions T k (z) = min{z, k}; Then T k (H) ∈ C 0 (R 4 ) and from the previous construction we know that On the other hand we have that thus, by monotone convergence theorem, we have that hence H is ν t,x -integrable but the integral can also be infinite. However, by the weak- * lower semi-continuity of the norm (ii) sup k∈N ν (·,·) ; T k (H) L 1 ((0,T )×Ω) ≤ c, applying Fatou's lemma we get that ν (·,·) ; H L 1 ((0,T )×Ω) ≤ c. Then ν t,x ; H is finite for a.e. (t, x) ∈ (0, T ) × Ω.
We can apply the lemma with

Dissipative measure-valued solution for the compressible Euler system with damping
Motivated by the previous discussion, we are ready to introduce the concept of dissipative measure-valued solution to the compressible Euler system with damping. It can be seen is a generalization of a similar concept introduced by Gwiazda et al. [7]. While the definition in [7] is based on the description of concentrations via the Alibert-Bouchitté defect measures [1], our approach is motivated by [2], where the mere inequality (37) is required postulating the domination of the concentrations by the energy dissipation defect. This strategy seems to fit better the studies of singular limits on general physical domains performed in the present paper.

Definition 1.7. A parametrized family of probability measures
, is a dissipative measure-valued solution of the problem (1), (2) with the initial condition {ν 0,x } x∈Ω if holds for a.e. τ ∈ (0, T ), where D ∈ L ∞ (0, T ), D ≥ 0 is called dissipation defect of the total energy; 4. there exists a constant C > 0 such that for a.e. τ ∈ (0, T ). Now, summarizing the discussion concerning the vanishing viscosity limit of the Navier-Stokes system, we can state the first result of the present paper. Theorem 1.8. Let Ω ⊂ R N , N = 2, 3 be a domain with compact Lipschitz boundary and ̺ ≥ 0 be a given far field density if Ω is unbounded. Suppose that γ > N 2 and let ̺ R , u R be a family of weak solutions to the Navier-Stokes system (7) - (12) in Let the corresponding initial data ̺ 0 , u 0 be independent of R satisfying Then the family {̺ R , m R = ̺ R u R } R>0 generates, as R → ∞, a Young measure {ν t,x } t∈(0,T );x∈Ω which is a dissipative measure-valued solution of the Euler system (1), (2).

Weak-strong uniqueness
Our next goal is to show that the dissipative measure-valued solutions introduced in the previous section satisfy an extended version of the energy inequality (40) known as relative energy inequality.
We introduce the relative energy functional : If ̺ → p(̺) is strictly increasing in (0, ∞), which is true in our case, then the pressure potential P is strictly convex; indeed For a differentiable function this is equivalent in saying that the function lies above all of its tangents: for all ̺, r ∈ (0, ∞), and the equality holds if and only if ̺ = r. Thus, we deduced that E ≥ 0, where equality holds if and only if ν t,x = δ r(t,x),r(t,x)U(t,x) for a.e. (t, x) ∈ (0, T ) × Ω.
We can now prove the following x)∈(0,T )×Ω be a dissipative measure-valued solution of the same system (in terms of ̺ and the momentum m), with a dissipation defect D and such that Then D = 0 and ν t,x = δ r(t,x),(rU)(t,x) for a.e. (t, x) ∈ (0, T ) × Ω.
Remark 2.2. Note that we must have ̺ > 0 if Ω is unbounded.
Proof. It is enough to prove that E(τ ) = 0 for all τ ∈ (0, T ). We can take U as a test function in the momentum equation (39) to obtain and 1 2 |U| 2 as a test function in the continuity equation (38) to get Finally, take P ′ (r) − P ′ (̺) as test function in (38) to get Then, from the energy inequality (40), summing up all these terms we get Notice that the term is well-defined and integrable. We have where, since P (̺) = 0, Then and knowing that the pressure potential satisfy the equation we can deduce that We obtain the relative energy inequality: Now we can use the fact that [r, U] is a strong solution: from the momentum equation we can deduce that substituting, we get From the continuity equation we also have and thus, knowing that rP ′′ (r) = p ′ (r), we get Finally, using the fact that the initial data are the same and thus E(ν|r, U)(0) = 0, we end up to Since U and P ′ (r) − P (̺) have compact support we can control the terms |∇ x U|, | div x U|, |U · ∇ x U| and |∇ x P ′ (r)| by some constants. It is also obvious that there exist a constant c 1 such that and a constant c 2 such that Thus By Gronwall lemma we obtain E(̺, m|r, U)(τ ) + D(τ ) ≤ 0 for all τ ∈ (0, T ).
Notice that the relative energy inequality (45) is true for general functions , not necessarily strong solutions to the Euler system. Then, using a density argument, we can prove the following result. Proof. We will first prove that the relative energy inequality (45) holds for [r, U] as in our hypothesis. By density, we can find two sequences If we now fix ε > 0, we know that there exists n 0 = n 0 (ε) such that, for every n ≥ n 0 From now on, let n ≥ n 0 ; for each t ∈ [0, T ] we havê Revoking notation introduced in Section 1.4, we focus on the last two lines line: we can rewrite the first term aŝ we can apply Hölder's inequality to get We also have that ν (t,·) ; [̺ − ̺] res U n ∈ L γ (K); R N ) with K compact and since γ > 2γ γ+1 we obtain ν (t,·) ; [m] res − [̺ − ̺] res U n ∈ L 2γ γ+1 (Ω; R N ); using the embedding of the Sobolev space into the Hölder one we get that (U − U n )(t, ·) ∈ L ∞ (Ω; R N ) and hence (U − U n )(t, ·) ∈ L p (Ω; R N ) for all p ∈ [2, ∞]. Since 2γ γ−1 > 2, we can again apply Hölder's inequality to get For the second term we can apply Hölder's inequality: Applying the same procedure as before to the third term we get For the last term we simply havê Similarly, We can now focus on the last two lines: the first term is simply bounded as followŝ The second term can be rewritten aŝ notice that, if γ ∈ (1, 2) we use the same argument as before while if γ ∈ [2, ∞) we have to use the Sobolev embedding in the L p -spaces. For the last term we can use Hölder inequality to get Repeating the same steps for each term that appears in the relative energy inequality and introducing the operator L(ν|r, U)(τ ) = aˆτ [ ν t,x ; (̺ − r)∂ t P ′ (r) + (m − rU) · ∇ x P ′ (r) dxdt we have [E(ν|r, U)(t)] t=τ t=0 + L(ν|r, U)(τ ) ≤ [E(ν|r n , U n )(t)] t=τ t=0 + L(ν|r n , U n )(τ ) + Cε ≤ Cε, for some positive constant C, since for a test function we already proved that the relative energy inequality holds which is equivalent in saying that [E(ν|r n , U n )(t)] t=τ t=0 + L(ν|r n , U n )(τ ) ≤ 0.
By the arbitrary of ε we can conclude that the relative energy inequality holds for [r, U] as in our hypothesis.
Repeating the same passages as we did in the proof of the previous theorem, we end up to the following inequality E(ν|r, U)(τ ) + aˆτ The thesis now follows as before -the only thing that changes is that in this case U and P (r) − P (̺) are L ∞ -functions, but still we can control the terms |∇ x U|, | div x U|, |U · ∇ x U| and |∇ x P (r)| by some constants.
Remark 2.4. This theorem applies to the already know results concerning strong solutions; in particular (i) if Ω is bounded, for local in time solutions see [12], and [11] for the global one; (ii) if Ω = R 3 , for local in time solution see for instance [8], [9], and [13] for the global one.