1 Introduction
The initial–boundary value problems to certain systems of nonlinear conservation laws are ill–posed in the class of weak solutions. An iconic example is the Euler system describing the time evolution of the density , the momentum , and the energy of a compressible inviscid fluid:
(1.1) 
Here, is the pressure related to through a suitable equation of state. The fluid is confined to a spatial domain , , on the boundary of which we impose the impermeability condition
(1.2) 
The initial state of the system is given through initial conditions
(1.3) 
The first numerical evidence that indicated ill–posedness of the Euler system was presented by Elling [21]. As proved later rigorously [14, 15, 16, 17, 19, 27], the initial–boundary value problem (1.1)–(1.3) admits infinitely many weak solutions on a given time interval for a rather vast class of initial data. Moreover, these solutions obey the standard entropy admissibility condition
(1.4) 
where is the total entropy of the system.
The ill–posedness of the Euler system is intimately related to the lack of compactness of the set of functions satisfying (1.1) and (1.4). Indeed bounded sequences of solutions may develop uncontrollable oscillations and/or concentrations. This phenomenon is then naturally inherited by their numerical approximations, see e.g. Fjordholm, Mishra, and Tadmor [30, 31]. This fact motivated the renewed interest in the measure–valued (MV) solutions introduced in the context of the incompressible Euler system by DiPerna and Majda [20]. The exact values of physical densities and fluxes, originally numerical functions of the physical variables , are replaced by a family of probability measures (Young measure)
The values of the observable fields are interpreted as the expected values:
In view of the observed fact that certain numerical schemes fail to provide convergent sequences of approximate solutions, Fjordholm et al. [30, 31] proposed a way how to compute the associated Young measure. Note that, in the context of numerical simulations, the convergence towards the limit (exact) solution is required to be pointwise (a.e.). On the other hand, however, the theoretical studies available so far in the literature, see e.g. [30, 31], provide only weak convergence in . This is of little practical interest as wildly oscillating output data are difficult to analyze in such a way.
In the present paper, we propose a new method to compute the Young measures associated to sequences of numerical solutions based on the concept of convergence. We start by showing a remarkable property of the Euler system (1.1) that can be roughly stated as follows: either a consistent numerical approximation converges strongly (pointwise a.e.) or its (weak) limit is not a weak solution of the Euler system, see Section 3. In the context of numerical analysis, this property might be seen as a sharp version of the wellknown LaxWendroff theorem. In view of these facts, the concept of measure–valued or other kind of generalized solution is necessary whenever the approximate sequence exhibits oscillations.
Following Balder [1], we associate to an approximate sequence the Young measure
where is the Dirac mass concentrated at . As already pointed out, observable limits of for must be given in terms of pointwise convergence. Unfortunately, the standard basic result of the theory of Young measures, see, e.g., Ball [3] or Pedregal [47], provides only the weak(*) convergence (up to a suitable subsequence):
(1.5) 
Thus, similarly to the approximate solutions , the family may exhibit oscillations with respect to the physical space . From the practical point of view, therefore, the piece of information provided by the convergence (1.5) is negligible.
The concept of convergence, developed in the framework of Young measures by Balder [1, 2], converts the weak(*) convergence to the desired pointwise one by replacing sequences by (suitable) Cesàro averages in the spirit of the original work by Komlós [39]:
A sequence of functions uniformly bounded in contains a subsequence such that
Any further subsequence of enjoys the same property.
Rephrased in terms of sequences of probability measures, Balder [2] obtained a remarkable extension of the Prokhorov theorem:
If a parametrized family of probability measures is tight, meaning,
then, for a suitable subsequence,
(1.6) 
We recall that a sequence of probability measures converges narrowly to a probability measure if
Here the symbol denotes the set of bounded continuous functions.
The aim of the present paper is to apply these ideas to sequences of numerical solutions of the Euler system (1.1)–(1.3). In addition to (1.6
), we show that the available stability estimates yield convergence in the standard Wasserstein metric
(1.7) 
Recall that the Wasserstein distance of th order of probability measures is defined as
(1.8) 
where is the set of probability measures on with marginals and . Indeed, this is a natural metric on the set of the Young measures generated by approximate sequences of numerical solutions in view of the available stability estimates. In comparison with the standard Lévy–Prokhorov metric induced by the weak(*) topology (on the space of measures), the Wasserstein distance includes a piece of information on large distances in the associated phase space .
For scalar conservation laws, the convergence of finite volume methods with respect to the Wasserstein distance has been investigated by Fjordholm and Solem [33]. They proved that monotone finite volume schemes being formally first order schemes show in special situations second order convergence rate in . On the other hand, the first order rate in has been proven optimal in general, see [49]. Such a result, however, seems to be out of reach for the Euler system as it would imply strong (a.e. pointwise) convergence of the expected values – the numerical solutions.
The paper is organized as follows. In Section 2 we introduce the concept of consistent approximate solutions. Section 3 summarizes available results on the strong (pointwise) convergence of the approximate solutions and the weak convergence in terms of Young measures. Section 4 is the heart of the paper. We use the theory of convergence adapted to the Young measures to show that the limiting Young measure can be effectively described by means of the Cesàro averages of the approximate solutions. As an added benefit with respect to available results, we therefore obtain strong convergence of the Cesàro averages in and the Wasserstein distance for some . Finally, in Section 5, we present numerical simulations obtained by two finite volume methods: a recently proposed finite volume method, see [25], that is (entropy)stable and consistent and thus yields the consistent approximate solutions required by the abstract theory and a more standard finite volume method based on the generalized Riemann problem, see, e.g. [4, 5, 6, 7, 13] and the references therein. The predicted strong convergence of the Cesàro averages when approximate solutions experience oscillations is confirmed by both numerical methods. Our numerical results clearly demonstrate robustness of the proposed convergence technique.
2 Approximate solutions
For the sake of simplicity, we focus on the perfect gas with the standard equation of state
where is the specific internal energy and the absolute temperature. Accordingly,
Finally, we introduce the specific entropy
Definition 2.1 (Consistent approximations).
We say that is a family of approximate solutions consistent with the Euler system (1.1) in if:

, , are measurable functions on ,

(2.1) for any , and any , where

(2.2) for any , and any , , where

(2.3)
In addition, we say that is admissible if for the entropy ,
the entropy inequality holds
(2.4) 
for any , any , , and any ,
where
A family of approximate solutions provides seemingly less information than the corresponding weak formulation of the Euler system (1.1)–(1.3), and (1.4). Nonetheless, as we shall see below, the approximate solutions in the sense of Definition 2.1 generate a measure–valued solution introduced in [10]. In this paper, we focus on the consistent approximate solutions generated by suitable numerical schemes. A specific example of such a scheme – the finite volume method (5.3)(5.5) – is given in Section 5. However, the concept presented here is quite general allowing to include the vanishing viscosity as well as other types of singular limit perturbations. In particular, may be weak solutions of the Euler system itself (the error terms being zero).
Note that, in contrast with the Euler system (1.1), the approximate solutions satisfy merely the total energy balance (2.3), meaning the last equation in (1.1) is integrated over the physical domain. It can be shown, however, that (2.3), together with the entropy inequality (2.4), give rise to the energy equation as soon as the all quantities are smooth and all error terms set to be zero. The proof is the same as for the Navier–Stokes–Fourier system and we refer the reader to [26] for details.
3 Strong vs. weak convergence
We discuss sequential stability of the set of approximate solutions introduced in Definition 2.1. To this end, we suppose that the corresponding initial data satisfy
(3.1) 
and
(3.2) 
The uniform bound (3.1) imposed on the initial energy, together with the total energy balance (2.3), yield
(3.3) 
in particular,
(3.4) 
Next, we suppose, in accordance with the fact that the entropy is transported along streamlines, that
(3.5) 
Remark 3.1.
Anticipating (3.5) we may rewrite the total energy in terms of as
In view of (3.3)–(3.5) we obtain
(3.6) 
and
(3.7) 
Finally, introducing the total entropy , we deduce
(3.8) 
and
(3.9) 
see [8, Section 3.2] for details.
3.1 Strong (pointwise) convergence
Note that all the uniform estimates established so far yield only boundedness of the approximate solutions in various spaces. The appropriate notion of convergence is therefore “weak”, or, more precisely, “weak(*)” convergence, meaning convergence in the sense of integral averages. As already pointed out, this is of little practical importance as the limits of oscillating quantities are difficult to identify.
However, the convergence is strong (pointwise a.e.) as soon as the limit Euler system (1.1)–(1.3) admits a (unique) strong solution. Indeed any admissible sequence of consistent approximate solutions generates a dissipative measure–valued (DMV) solution in the sense of [10]. The (DMV) solutions enjoy the weak–strong uniqueness property yielding the desired conclusion. The relevant result can be stated as follows, cf. [10, Theorem 3.3]:
Proposition 3.2.
Let , be a bounded Lipschitz domains. Let be a family of admissible approximate solutions consistent with the Euler system in the sense of Definition 2.1. Let the approximate initial data satisfy (3.1), (3.2), and suppose that (3.5) holds. Let
where
Finally, suppose that the Euler system (1.1), (1.2), with the initial data (1.3), admits a strong solution Lipschitz continuous in .
Then
Note that the convergence stated in Proposition 3.2 is unconditional, meaning it is not necessary to consider subsequences, as the limit solution is unique.
3.2 Weak convergence
If the limit system fails to possess a classical solution, the convergence might not be strong. In such a case, the nonlinear composition operators do not commute with weak limits and the limit object may not be even a weak solution of the Euler system.
To analyze the weak convergence, it is convenient to work with the variables . In view of the uniform bounds (3.6)–(3.8) we obtain that
(3.10) 
passing to suitable subsequences as the case may be. The key quantity is the total energy
where . Note that
is a convex lower semicontinuous function on , smooth and strictly convex for and , see [8, Lemma 3.1]. Considering a subsequence if necessary, we conclude that
In addition we introduce a marginal , such that
Moreover, we have
where the first inequality is Jensen’s inequality, while the second follows from from the arguments of [23, Lemma 2.1] applied to , and Using the energy conservation (2.3) we obtain
We proceed by introducing the oscillation defect
the concentration defect
and the total energy defect
Note that , while
The following observations are standard:
for any Borel set .
Finally, we report the following result, see Chaudhuri [12], which can be obtained in analogously way as in [24] due to the convexity of pressure.
Proposition 3.3.
Let , be a bounded Lipschitz domain. Let be a family of admissible approximate solutions consistent with the Euler system in the sense of Definition 2.1 such that
and
Suppose that there is an open neighborhood of such that
Finally, suppose that the limit is a weak solution of the Euler system in , in particular,
Then
and
Proposition 3.3 asserts that as long as the convergence of approximate solutions is strong in a neighborhood of the boundary and the limit is a weak solution of the Euler system, then the convergence must be strong everywhere. A very rough extrapolation might be that either the convergence is strong or the limit object is not a weak solution of the limit system. Such a result might be seen as a sharp version of the celebrated LaxWendroff theorem [41] which states that a bounded sequence of pointwisely convergent numerical solutions, that are generated by a consistent, conservative and entropy stable numerical scheme, converges to a weak entropy solution.
4 convergence of Young measures
Accepting the conclusion of Proposition 3.3 as an evidence that oscillating (weakly converging) sequences of approximate solutions may give rise to “truly” measure–valued solutions, we might want to compute the distribution of the associated Young measure. Unfortunately, the method proposed in [30, 31] asserts only the weak(*) convergence of these objects featuring the same difficulties to be captured numerically as oscillating solutions. Here, we propose a new method based on the concept of convergence developed in the context of Young measures by Balder [1, 2].
Evoking the situation described in Section 3, we consider a Young measure generated by a sequence of admissible approximate solutions consistent with the Euler system. Note that we have deliberately included the conservative variables in order to capture correctly shock positions and to provide a direct comparison with the results [30, 31]. The Young measure adapted variant of the celebrated Prokhorov theorem proved by Balder [2], [1, Theorem 3.15] asserts the existence of a suitable subsequence such that
(4.1) 
Here and hereafter, the symbol denotes the Dirac mass supported at the point . Moreover, in view of Castaign, de Fitte, and Valadier [11, Lemma 6.5.17] and the Komlós theorem [39], the above sequence can be chosen in such a way that the barycenters converge:
(4.2) 
for a.a. . Finally, in view of (3.10) and the standard Banach–Sacks theorem, the convergence of the density, momentum and entropy can be strengthened to
(4.3) 
As observed in Section 3, we have , where is the weak(*) limit of in . Moreover, from the FatouVitali theorem for the Young measures [1, Theorem 3.13] we also deduce
(4.4) 
Lemma 4.1.
Then
(4.5) 
for any function
Proof.
Decomposing into positive and negative part, we may assume, without loss of generality, that Let be a family of cutoff functions,
We write
In virtue of (4.1) we have
On the other hand,
Seeing that
we may infer there exists small enough so that
Consequently using the convergence stated in (4.2) and the Levy monotone convergence theorem we may pass to the limit obtaining the desired conclusion. ∎
4.1 Strong convergence in the Wasserstein distance
Relation (4.1) asserts narrow convergence of the Cesàro averages to the Young measure that is (a.e.) pointwise with respect to the parameter in the physical space . It is legitimate to ask whether this can be strengthened to the convergence in the Wasserstein metric. We first observe that
(4.6) 