On the density of wild initial data for the compressible Euler system

We consider a class of wild initial data to the compressible Euler system that give rise to infinitely many admissible weak solutions via the method of convex integration. We identify the closure of this class in the natural $L^1$-topology and show that its complement is rather large, specifically it is an open dense set.


Smooth and weak solutions
Smooth solutions are at least continuously differentiable on the set [0, T ) × Ω, ̺ > 0, ϑ > 0, and satisfy (1.1-1.3) pointwise. Smooth solutions are known to exist locally on a maximal interval [0, T max ) as long as the initial data are smooth enough and satisfy the corresponding compatibility conditions on ∂Ω, see last part of Section 4 for details. Moreover, it is well known that T max < ∞ for a "generic" class of data. The weak solutions satisfy (1.1), (1.2) in the sense of distributions, the conservative variables are weakly continuous in time so that the initial conditions are well defined. The weak solutions are called admissible if the entropy inequality (1.5) holds in the weak sense. The existence of global in time weak (admissible) solutions for given initial data is an open problem. Recently, however, the method of convex integration developed in the context of the incompressible Euler system by De Lellis and Székelyhidi [7] has been adapted to identify a class of initial data for which the problem (1.1-1.3) admits infinitely many admissible weak solutions defined on a given time interval (0, T ), [9]. Similar results have been obtained also for the associated Riemann problem in [1].
The initial data for which the problem admits local smooth solution will be termed regular, the data giving rise to infinitely many admissible weak solutions are called wild. Our goal is to identify the class of regular data that can be obtained as limits of wild data. Note that for the incompressible Euler system, the wild data (velocities) are dense in the Lebesgue space L 2 (Ω; R N ), see Székelyhidi and Wiedemann [13].
In the present paper, we restrict ourselves to the class of wild solutions resulting from a rather general splitting method which goes back to [8]. To the best of our knowledge, all convex integration results for the Euler system (1.6) in the literature use such a splitting method.
A vector field m can be decomposed by means of Helmholtz decomposition, i. e. it can be written as the sum of a solenoidal vector field H[m] and the gradient of a scalar field.
More precisely, the Helmholtz projection operator H is defined as , the system (1.6) can be written in the form (1.9) To the best of our knowledge, there are only two convex integration ansatzes for obtaining wild solutions to the Euler system (1.6) in the literature. Let us begin with the first one, which is used in [9]. This ansatz is based on replacing equations (1.9) 2 , (1.9) 3 by the system (1.10) Equation (1.10) 1 represents the volume preserving part of the motion, while (1.10) 2 may be seen as a wave equation governing the propagation of acoustic waves. Note that the fields v and ∇ x Φ as well as the fluxes in (1.10) are orthogonal with respect to the L 2 scalar product, specifically, where the latter is true since it is a product of a traceless matrix and a multiple of the identity matrix.
Although (1.10) 1 is apparently overdetermined, it admits (infinitely many) weak solutions for any fixed ̺ and Φ, cf. Appendix A. Note that a technique similar to (1.10) has been used also for the simplified isentropic Euler system by Chiodaroli [4].
The second convex integration ansatz, that is available in the literature, is based on the analysis of the corresponding Riemann problem, see [1], [6], [10] among others. We will first focus on the ansatz (1.10) and afterwards extend our results to the wild solutions obtained via the Riemann problem in Section 5.

Main result
We are ready to formulate our main result. To avoid the situation when the temperature approaches absolute zero, we restrict ourselves to the phase space Note that this is not very restrictive as any admissible weak solution satisfies the minimum principle We say that a sequence of data {̺ 0,n , u 0,n , ϑ 0,n } ∞ n=1 , or, equivalently, • the initial data [̺ 0,n , m 0,n , E 0,n ] give rise to a sequence of admissible weak solutions [̺ n , m n , E n ] satisfying Note that, in view of (1.10), whenever [̺ n , m n , E n ] are wild solutions in the sense of Definition 1.1. In particular, (1.12) is trivially satisfied for any sequence of wild initial data in the sense of Definition 1.1. Finally, note that the sequence of solutions {̺ n , m n , E n } ∞ n=1 need not a priori converge strongly, and, consequently, (1.12) does not impose any similar restriction on a possible limit solution emanating from Our goal is to show that the set of reachable data is "small" in the sense that its complement is an open dense set in L 1 +,s 0 (Ω; R N +2 ). For the sake of simplicity, we focus on the equations of gas dynamics, where the pressure is given by Boyle-Mariotte law p(̺, ϑ) = ̺ϑ, and e(̺, ϑ) = c v ϑ, c v > 0.
Equivalently, in the conservative variables, Here the kinetic energy is defined as Here is our main result.
Then the complement of the set of reachable data is an open dense set in L 1 +,s 0 (Ω; R N +2 ). contains an open dense set in L 1 +,s 0 (Ω; R N +2 ). The main part of the paper is devoted to the proof of Theorem 1.3. Our strategy is to identify a large (dense) set of regular data for the Euler system that are not reachable. To see this, we consider initial data [̺ 0 , m 0 , E 0 ] giving rise to a smooth solution [̺, m, E] defined on a maximal time interval [0, T max ). Then we proceed in several steps: • Assuming the data [̺ 0 , m 0 , E 0 ] are reachable we show that the associated sequence of solutions {̺ n , m n , E n } ∞ n=1 generates a Young measure that can be identified with a dissipative measure valued (DMV) solution to the Euler system in the sense of [2], see Section 2.
• Next we realize that, thanks to the strong convergence required in (1.11), the limit DMV solution starts from the initial data represented by the parameterized family of Dirac masses, Thanks to the general weak-strong uniqueness principle, we conclude that the DMV solution coincides with the strong solution on the time interval [0, T max ), more specifically, the DMV solution is represented by the parameterized family of Dirac masses, see Section 3. In particular, the solutions [̺ n , m n , E n ] converge strongly and we may assume (1.14) • Thanks to the strong convergence (1.14), condition (1.12), and smoothness of the limit solution, we deduce that see Section 3 for details. In particular, as the limit solution is continuous up to t = 0, • In Section 4, we show that (1.15) can be satisfied for a very narrow class of the initial data.
In particular, we complete the proof of Theorem 1.3.
A related result for the wild solutions obtained via the Riemann problem is shown in Section 5.

Definitions
First we recall the standard definition of the admissible weak solutions to the Euler system: and the following holds: for any ϕ ∈ C 1 c ([0, T ) × Ω), ϕ ≥ 0, and Z(s) ≤ Z, Z ′ (s) ≥ 0.

Remark 2.2.
Here the entropy inequality is satisfied in the renormalized sense, see e. g. [3].
To carry out the programme delineated at the end of the preceding section, we introduce the concept of dissipative measure valued (DMV) solution, see [2]. Let be the phase space associated to the Euler system.
The reader will have noticed that Definition 2.3 is not a mere measure-valued variant of Definition 2.1. In particular, the energy equation (2.1) has been replaced by its integrated version (2.3). The reader may consult [2] for a thorough discussion of this new concept of solution.

Generating a DMV solution
Suppose now that where [̺ 0,n , m 0,n , E 0,n ] are initial data of admissible weak solutions [̺ n , m n , E n ] defined on (0, T ) × Ω.
Passing to a suitable subsequence, we may suppose that {̺ n , m n , E n } ∞ n=1 generates a Young measure As shown in [2, Section 2.1], the Young measure {V t,x } (t,x)∈(0,T )×Ω) is a DMV solution of the Euler system in the sense of Definition 2.3. Moreover, in view of (2.4), the initial measure {V 0,x } x∈Ω reads V 0,x = δ ̺ 0 (x),m 0 (x),E 0 (x) for a.a. x ∈ Ω.

Application of the weak-strong uniqueness principle
The weak-strong uniqueness principle (Theorem 3.3 in [2]) asserts that a DMV solution coincides with the strong solution emanating from the same initial data at least on the life span of the latter. Evoking the situation from Section 2.2, we suppose now that Finally, as the limit solution is continuous up to the time t = 0, we may infer that in particular, For C 2 initial data, relation (3.2) can be rewritten as a non-linear differential equation

Reachability
In this section we finish the proof of Theorem 1.3. Introducing w = m 0 √ ̺ 0 we may write (3.2) in a concise form We consider solutions of (4.1) in the space L 2 (Ω; R N ), S = w ∈ L 2 (Ω; R N ) w solves (4.1) .
It is easy to check that: • the set S is closed in L 2 (Ω; R N ); • if w ∈ S, then λw ∈ S for any λ ∈ R. Proof. Arguing by contradiction, we suppose that there is w 0 ∈ L 2 (Ω; R N ) and r > 0 such that the ball centered at w 0 with radius r is contained in S. Hence and any w, w L 2 (Ω;R N ) ≤ r. Next, we show that this implies that the ball centered at 0 with radius r is contained in S, too. To this end, we write Thus using the fact that w 0 solves (4.1) and (4.2) we conclude for any ϕ ∈ C ∞ c (Ω) and any w, w L 2 (Ω;R N ) ≤ r. If relation (4.3) holds for any w, w L 2 (Ω;R N ) ≤ r, it must hold for λw, 0 ≤ λ ≤ 1. Consequently, we get from (4.3), and any w, w L 2 (Ω;R N ) ≤ r.
Thus the set S contains the ball of radius r > 0 centered at 0 but as it is invariant with respect to multiplication by any real constant, we conclude which is obviously false as soon as N ≥ 2. Proof. Let [̺ 0,n , m 0,n , E 0,n ] be a sequence of reachable data such that To see that the limit is reachable, we have to find a generating sequence satisfying ( for the corresponding solutions ̺ n,m , m n,m and any fixed n. Now, let be a countable dense set in the Sobolev space × Ω) as soon as ℓ > N + 5 2 . Indeed, for given j ≥ 1, we find n(j) such that and m = m(n(j)) such that [̺ 0,n(j),m(n(j)) , m 0,n(j),m(n(j)) , E 0,n(j),m(n(j)) ] − [̺ 0,n(j) , m 0,n(j) , E 0,n(j) ] < 1 j , T 0 Ω m n(j),m(n(j)) ⊗ m n(j),m(n(j)) ̺ n(j),m(n(j)) − 1 N |m n(j),m(n(j)) | 2 ̺ n(j),m(n(j)) I : Finally, by virtue of (4.4), any ∇ 2 x ϕ can be uniformly approximated by ∇ 2 x ϕ k , and, as the energy E n(j),m(j) are bounded, we conclude that (4.5) holds for any ϕ ∈ C ∞ c ((0, T ) × Ω). Thus we have shown that [̺ 0,n(j),m(j) , m 0,n(j),m(j) , E 0,n(j),m(j) ] We are ready to complete the proof of Theorem 1.3. The set of reachable data being closed (cf. Lemma 4.2), its complement is open. In view of Lemma 4.1, we have only to show that the set of the smooth initial data [̺ 0 , m 0 , E 0 ] giving rise to local-in-time regular solutions is dense in the space L 1 +,s 0 (Ω; R N +2 ). To this end, we record the following result by Schochet [12, Theorem 1] that asserts the existence of smooth solutions whenever: • Ω ⊂ R N is a bounded domain with a sufficiently smooth boundary, say ∂Ω of class C ∞ ; • the initial data [̺ 0,E , ϑ 0,E , u 0,E ] belong to the class • the compatibility conditions ∂ k t u 0,E · n| ∂Ω = 0 hold for k = 0, 1, . . . , m.
In particular, any trio of initial data, (4.6) gives rise to a smooth solution defined on a maximal time interval [0, T max ). Seeing that the set (4.6) is dense in L 1 +,s 0 (Ω; R N +2 ), we have completed the proof of Theorem 1.3.

Reachability for the Riemann problem
To the best of our knowledge there are only two methods available in the literature that give rise to infinitely many solutions of the compressible Euler system, which are both based on a splitting as in (1.9). The first method [9] was explained above, see (1.10). The second method considers initial data constant in each of the two half spaces (Riemann data), see [1] and also [6], [5], [10] for the isentropic case. Here -instead of (1.10) -equations (1.9) 2 and (1.9) 3 are replaced by with a suitable matrix field U. The density ̺, the acoustic potential ∇ x Φ as well as the field U are piecewise constant in (0, T ) × Ω. In particular, there exists a partition of the set (0, T ) × Ω into a finite number of sectors Q i , i = 1, . . . m, such that the wild solutions satisfy Consequently, we may accommodate the data reachable by this method replacing (1.12) by The above observation motivates the following extension of the concept of reachability. We say that Q is a partition of the domain (0, T ) × Ω if Next, we introduce the following generalization of (W )−convergence. Let Q be a closed partition set. We say that a sequence of data {̺ 0,n , m 0,n , E 0,n } ∞ n=1 (W [Q])-converges to [̺ 0 , m 0 , E 0 ], • the initial data [̺ 0,n , m 0,n , E 0,n ] give rise to a sequence of admissible weak solutions [̺ n , m n , E n ] satisfying for some Q ∈ Q.
Obviously any reachable data in the sense of Definition 1.2 are Q-reachable so the set of Q−reachable data is always larger for any closed partition set.
In order to adapt the arguments used in Sections 3, 4 we need the following result.
Lemma 5.2. Let Q be a closed partition set. Then the set of Q−reachable data is closed in L 1 ((0, T ) × Ω; R N +2 ).
Proof. Let [̺ 0,n , m 0,n , E 0,n ] be a sequence of Q−reachable data such that for some Q n ∈ Q and any fixed n.
As the partition set Q is closed, there exists a partition Q ∈ Q such that K ⊂ Q a compact set ⇒ there exist n(K) such that K ⊂ Q n for all n ≥ n(K).
at least for a suitable subsequence (not relabeled). Let ϕ k ∞ k=1 , ϕ ∈ C ∞ c (Q) be a countable dense subset of the Sobolev space Using property (5.3), we can find a subsequence By virtue of (5.4), any ∇ 2 x ϕ can be uniformly approximated by ∇ 2 x ϕ k , and, as the energies E n(j),m(j) are bounded, we conclude that (5.5) holds for any ϕ ∈ C ∞ c (Q). We may infer that whence [̺ 0 , m 0 , E 0 ] is Q−reachable.
Finally, we observe that any regular initial data [̺ 0 , m 0 , E 0 ] satisfy (3.2). To see this, we first observe, similarly to Section 3, that As ̺, m are smooth, we get However, as Q = [0, T ] × Ω and the second derivatives of ρ, m are continuous, this implies In particular, we obtain (3.2) for the initial data. We have shown the following extension of Theorem 1.3.
Theorem 5.3. Let s 0 ∈ R be given. Let Ω ⊂ R N , N = 2, 3 be a bounded smooth domain. Let Q be a closed partition set in (0, T ) × Ω.
Then the complement of the set of Q−reachable data is an open dense set in L 1 +,s 0 (Ω; R N +2 ).
As already pointed out, Theorem 5.3 accommodates the limits of the wild initial data constructed via the Riemann problem with M fans in the sense of [5].

A Wild solutions
The existence of weak solutions for the Euler system in higher space dimensions is an outstanding open problem. Nevertheless, the recently developed adaptation of the convex integration technique to the incompressible Euler system provided certain results also for the compressible case. We report the following result proved in [9].
Theorem A.1. Let the initial data ̺ 0 and ϑ 0 be piecewise constant strictly positive functions defined on Ω. And let T > 0 be given.
In addition, these solutions enjoy the following properties: and the renormalized entropy inequality (2.2) holds as equality.