1 Introduction

The motion of a fluid in gas dynamics is described by means of the standard variables: the fluid density \(\varrho = \varrho (t,x)\), the macroscopic velocity \({{\mathbf {u}}}= {{\mathbf {u}}}(t,x)\), and the absolute temperature \(\vartheta = \vartheta (t,x)\) satisfying the Euler system:

$$\begin{aligned} \begin{aligned}&\partial _t \varrho + \mathrm{div}_x(\varrho {{\mathbf {u}}}) = 0,\\&\partial _t (\varrho {{\mathbf {u}}}) + \mathrm{div}_x(\varrho {{\mathbf {u}}}\otimes {{\mathbf {u}}}) + \nabla _xp(\varrho , \vartheta ) = 0,\\&\partial _t \left( \frac{1}{2} \varrho |{{\mathbf {u}}}|^2 + \varrho e(\varrho , \vartheta ) \right) + \mathrm{div}_x\left[ \left( \frac{1}{2} \varrho |{{\mathbf {u}}}|^2 + \varrho e(\varrho , \vartheta ) \right) {{\mathbf {u}}}\right] + \mathrm{div}_x(p(\varrho , \vartheta ) {{\mathbf {u}}}) =0. \end{aligned} \end{aligned}$$
(1)

We suppose the fluid is confined to a smooth bounded domain \(\varOmega \subset R^N\), \(N=2,3\), with the impermeable boundary:

$$\begin{aligned} {{\mathbf {u}}}\cdot {{\mathbf {n}}}|_{\partial \varOmega } = 0. \end{aligned}$$
(2)

The state of the system at the reference time \(t = 0\) is given by the initial conditions:

$$\begin{aligned} \varrho (0, \cdot ) = \varrho _0, \ {{\mathbf {u}}}(0, \cdot ) = {{\mathbf {u}}}_0, \ \vartheta (0, \cdot ) = \vartheta _0. \end{aligned}$$
(3)

1.1 Thermodynamics

The system (1) contains two thermodynamics functions: the pressure \(p = p(\varrho , \vartheta )\) and the internal energy \(e = e(\varrho , \vartheta )\). In accordance with the Second law, we postulate the existence of the entropy \(s = s(\varrho , \vartheta )\) related to p and e via Gibbs’ equation

$$\begin{aligned} \vartheta Ds = De + D \left( \frac{1}{\varrho } \right) p. \end{aligned}$$

If the functions \(\varrho \), \({{\mathbf {u}}}\), and \(\vartheta \) are continuously differentiable, the relations (1) give rise to the entropy balance:

$$\begin{aligned} \partial _t (\varrho s(\varrho ,\vartheta )) + \mathrm{div}_x(\varrho s(\varrho , \vartheta ) {{\mathbf {u}}}) = 0. \end{aligned}$$
(4)

In the context of weak solutions considered in this paper, it is customary to relax (4) to inequality

$$\begin{aligned} \partial _t (\varrho s(\varrho ,\vartheta )) + \mathrm{div}_x(\varrho s(\varrho , \vartheta ) {{\mathbf {u}}}) \ge 0. \end{aligned}$$
(5)

The Euler system (1) can be written in the conservative variables: the density \(\varrho \), the momentum \({{\mathbf {m}}} = \varrho {{\mathbf {u}}}\), and the energy \(E = \frac{1}{2} \varrho |{{\mathbf {u}}}|^2 + \varrho e(\varrho , \vartheta )\):

$$\begin{aligned} \begin{aligned}&\partial _t \varrho + \mathrm{div}_x{{\mathbf {m}}} = 0,\\&\partial _t {{\mathbf {m}}} + \mathrm{div}_x\left( \frac{{{\mathbf {m}}} \otimes {{\mathbf {m}}} }{\varrho } \right) + \nabla _xp(\varrho , {{\mathbf {m}}}, E) = 0,\\&\partial _t E + \mathrm{div}_x\left( E \frac{{{\mathbf {m}}}}{\varrho } \right) + \mathrm{div}_x\left( p(\varrho , {{\mathbf {m}}}, E) \frac{{{\mathbf {m}}}}{\varrho } \right) =0. \end{aligned} \end{aligned}$$
(6)

Although we divide by \(\varrho \) when passing to the formulation in conservative variables, we can still treat vacuum by requiring the momentum to be zero in vacuum zones, see Sect. 2.

The forthcoming analysis leans essentially on the hypothesis of thermodynamics stability. The latter can be formulated either in the standard variables:

$$\begin{aligned} \frac{\partial p(\varrho , \vartheta )}{\partial \varrho }> 0, \ \frac{\partial e(\varrho , \vartheta )}{\partial \vartheta } > 0; \end{aligned}$$
(7)

or in the conservative variables:

$$\begin{aligned} \begin{aligned}&(\varrho , {{\mathbf {m}}}, E) \mapsto S(\varrho , {{\mathbf {m}}}, E) \equiv \varrho s (\varrho , {{\mathbf {m}}}, E) \\&\quad \hbox {is a strictly concave function of}\ (\varrho , {{\mathbf {m}}}, E). \end{aligned} \end{aligned}$$
(8)

1.2 Smooth and weak solutions

Smooth solutions are at least continuously differentiable on the set \([0,T) \times \overline{\varOmega }\), \(\varrho > 0\), \(\vartheta > 0\), and satisfy (13) pointwise. Smooth solutions are known to exist locally on a maximal interval \([0,T_\mathrm{max})\) as long as the initial data are smooth enough and satisfy the corresponding compatibility conditions on \(\partial \varOmega \), see last part of Sect. 4 for details. Moreover, it is well known that \(T_\mathrm{max} < \infty \) for a “generic” class of data.

The weak solutions satisfy (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 (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 (13) 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 \(L^2(\varOmega ; R^N)\), 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 (6) in the literature use such a splitting method.

A vector field \({{\mathbf {m}}}\) can be decomposed by means of Helmholtz decomposition, i. e. it can be written as the sum of a solenoidal vector field \({{\mathbf {H}}}[{{\mathbf {m}}}]\) and the gradient of a scalar field.

More precisely, the Helmholtz projection operator \({{\mathbf {H}}}\) is defined as

$$\begin{aligned} \begin{aligned}&{{\mathbf {m}}} = {{\mathbf {H}}}[ {{\mathbf {m}}} ] + {{\mathbf {H}}}^\perp [ {{\mathbf {m}}} ], \\&\quad \hbox {where}\; {{\mathbf {H}}}^\perp = \nabla _x\varPhi ,\ \varDelta _x\varPhi = \mathrm{div}_x{{\mathbf {m}}},\ (\nabla _x\varPhi - {{\mathbf {m}}}) \cdot {{\mathbf {n}}}|_{\partial \varOmega } = 0,\ \int _{\varOmega } \varPhi \ {\mathrm{d}} {x} = 0. \end{aligned} \end{aligned}$$

Accordingly, for \({{\mathbf {m}}} = {{\mathbf {v}}} + \nabla _x\varPhi \), \({{\mathbf {v}}} = {{\mathbf {H}}} [{{\mathbf {m}}}]\), the system (6) can be written in the form

$$\begin{aligned} \begin{aligned}&\partial _t \varrho + \varDelta _x\varPhi = 0,\\&\partial _t {{\mathbf {v}}} + {{\mathbf {H}}} \left[ \mathrm{div}_x\left( \frac{{{\mathbf {m}}} \otimes {{\mathbf {m}}} }{\varrho } - \frac{1}{N} \frac{|{{\mathbf {m}}}|^2}{\varrho } {\mathbb {I}} \right) \right] = 0,\\&\partial _t (\nabla _x\varPhi ) + {{\mathbf {H}}}^\perp \left[ \mathrm{div}_x\left( \frac{{{\mathbf {m}}} \otimes {{\mathbf {m}}} }{\varrho } - \frac{1}{N} \frac{|{{\mathbf {m}}}|^2}{\varrho } {\mathbb {I}}\right) \right] \\&\quad + \nabla _x\left( \frac{1}{N} \frac{|{{\mathbf {m}}}|^2}{\varrho } \right) + \nabla _xp(\varrho , {{\mathbf {m}}}, E) =0,\\&\partial _t E + \mathrm{div}_x\left( E \frac{{{\mathbf {m}}}}{\varrho } \right) + \mathrm{div}_x\left( p(\varrho , {{\mathbf {m}}}, E) \frac{{{\mathbf {m}}}}{\varrho } \right) =0. \end{aligned} \end{aligned}$$
(9)

To the best of our knowledge, there are only two convex integration ansatzes for obtaining wild solutions to the Euler system (6) in the literature. Let us begin with the first one, which is used in [9]. This ansatz is based on replacing Eqs. (9)\(_2\) and (9)\(_3\) by the system

$$\begin{aligned} \begin{aligned}&\partial _t {{\mathbf {v}}} +\mathrm{div}_x\left( \frac{{{\mathbf {m}}} \otimes {{\mathbf {m}}} }{\varrho } - \frac{1}{N} \frac{|{{\mathbf {m}}}|^2}{\varrho } {\mathbb {I}} \right) = 0, \ \Big ( \mathrm{div}_x{{\mathbf {v}}} = 0 \Big ), \\&\partial _t (\nabla _x\varPhi )+ \nabla _x\left( \frac{1}{N} \frac{|{{\mathbf {m}}}|^2}{\varrho } \right) + \nabla _xp(\varrho , {{\mathbf {m}}}, E) =0,\ \Big ( \partial _t \varrho + \varDelta _x\varPhi = 0 \Big ). \end{aligned} \end{aligned}$$
(10)

Equation (10)\(_1\) represents the volume preserving part of the motion, while (10)\(_2\) may be seen as a wave equation governing the propagation of acoustic waves. Note that the fields \({{\mathbf {v}}}\) and \(\nabla _x\varPhi \) as well as the fluxes in (10) are orthogonal with respect to the \(L^2\) scalar product, specifically,

$$\begin{aligned} \int _{\varOmega } {{\mathbf {v}}} \cdot \nabla _x\varPhi \ {\mathrm{d}} {x} = 0,\ \left[ \frac{{{\mathbf {m}}} \otimes {{\mathbf {m}}} }{\varrho } - \frac{1}{N} \frac{|{{\mathbf {m}}}|^2}{\varrho } {\mathbb {I}} \right] : \left[ \left( \frac{1}{N} \frac{|{{\mathbf {m}}}|^2}{\varrho } + p(\varrho , {{\mathbf {m}}}, E) \right) {\mathbb {I}} \right] = 0, \end{aligned}$$

where the latter is true since it is a product of a traceless matrix and a multiple of the identity matrix.

Although (10)\(_1\) is apparently overdetermined, it admits (infinitely many) weak solutions for any fixed \(\varrho \) and \(\varPhi \), cf. “Appendix A”.

Definition 1.1

A weak solution \([\varrho , {{\mathbf {m}}}, E]\) of the Euler system (6) is called wild solution if \([\varrho , {{\mathbf {m}}}, E]\) satisfy (10), with \({{\mathbf {v}}} = {{\mathbf {H}}}[{{\mathbf {m}}}]\), \(\nabla _x\varPhi = {{\mathbf {H}}}^\perp [{{\mathbf {m}}}]\). The corresponding initial data \([\varrho _0, {{\mathbf {m}}}_0, E_0]\) are called wild initial data.

As indicated above, Theorem A.1 yields solutions which are wild in the sense of Definition 1.1. This guarantees that the set of wild solutions, as well as the set of wild initial data are non-empty. However Definition 1.1 is more general, which means that there might be wild solutions (and hence wild initial data) in the sense of Definition 1.1 that cannot be obtained by Theorem A.1.

Note that a technique similar to (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 (10) and afterwards extend our results to the wild solutions obtained via the Riemann problem in Sect. 5.

1.3 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

$$\begin{aligned} \begin{aligned}&\texttt {L}^1_{+,s_0} (\varOmega ; R^{N + 2}) \\&= \left\{ [\varrho , {{\mathbf {m}}}, E] \in L^1(\varOmega ; R^{N+2}) \ \Big | \ \varrho \ge 0, \ E \ge 0, \ s(\varrho , {{\mathbf {m}}}, E) \ge s_0 > - \infty \right\} . \end{aligned} \end{aligned}$$

Note that this is not very restrictive as any admissible weak solution satisfies the minimum principle

$$\begin{aligned} s(\varrho , {{\mathbf {m}}}, E)(t,x) \ge \mathrm{ess}\inf _{y \in \varOmega } s(\varrho _0(y), {{\mathbf {m}}}_0(y), E_0(y)) \ \quad \hbox {a.a. in}\ (0,T) \times \varOmega , \end{aligned}$$

cf. [2, Section 2.1.1].

We say that a sequence of data \(\left\{ \varrho _{0,n}, {{\mathbf {u}}}_{0,n}, \vartheta _{0,n} \right\} _{n=1}^\infty \), or, equivalently,

\(\left\{ \varrho _{0,n}, {{\mathbf {m}}}_{0,n}, E_{0,n} \right\} _{n=1}^\infty \), (W)-converges to \([\varrho _0, {{\mathbf {m}}}_0, E_0]\),

$$\begin{aligned}{}[ \varrho _{0,n}, {{\mathbf {m}}}_{0,n}, E_{0,n} ] {\mathop {\rightarrow }\limits ^{(W)}} [\varrho _0, {{\mathbf {m}}}_0, E_0], \end{aligned}$$

if

  • $$\begin{aligned} \varrho _{0,n}> 0,\quad \ s(\varrho _{0,n}, {{\mathbf {m}}}_{0,n}, E_{0,n}) \ge s_0 > -\infty ; \end{aligned}$$
  • $$\begin{aligned}{}[ \varrho _{0,n}, {{\mathbf {m}}}_{0,n}, E_{0,n} ] \rightarrow [\varrho _0, {{\mathbf {m}}}_0, E_0] \quad \ \hbox {in}\ L^1(\varOmega ; R^{N+2}); \end{aligned}$$
    (11)
  • the initial data \([\varrho _{0,n}, {{\mathbf {m}}}_{0,n}, E_{0,n}]\) give rise to a sequence of admissible weak solutions \([\varrho _n, {{\mathbf {m}}}_n, E_n]\) satisfying

    $$\begin{aligned} \begin{aligned} \int _0^T \int _{\varOmega } \left( \frac{{{\mathbf {m}}}_n \otimes {{\mathbf {m}}}_n}{\varrho _n} - \frac{1}{N} \frac{ |{{\mathbf {m}}}_n|^2}{\varrho _n} {\mathbb {I}} \right) : \nabla _x^2 \varphi \ {\mathrm{d}} {x} \ {\mathrm{d}} t \rightarrow 0 \quad \ \hbox {as}\ n \rightarrow \infty \\ \quad \hbox {for any}\ \varphi \in C^\infty _c((0,T) \times \varOmega ). \end{aligned} \end{aligned}$$
    (12)

Note that, in view of (10),

$$\begin{aligned} {{\mathbf {H}}}^\perp \left[ \mathrm{div}_x\left( \frac{{{\mathbf {m}}}_n \otimes {{\mathbf {m}}}_n}{\varrho _n} - \frac{1}{N} \frac{ |{{\mathbf {m}}}_n|^2}{\varrho _n} {\mathbb {I}} \right) \right] = 0 \end{aligned}$$

whenever \([\varrho _n, {{\mathbf {m}}}_n, E_n]\) are wild solutions in the sense of Definition 1.1. In particular, (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 \(\left\{ \varrho _{n}, {{\mathbf {m}}}_{n}, E_{n} \right\} _{n=1}^\infty \) need not a priori converge strongly, and, consequently, (12) does not impose any similar restriction on a possible limit solution emanating from \([\varrho _0, {{\mathbf {m}}}_0, E_0]\).

Definition 1.2

We say that a trio \([\varrho _0, {{\mathbf {m}}}_0, E_0]\) is reachable if there exists a sequence of initial data \(\left\{ \varrho _{0,n}, {{\mathbf {m}}}_{0,n}, E_{0,n} \right\} _{n=1}^\infty \) such that

$$\begin{aligned}{}[ \varrho _{0,n}, {{\mathbf {m}}}_{0,n}, E_{0,n} ] {\mathop {\rightarrow }\limits ^{(W)}} [\varrho _0, {{\mathbf {m}}}_0, E_0]. \end{aligned}$$

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 \(\texttt {L}^1_{+,s_0} (\varOmega ; R^{N+2})\). For the sake of simplicity, we focus on the equations of gas dynamics, where the pressure is given by Boyle–Mariotte law

$$\begin{aligned} p(\varrho , \vartheta ) = \varrho \vartheta , \quad \ \ e(\varrho , \vartheta ) = c_v \vartheta , \ c_v > 0. \end{aligned}$$

Equivalently, in the conservative variables,

$$\begin{aligned} p = \frac{1}{c_v} \left( E - \frac{1}{2} \frac{|{{\mathbf {m}}}|^2}{\varrho } \right) ,\ \quad \vartheta = \frac{1}{c_v \varrho } \left( E - \frac{1}{2} \frac{|{{\mathbf {m}}}|^2}{\varrho } \right) > 0. \end{aligned}$$
(13)

Here the kinetic energy is defined as

$$\begin{aligned} \frac{|{{\mathbf {m}}}|^2}{\varrho } = \left\{ \begin{array}{ll} 0 \ &{}\quad \hbox {if}\quad {{\mathbf {m}}}= 0 \\ \infty \ &{}\quad \hbox {if}\quad \varrho = 0 \\ \frac{|{{\mathbf {m}}}|^2}{\varrho } \ &{}\quad \hbox {otherwise.} \end{array} \right. \end{aligned}$$

The entropy reads

$$\begin{aligned} S (\varrho , {{\mathbf {m}}}, E) = \varrho c_v \log \left( \frac{ E - \frac{1}{2} \frac{|{{\mathbf {m}}}|^2}{\varrho } }{ c_v \varrho ^{1 + \frac{1}{c_v}}}. \right) \end{aligned}$$

Here is our main result.

Theorem 1.3

Let \(s_0 \in R\) be given. Let \(\varOmega \subset R^N\), \(N=2,3\) be a bounded smooth domain.

Then the complement of the set of reachable data is an open dense set in \(\texttt {L}^1_{+,s_0} (\varOmega ; R^{N + 2})\).

Corollary 1.4

The complement of the set of wild initial data (in the sense of Definition 1.1) contains an open dense set in \(\texttt {L}^1_{+,s_0} (\varOmega ; R^{N + 2})\).

Proof

As mentioned above, wild initial data are reachable and hence the complement of the set of wild initial data contains the complement of the set of reachable data. Thus Theorem 1.3 yields the claim. \(\square \)

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 \([\varrho _0, {{\mathbf {m}}}_0, E_0]\) giving rise to a smooth solution \([\varrho , {{\mathbf {m}}}, E]\) defined on a maximal time interval \([0, T_{\mathrm{max}})\). Then we proceed in several steps:

  • Assuming the data \([\varrho _0, {{\mathbf {m}}}_0, E_0]\) are reachable we show that the associated sequence of solutions \(\{ \varrho _n, {{\mathbf {m}}}_n, E_n \}_{n=1}^\infty \) 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 Sect. 2.

  • Next we realize that, thanks to the strong convergence required in (11), the limit DMV solution starts from the initial data represented by the parameterized family of Dirac masses,

    $$\begin{aligned} \left\{ \delta _{\varrho _0(x),{{\mathbf {m}}}_0(x), E_0(x) } \right\} _{x \in \varOmega }. \end{aligned}$$

    Thanks to the general weak–strong uniqueness principle, we conclude that the DMV solution coincides with the strong solution on the time interval \([0, T_\mathrm{max})\), more specifically, the DMV solution is represented by the parameterized family of Dirac masses,

    $$\begin{aligned} \left\{ \delta _{\varrho (t,x),{{\mathbf {m}}}(t,x), E(t,x) } \right\} _{(t,x) \in (0, T_{\mathrm{max}}) \times \varOmega }, \end{aligned}$$

    see Sect. 3. In particular, the solutions \([\varrho _n, {{\mathbf {m}}}_n, E_n]\) converge strongly and we may assume

    $$\begin{aligned} \varrho _n \rightarrow \varrho , \ {{\mathbf {m}}}_n \rightarrow {{\mathbf {m}}},\ E_n \rightarrow E \ \quad \hbox {a.a. in}\ (0,T_{\mathrm{max}}) \times \varOmega . \end{aligned}$$
    (14)
  • Thanks to the strong convergence (14), condition (12), and smoothness of the limit solution, we deduce that

    $$\begin{aligned} \mathrm{div}_x\mathrm{div}_x\left( \frac{ {{\mathbf {m}}} \otimes {{\mathbf {m}}} }{\varrho } \right) - \frac{1}{N} \varDelta _x\left( \frac{|{{\mathbf {m}}}|^2}{\varrho } \right) = 0 \ \quad \hbox {in}\ (0,T) \times \varOmega . \end{aligned}$$

    see Sect. 3 for details. In particular, as the limit solution is continuous up to \(t= 0\),

    $$\begin{aligned} \mathrm{div}_x\mathrm{div}_x\left( \frac{ {{\mathbf {m}}}_0 \otimes {{\mathbf {m}}}_0 }{\varrho _0} \right) - \frac{1}{N} \varDelta _x\left( \frac{|{{\mathbf {m}}}_0|^2}{\varrho _0} \right) = 0 \ \quad \hbox {in}\ \varOmega . \end{aligned}$$
    (15)
  • In Sect. 4, we show that (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 Sect. 5.

2 Dissipative measure valued (DMV) solutions

2.1 Definitions

First we recall the standard definition of the admissible weak solutions to the Euler system:

Definition 2.1

(Admissible weak solution) Let \(\gamma = 1 + \frac{1}{c_v}\). We say that \([\varrho , {{\mathbf {m}}}, E]\) is an admissible weak solution to the Euler system (13) if

$$\begin{aligned} \varrho&\in C_{\mathrm{weak}}([0,T]; L^\gamma (\varOmega )), \ \varrho \ge 0,\ \\ {{\mathbf {m}}}&\in C_{\mathrm{weak}}([0,T]; L^{\frac{2 \gamma }{\gamma + 1}}(\varOmega ; R^N)), \ {{\mathbf {m}}}=0\ \text {whenever}\ \varrho =0,\\ E&\in C_{\mathrm{weak}} ([0,T]; L^1(\varOmega )), \ E\ge 0, \end{aligned}$$

and the following holds:

  • $$\begin{aligned} \int _0^T \int _{\varOmega } \left[ \varrho \partial _t \varphi + {{\mathbf {m}}}\cdot \nabla _x\varphi \right] \ {\mathrm{d}} {x} \ {\mathrm{d}} t = - \int _{\varOmega } \varrho _0 \varphi (0, \cdot ) \ {\mathrm{d}} {x} \end{aligned}$$

    for any \(\varphi \in C^1_c([0,T) \times \overline{\varOmega })\);

  • $$\begin{aligned} \begin{aligned}&\int _0^T \int _{\varOmega } \left[ {{\mathbf {m}}}\cdot \partial _t\varvec{\varphi }+ \left( \frac{{{\mathbf {m}}}\otimes {{\mathbf {m}}}}{\varrho } \right) : \nabla _x\varvec{\varphi }+ p \left( \varrho , {{\mathbf {m}}}, E \right) \mathrm{div}_x\varvec{\varphi }\right] \ {\mathrm{d}} {x} \ {\mathrm{d}} t \\&\quad = - \int _{\varOmega } {{\mathbf {m}}}_0 \cdot \varvec{\varphi }(0, \cdot ) \ {\mathrm{d}} {x} \end{aligned} \end{aligned}$$

    for any \(\varvec{\varphi }\in C^1_c([0,T) \times \overline{\varOmega }; R^N)\), \(\varvec{\varphi }\cdot {{\mathbf {n}}}|_{\partial \varOmega } = 0\);

  • $$\begin{aligned} \int _0^T \int _{\varOmega } \left[ E \partial _t \varphi + \left( E + p(\varrho , {{\mathbf {m}}}, E) \right) \left( \frac{{{\mathbf {m}}}}{\varrho } \right) \cdot \nabla _x\varphi \right] \ {\mathrm{d}} {x} \ {\mathrm{d}} t = - \int _{\varOmega } E_0 \varphi (0, \cdot ) \ {\mathrm{d}} {x} \end{aligned}$$
    (16)

    for any \(\varphi \in C^1_c ([0,T) \times \overline{\varOmega })\);

  • $$\begin{aligned} \begin{aligned} -&\int _{\varOmega } \varrho _0 Z \left( s \left( \varrho _0, {{\mathbf {m}}}_0, E_0 \right) \right) \varphi (0, \cdot ) \ {\mathrm{d}} {x} \\&\ge \int _0^T \int _{\varOmega } \left[ \varrho Z \left( s \left( \varrho , {{\mathbf {m}}}, E \right) \right) \partial _t \varphi + Z \left( s \left( \varrho , {{\mathbf {m}}}, E \right) \right) {{\mathbf {m}}} \cdot \nabla _x\varphi \right] \ {\mathrm{d}} {x} \ {\mathrm{d}} t \end{aligned} \end{aligned}$$
    (17)

    for any \(\varphi \in C^1_c([0,T) \times \overline{\varOmega })\), \(\varphi \ge 0\), and \(Z(s) \le \overline{Z}\), \(Z'(s) \ge 0\).

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

$$\begin{aligned} {\mathcal {P}} = \left\{ [\varrho , {{\mathbf {m}}}, E] \ \Big | \ \varrho \ge 0, \ {{\mathbf {m}}}\in R^N, \ E \ge 0 \right\} \end{aligned}$$

be the phase space associated to the Euler system.

Definition 2.3

(DMV solution) A parametrized family of probability measures \(\{ {\mathcal {V}}_{t,x} \}_{(t,x) \in (0,T) \times \varOmega }\) on the set \({\mathcal {P}}\),

$$\begin{aligned} {\mathcal {V}} \in L^\infty _\mathrm{weak-(*)} ((0,T) \times \varOmega ; \mathrm{prob} [ {\mathcal {P}} ] ) \end{aligned}$$

is called a dissipative measure–valued (DMV) solution to the compressible Euler system with the initial data \(\{ {\mathcal {V}}_{0,x} \}_{x \in \varOmega }\), if the following holds:

  • $$\begin{aligned} \int _0^T \int _{\varOmega } \left[ \left\langle {\mathcal {V}}_{t,x}; \varrho \right\rangle \partial _t \varphi + \left\langle {\mathcal {V}}_{t,x}; {{\mathbf {m}}}\right\rangle \cdot \nabla _x\varphi \right] \ {\mathrm{d}} {x} \ {\mathrm{d}} t = - \int _{\varOmega } \left\langle {\mathcal {V}}_{0,x}; \varrho \right\rangle \varphi (0, \cdot ) \ {\mathrm{d}} {x} \end{aligned}$$

    for any \(\varphi \in C^1_c([0,T) \times \overline{\varOmega })\);

  • $$\begin{aligned} \begin{aligned}&\int _0^T \int _{\varOmega } \bigg [ \left\langle {\mathcal {V}}_{t,x}; {{\mathbf {m}}}\right\rangle \cdot \partial _t\varvec{\varphi }+ \left\langle {\mathcal {V}}_{t,x} ; \frac{ {{\mathbf {m}}}\otimes {{\mathbf {m}}}}{\varrho } \right\rangle : \nabla _x\varvec{\varphi }\\&\qquad + \left\langle {\mathcal {V}}_{t,x}; p \left( \varrho , {{\mathbf {m}}}, E \right) \right\rangle \mathrm{div}_x\varvec{\varphi }\bigg ] \ {\mathrm{d}} {x} \ {\mathrm{d}} t \\&\quad = \int _0^T \int _{\varOmega } \nabla _x\varvec{\varphi }: \mathrm{d}\mu _C - \int _{\varOmega } \left\langle {\mathcal {V}}_{0,x}; {{\mathbf {m}}} \right\rangle \cdot \varvec{\varphi }(0, \cdot ) \ {\mathrm{d}} {x} \end{aligned} \end{aligned}$$

    for any \(\varvec{\varphi }\in C^1_c([0,T) \times \overline{\varOmega }; R^N)\), \(\varvec{\varphi }\cdot {{\mathbf {n}}}|_{\partial \varOmega } = 0\), where \(\mu _C\) is a (vectorial) signed measure on \([0,T] \times \overline{\varOmega }\);

  • $$\begin{aligned} \int _{\varOmega } \left\langle {\mathcal {V}}_{\tau ,x}; E \right\rangle \ {\mathrm{d}} {x} + {\mathcal {D}}(\tau ) \le \int _{\varOmega } \left\langle {\mathcal {V}}_{0,x}; E \right\rangle \ {\mathrm{d}} {x} \end{aligned}$$
    (18)

    for a.a. \(\tau \in (0,T)\), where \({\mathcal {D}} \in L^\infty (0,T)\), \({\mathcal {D}} \ge 0\) is called dissipation defect;

  • $$\begin{aligned} \begin{aligned}&- \int _{\varOmega } \left\langle {\mathcal {V}}_{0,x}; \varrho Z \left( s \left( \varrho , {{\mathbf {m}}}, E \right) \right) \right\rangle \varphi (0, \cdot ) \ {\mathrm{d}} {x} \\&\quad \ge \int _0^T \int _{\varOmega } \left[ \left\langle {\mathcal {V}}_{t,x} ; \varrho Z \left( s \left( \varrho , {{\mathbf {m}}}, E \right) \right) \right\rangle \right] \partial _t \varphi \ {\mathrm{d}} {x} \ {\mathrm{d}} t \\&\qquad + \int _0^T \int _{\varOmega } \left[ \left\langle {\mathcal {V}}_{t,x} ; Z \left( s \left( \varrho , {{\mathbf {m}}}, E \right) \right) {{\mathbf {m}}} \right\rangle \cdot \nabla _x\varphi \right] \ {\mathrm{d}} {x} \ {\mathrm{d}} t \end{aligned} \end{aligned}$$

    for any \(\varphi \in C^1_c([0,T) \times \overline{\varOmega })\), \(\varphi \ge 0\), and \(Z(s) \le \overline{Z}\), \(Z'(s) \ge 0\);

  • $$\begin{aligned} \left\| \mu _C \right\| _{{\mathcal {M}}([0, \tau ] \times \varOmega ) } \le c \int _0^\tau {\mathcal {D}}(t) \ {\mathrm{d}} t \ \quad \hbox {for a.a.}\ \tau \in (0,T). \end{aligned}$$

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 (16) has been replaced by its integrated version (18). The reader may consult [2] for a thorough discussion of this new concept of solution. Note in particular, that (18) implies

$$\begin{aligned} \mathrm{supp}[{\mathcal {V}}_{t,x}] \cap \{ \varrho =0,\ {{\mathbf {m}}}\ne 0\} = \emptyset , \end{aligned}$$

see [2, Remark 2.7]. Hence DMV solutions allow for treating vacuum.

2.2 Generating a DMV solution

Suppose now that

$$\begin{aligned} \begin{aligned}&[ \varrho _{0,n}, {{\mathbf {m}}}_{0,n}, E_{0,n} ] \rightarrow [\varrho _0, {{\mathbf {m}}}_0, E_0] \quad \ \hbox {in}\ L^1(\varOmega ; R^{N+2}),\\&\varrho _{0,n}> 0 \ \hbox {a.a. in}\ \varOmega ,\quad \ s(\varrho _{0,n}, {{\mathbf {m}}}_{0,n}, E_{0,n}) \ge s_0 > -\infty , \end{aligned} \end{aligned}$$
(19)

where \([\varrho _{0,n}, {{\mathbf {m}}}_{0,n}, E_{0,n}]\) are initial data of admissible weak solutions \([\varrho _n, {{\mathbf {m}}}_n, E_n]\) defined on \((0,T) \times \varOmega \).

Passing to a suitable subsequence, we may suppose that \(\{ \varrho _n, {{\mathbf {m}}}_n, E_n \}_{n=1}^\infty \) generates a Young measure

$$\begin{aligned} \left\{ {\mathcal {V}}_{t,x} \right\} _{(t,x) \in (0,T) \times \varOmega )},\ {\mathcal {V}}_{t,x} \in \mathrm{prob} [{\mathcal {P}}]. \end{aligned}$$

As shown in [2, Section 2.1], the Young measure \(\left\{ {\mathcal {V}}_{t,x} \right\} _{(t,x) \in (0,T) \times \varOmega )}\) is a DMV solution of the Euler system in the sense of Definition 2.3. Moreover, in view of (19), the initial measure \(\{ {\mathcal {V}}_{0,x} \}_{x \in \varOmega }\) reads

$$\begin{aligned} {\mathcal {V}}_{0,x} = \delta _{\varrho _0(x), {{\mathbf {m}}}_0(x), E_{0}(x)} \ \quad \hbox {for a.a.}\ x \in \varOmega . \end{aligned}$$

3 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 Sect. 2.2, we suppose now that

$$\begin{aligned}{}[\varrho _{0,n}, {{\mathbf {m}}}_{0,n}, E_{0,n}] {\mathop {\rightarrow }\limits ^{(W)}} [\varrho _0, {{\mathbf {m}}}_0, E_0], \end{aligned}$$

where \([\varrho _0, {{\mathbf {m}}}_0, E_0]\) are now regular initial data generating a \(C^1-\)solution \([\varrho , {{\mathbf {m}}}, E]\) of the Euler system (13) on a maximal time interval \([0, T_{\mathrm{max}})\). Without loss of generality, we may suppose that \(0< T < T_{\mathrm{max}}\).

Applying the weak strong uniqueness principle we obtain

$$\begin{aligned} {\mathcal {V}}_{t,x} = \delta _{\varrho (t,x), {{\mathbf {m}}}(t,x), E(t,x)} \ \quad \hbox {for a.a.}\ (t,x) \in (0,T) \times \varOmega . \end{aligned}$$
(20)

In particular, we may assume

$$\begin{aligned} \varrho _n \rightarrow \varrho , \ {{\mathbf {m}}}_n \rightarrow {{\mathbf {m}}},\ E_n \rightarrow E \ \quad \hbox {a.a. in}\ (0,T) \times \varOmega \end{aligned}$$

for the associated weak solutions \([\varrho _n, {{\mathbf {m}}}_n, E_n]\). Consequently, as \([\varrho _n, {{\mathbf {m}}}_n, E_n]\) satisfy (12), we get

$$\begin{aligned} \int _0^T \int _{\varOmega } \left[ \left( \frac{{{\mathbf {m}}} \otimes {{\mathbf {m}}} }{\varrho } \right) : \nabla _x^2 \varphi - \frac{1}{N} \frac{ |{{\mathbf {m}}}|^2 }{\varrho } \varDelta _x\varphi \right] \ {\mathrm{d}} {x} \ {\mathrm{d}} t = 0 \ \hbox {for any} \ \varphi \in C^\infty _c((0,T) \times \varOmega ). \end{aligned}$$

Finally, as the limit solution is continuous up to the time \(t = 0\), we may infer that in particular,

$$\begin{aligned} \int _{\varOmega } \left[ \left( \frac{{{\mathbf {m}}}_0 \otimes {{\mathbf {m}}}_0 }{\varrho _0} \right) : \nabla _x^2 \varphi - \frac{1}{N} \frac{ |{{\mathbf {m}}}_0|^2 }{\varrho _0} \varDelta _x\varphi \right] \ {\mathrm{d}} {x} = 0 \quad \ \hbox {for any} \ \varphi \in C^\infty _c(\varOmega ). \end{aligned}$$
(21)

For \(C^2\) initial data, relation (21) can be rewritten as a non–linear differential equation

$$\begin{aligned} \mathrm{div}_x\mathrm{div}_x\left( \frac{{{\mathbf {m}}}_0 \otimes {{\mathbf {m}}}_0 }{\varrho _0} \right) - \varDelta _x\left( \frac{1}{N} \frac{ |{{\mathbf {m}}}_0|^2 }{\varrho _0} \right) = 0 \quad \ \hbox {in}\ \varOmega . \end{aligned}$$

4 Reachability

In this section we finish the proof of Theorem 1.3. Introducing \({{\mathbf {w}}} = \frac{ {{\mathbf {m}}}_0 }{\sqrt{\varrho _0}}\) we may write (21) in a concise form

$$\begin{aligned} \int _{\varOmega } \left[ {{\mathbf {w}}} \otimes {{\mathbf {w}}} : \nabla _x^2 \varphi - \frac{1}{N} |{{\mathbf {w}}}|^2 \varDelta _x\varphi \right] \ {\mathrm{d}} {x} = 0 \quad \ \hbox {for any} \ \varphi \in C^\infty _c(\varOmega ). \end{aligned}$$
(22)

We consider solutions of (22) in the space \(L^2(\varOmega ; R^N)\),

$$\begin{aligned} {\mathcal {S}} = \left\{ {{\mathbf {w}}} \in L^2(\varOmega ; R^N)\ \Big | \ {{\mathbf {w}}} \ \hbox {solves}\ (22) \right\} . \end{aligned}$$

It is easy to check that:

  • the set \({\mathcal {S}}\) is closed in \(L^2(\varOmega ; R^N)\);

  • if \({{\mathbf {w}}} \in {\mathcal {S}}\), then \(\lambda {{\mathbf {w}}} \in {\mathcal {S}}\) for any \(\lambda \in R\).

Lemma 4.1

The set \({\mathcal {S}}\) is nowhere dense in \(L^2(\varOmega ; R^N)\), meaning the (closure of the) set \({\mathcal {S}}\) does not contain any ball in \(L^2(\varOmega ; R^N)\).

Proof

Arguing by contradiction, we suppose that there is \({{\mathbf {w}}}_0 \in L^2(\varOmega ; R^N)\) and \(r > 0\) such that the ball centered at \({{\mathbf {w}}}_0\) with radius r is contained in \({\mathcal {S}}\). Hence

$$\begin{aligned} \begin{aligned} \int _{\varOmega } \left[ ({{\mathbf {w}}} + {{\mathbf {w}}}_0) \otimes ({{\mathbf {w}}} + {{\mathbf {w}}}_0) : \nabla _x^2 \varphi - \frac{1}{N} |{{\mathbf {w}}} + {{\mathbf {w}}}_0|^2 \varDelta _x\varphi \right] \ {\mathrm{d}} {x} = 0 \\ \quad \hbox {for any} \ \varphi \in C^\infty _c(\varOmega ), \hbox {and any}\ {{\mathbf {w}}},\ \Vert {{\mathbf {w}}} \Vert _{L^2(\varOmega ; R^N)} \le r. \end{aligned} \end{aligned}$$
(23)

Next, we show that this implies that the ball centered at 0 with radius r is contained in \({\mathcal {S}}\), too. To this end, we write

$$\begin{aligned} \begin{aligned}&\int _{\varOmega } \left[ ({{\mathbf {w}}} + {{\mathbf {w}}}_0) \otimes ({{\mathbf {w}}} + {{\mathbf {w}}}_0) : \nabla _x^2 \varphi - \frac{1}{N} |{{\mathbf {w}}} + {{\mathbf {w}}}_0|^2 \varDelta _x\varphi \right] \ {\mathrm{d}} {x}\\&\quad = \int _{\varOmega } \left[ {{\mathbf {w}}} \otimes {{\mathbf {w}}} : \nabla _x^2 \varphi - \frac{1}{N} |{{\mathbf {w}}}|^2 \varDelta _x\varphi \right] \ {\mathrm{d}} {x} \\&\qquad + \int _{\varOmega } \left[ ({{\mathbf {w}}} \otimes {{\mathbf {w}}}_0 + {{\mathbf {w}}}_0 \otimes {{\mathbf {w}}}) : \nabla _x^2 \varphi - \frac{2}{N} {{\mathbf {w}}} \cdot {{\mathbf {w}}}_0 \varDelta _x\varphi \right] \ {\mathrm{d}} {x}\\&\qquad +\int _{\varOmega } \left[ {{\mathbf {w}}}_0 \otimes {{\mathbf {w}}}_0 : \nabla _x^2 \varphi - \frac{1}{N} |{{\mathbf {w}}}_0|^2 \varDelta _x\varphi \right] \ {\mathrm{d}} {x}. \end{aligned} \end{aligned}$$

Thus using the fact that \({{\mathbf {w}}}_0\) solves (22) and (23) we conclude

$$\begin{aligned} \begin{aligned}&\int _{\varOmega } \left[ {{\mathbf {w}}} \otimes {{\mathbf {w}}} : \nabla _x^2 \varphi - \frac{1}{N} |{{\mathbf {w}}}|^2 \varDelta _x\varphi \right] \ {\mathrm{d}} {x} \\&\qquad + \int _{\varOmega } \left[ ({{\mathbf {w}}} \otimes {{\mathbf {w}}}_0 + {{\mathbf {w}}}_0 \otimes {{\mathbf {w}}}) : \nabla _x^2 \varphi - \frac{2}{N} {{\mathbf {w}}} \cdot {{\mathbf {w}}}_0 \varDelta _x\varphi \right] \ {\mathrm{d}} {x} \ =\ 0\\&\quad \qquad \hbox {for any} \ \varphi \in C^\infty _c(\varOmega )\ \hbox {and any}\ {{\mathbf {w}}},\ \Vert {{\mathbf {w}}} \Vert _{L^2(\varOmega ; R^N)} \le r. \end{aligned} \end{aligned}$$
(24)

If relation (24) holds for any \({{\mathbf {w}}}\), \(\Vert {{\mathbf {w}}} \Vert _{L^2(\varOmega ; R^N)} \le r\), it must hold for \(\lambda {{\mathbf {w}}}\), \(0 \le \lambda \le 1\). Consequently, we get from (24),

$$\begin{aligned} \begin{aligned} 0&= (\lambda ^2 - \lambda )\int _{\varOmega } \left[ {{\mathbf {w}}} \otimes {{\mathbf {w}}} : \nabla _x^2 \varphi - \frac{1}{N} |{{\mathbf {w}}}|^2 \varDelta _x\varphi \right] \ {\mathrm{d}} {x} \\&\quad +\, \lambda \int _{\varOmega } \left[ {{\mathbf {w}}} \otimes {{\mathbf {w}}} : \nabla _x^2 \varphi - \frac{1}{N} |{{\mathbf {w}}}|^2 \varDelta _x\varphi \right] \ {\mathrm{d}} {x} \\&\quad +\, \lambda \int _{\varOmega } \left[ ({{\mathbf {w}}} \otimes {{\mathbf {w}}}_0 + {{\mathbf {w}}}_0 \otimes {{\mathbf {w}}}) : \nabla _x^2 \varphi - \frac{2}{N} {{\mathbf {w}}} \cdot {{\mathbf {w}}}_0 \varDelta _x\varphi \right] \ {\mathrm{d}} {x} \\&= (\lambda ^2 - \lambda )\int _{\varOmega } \left[ {{\mathbf {w}}} \otimes {{\mathbf {w}}} : \nabla _x^2 \varphi - \frac{1}{N} |{{\mathbf {w}}}|^2 \varDelta _x\varphi \right] \ {\mathrm{d}} {x} \ \quad \hbox {for any}\ 0 \le \lambda \le 1. \end{aligned} \end{aligned}$$

We deduce

$$\begin{aligned} \begin{aligned}&\int _{\varOmega } \left[ {{\mathbf {w}}} \otimes {{\mathbf {w}}} : \nabla _x^2 \varphi - \frac{1}{N} |{{\mathbf {w}}}|^2 \varDelta _x\varphi \right] \ {\mathrm{d}} {x} = 0 \ \\&\quad \hbox {for any} \ \varphi \in C^\infty _c(\varOmega ), \hbox {and any}\ {{\mathbf {w}}},\ \Vert {{\mathbf {w}}} \Vert _{L^2(\varOmega ; R^N)} \le r. \end{aligned} \end{aligned}$$

Thus the set \({\mathcal {S}}\) 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

$$\begin{aligned} {\mathcal {S}} = L^2(\varOmega ; R^N), \end{aligned}$$

which is obviously false as soon as \(N \ge 2\). \(\square \)

Lemma 4.2

The set of reachable data in the sense of Definition 1.2 is closed in \(L^1((0,T) \times \varOmega )\).

Proof

Let \([\varrho _{0,n}, {{\mathbf {m}}}_{0,n}, E_{0,n}]\) be a sequence of reachable data such that

$$\begin{aligned}{}[\varrho _{0,n}, {{\mathbf {m}}}_{0,n}, E_{0,n}] \rightarrow [\varrho _0, {{\mathbf {m}}}_0, E_0] \ \quad \hbox {in}\ L^1((0,T) \times \varOmega ; R^{N + 2}). \end{aligned}$$

To see that the limit is reachable, we have to find a generating sequence satisfying (11), (12). To this end, consider the generating sequences of data \([\varrho _{0,n,m}, {{\mathbf {m}}}_{0,n,m}, E_{0,n,m}]\) satisfying

$$\begin{aligned}{}[\varrho _{0,n,m}, {{\mathbf {m}}}_{0,n,m}, E_{0,n,m}] {\mathop {\rightarrow }\limits ^{(W)}} [\varrho _{0,n}, {{\mathbf {m}}}_{0,n}, E_{0,n}] \ \quad \hbox {as}\ m \rightarrow \infty , \end{aligned}$$

meaning, in particular,

$$\begin{aligned} \begin{aligned} \int _0^T \int _{\varOmega } \left( \frac{{{\mathbf {m}}}_{n,m} \otimes {{\mathbf {m}}}_{n,m}}{\varrho _{n,m}} - \frac{1}{N} \frac{ |{{\mathbf {m}}}_{n,m}|^2}{\varrho _{n,m}} {\mathbb {I}} \right) : \nabla _x^2 \varphi \ {\mathrm{d}} {x} \ {\mathrm{d}} t \rightarrow 0 \\ \quad \ \hbox {as}\ m \rightarrow \infty \ \hbox {for any}\ \varphi \in C^\infty _c((0,T) \times \varOmega ) \end{aligned} \end{aligned}$$

for the corresponding solutions \(\varrho _{n,m}\), \({{\mathbf {m}}}_{n,m}\) and any fixed n. Now, let

$$\begin{aligned} \left\{ \varphi ^k \right\} _{k = 1}^\infty , \ \varphi ^k \in C^\infty _c((0,T) \times \varOmega ) \end{aligned}$$

be a countable dense set in the Sobolev space

$$\begin{aligned} \begin{aligned} W^{\ell ,2}_0 ((0,T) \times \varOmega ) \equiv \overline{ C^\infty _c((0,T) \times \varOmega ) }^{\Vert \cdot \Vert _{\ell ,2} } \hookrightarrow C^2([0,T] \times \overline{\varOmega }) \ \hbox {as soon as} \ \ell > \frac{N + 5}{2}. \end{aligned}\nonumber \\ \end{aligned}$$
(25)

By the diagonal method, we can find a subsequence

$$\begin{aligned} \begin{aligned} {[}\varrho _{0,(n, m)(j)}, {{\mathbf {m}}}_{0, (n,m)(j)}, E_{0, (n,m)(j)} ] \rightarrow [\varrho _0, {{\mathbf {m}}}_0, E_0] \ \quad \text{ as }\ j \rightarrow \infty \ \\ \text{ in }\ L^1((0,T) \times \varOmega ; R^{N+2}) \end{aligned} \end{aligned}$$

such that

$$\begin{aligned} \begin{aligned} \int _0^T \int _{\varOmega } \left( \frac{{{\mathbf {m}}}_{(n,m)(j)} \otimes {{\mathbf {m}}}_{(n,m)(j)}}{\varrho _{(n,m)(j)}} - \frac{1}{N} \frac{ |{{\mathbf {m}}}_{(n,m)(j)}|^2}{\varrho _{(n,m)(j)}} {\mathbb {I}} \right) : \nabla _x^2 \varphi ^k \ {\mathrm {d}} {x} \ {\mathrm {d}} t \rightarrow 0 \\ \ \text{ as }\ j \rightarrow \infty \ \text{ for } \text{ any }\ k. \end{aligned} \end{aligned}$$
(26)

Indeed, for given \(j \ge 1\), we find n(j) such that

$$\begin{aligned} \left\| [\varrho _{0,n(j)}, {{\mathbf {m}}}_{0,n(j)}, E_{0,n(j)}] - [\varrho _0, {{\mathbf {m}}}_0, E_0] \right\| < \frac{1}{j}, \end{aligned}$$

and \(m = m(n(j))\) such that

$$\begin{aligned} \left\| [\varrho _{0,n(j), m(n(j))}, {{\mathbf {m}}}_{0,n(j), m(n(j))}, E_{0,n(j), m(n(j))}] - [\varrho _{0,n(j)}, {{\mathbf {m}}}_{0, n(j)}, E_{0, n(j)}] \right\|< \frac{1}{j}, \\ \begin{aligned} \left| \int _0^T \int _{\varOmega } \left( \frac{{{\mathbf {m}}}_{n(j),m(n(j))} \otimes {{\mathbf {m}}}_{n(j),m(n(j))}}{\varrho _{n(j),m(n(j))}} - \frac{1}{N} \frac{ |{{\mathbf {m}}}_{n(j),m(n(j))}|^2}{\varrho _{n(j),m(n(j))}} {\mathbb {I}} \right) : \nabla _x^2 \varphi ^k \ {\mathrm{d}} {x} \ {\mathrm{d}} t \right| < \frac{1}{j} \\ \hbox {for all}\ \varphi ^k \in C^\infty _c((0,T) \times \varOmega ), \ k \le j. \end{aligned} \end{aligned}$$

Finally, by virtue of (25), any \(\nabla ^2_x \varphi \) can be uniformly approximated by \(\nabla ^2_x \varphi ^k\), and, as the energy \(E_{n(j), m(j)}\) are bounded, we conclude that (26) holds for any \(\varphi \in C^\infty _c((0,T) \times \varOmega )\). Thus we have shown that

$$\begin{aligned}{}[\varrho _{0,n(j), m(j)}, {{\mathbf {m}}}_{0, n(j), m(j)}, E_{0, n(j), m(j)} ] {\mathop {\rightarrow }\limits ^{(W)}} [\varrho _0, {{\mathbf {m}}}_0, E_0] \ \quad \hbox {as}\ j \rightarrow \infty ; \end{aligned}$$

whence \([\varrho _0, {{\mathbf {m}}}_0, E_0]\) is reachable.

\(\square \)

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 \([\varrho _0, {{\mathbf {m}}}_0, E_0]\) giving rise to local-in-time regular solutions is dense in the space \(\texttt {L}^1_{+, s_0}(\varOmega ; R^{N+2})\). To this end, we record the following result by Schochet [12, Theorem 1] that asserts the existence of smooth solutions whenever:

  • \(\varOmega \subset R^N\) is a bounded domain with a sufficiently smooth boundary, say \(\partial \varOmega \) of class \(C^\infty \);

  • the initial data \([\varrho _{0,E}, \vartheta _{0,E}, {{\mathbf {u}}}_{0,E}]\) belong to the class

    $$\begin{aligned} \varrho _{0, E}, \vartheta _{0, E} \in W^{m,2}(\varOmega ), \ \quad {{\mathbf {u}}}_{0, E} \in W^{m,2}(\varOmega ; R^N),\quad \ \varrho _{0, E}, \ \vartheta _{0,E} > 0 \ \hbox {in}\ \overline{\varOmega }, \end{aligned}$$

    where \( m > N\);

  • the compatibility conditions

    $$\begin{aligned} \partial ^k_t {{\mathbf {u}}}_{0,E} \cdot {{\mathbf {n}}}|_{\partial \varOmega } = 0 \end{aligned}$$

    hold for \(k=0,1,\dots ,m\).

In particular, any trio of initial data,

$$\begin{aligned} \begin{aligned}&[\varrho _0, {{\mathbf {m}}}_0, E_0] \in C^\infty (\overline{\varOmega }; R^{N+2}), \ \\&{{\mathbf {m}}}_0 = 0, \ \varrho _0 = \overline{\varrho }> 0,\ \vartheta _0 = \overline{\vartheta } > 0\ \quad \hbox {on a neighborhood of}\ \partial \varOmega , \end{aligned} \end{aligned}$$
(27)

gives rise to a smooth solution defined on a maximal time interval \([0, T_{\mathrm{max}})\). Seeing that the set (27) is dense in \(\texttt {L}^1_{+, s_0}(\varOmega ; R^{N+2})\), we have completed the proof of Theorem 1.3.

Remark 4.3

Note that the result is basically independent of the choice of boundary conditions. Similarly, it can be easily extended to more general equation of state satisfying (8), including the isentropic case. Of course, the wild solution class is restricted only to those solutions that can be obtained by the splitting of the incompressible and acoustic part as specified in (10).

5 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 (9). The first method [9] was explained above, see (10). The second method considers initial data constant in each of the two half spaces (Riemann data), see [1] and also [5, 6, 10] for the isentropic case. Here—instead of (10)—Eqs. (9)\(_2\) and (9)\(_3\) are replaced by

$$\begin{aligned} \begin{aligned}&\partial _t {{\mathbf {v}}} +\mathrm{div}_x\left( \frac{{{\mathbf {m}}} \otimes {{\mathbf {m}}} }{\varrho } - \frac{1}{N} \frac{|{{\mathbf {m}}}|^2}{\varrho } {\mathbb {I}} - \varrho {\mathbb {U}} \right) = 0, \ \Big ( \mathrm{div}_x{{\mathbf {v}}} = 0 \Big ), \\&\partial _t (\nabla _x\varPhi ) + \mathrm{div}_x(\varrho {\mathbb {U}}) + \nabla _x\left( \frac{1}{N} \frac{|{{\mathbf {m}}}|^2}{\varrho } \right) + \nabla _xp(\varrho , {{\mathbf {m}}}, E) =0,\ \Big ( \partial _t \varrho + \varDelta _x\varPhi = 0 \Big ), \end{aligned} \end{aligned}$$

with a suitable matrix field \({\mathbb {U}}\). The density \(\varrho \), the acoustic potential \(\nabla _x\varPhi \) as well as the field \({\mathbb {U}}\) are piecewise constant in \((0,T) \times \varOmega \). In particular, there exists a partition of the set \((0,T) \times \varOmega \) into a finite number of sectors \(Q_i, i=1, \dots m,\) such that the wild solutions satisfy

$$\begin{aligned} \begin{aligned} \int _{Q_i} \left[ \frac{{{\mathbf {m}}} \otimes {{\mathbf {m}}} }{\varrho } - \frac{1}{N} \frac{ |{{\mathbf {m}}}|^2 }{\varrho } {\mathbb {I}} \right] : \nabla _x^2 \varphi \ {\mathrm{d}} {x}\ {\mathrm{d}} t = 0 \quad \hbox {for any} \ \varphi \in C^\infty _c(Q_i), \ i = 1,\dots , m. \end{aligned} \end{aligned}$$

Consequently, we may accommodate the data reachable by this method replacing (12) by

$$\begin{aligned} \begin{aligned} \int _0^T \int _{\varOmega } \left( \frac{{{\mathbf {m}}}_n \otimes {{\mathbf {m}}}_n}{\varrho _n} - \frac{1}{N} \frac{ |{{\mathbf {m}}}_n|^2}{\varrho _n} {\mathbb {I}} \right) : \nabla _x^2 \varphi \ {\mathrm{d}} {x} \ {\mathrm{d}} t \rightarrow 0 \ \quad \hbox {as}\ n \rightarrow \infty \quad \hbox {for any}\ \varphi \in C^\infty _c(Q), \end{aligned} \end{aligned}$$

where \(Q \subset (0,T) \times \varOmega \) is an open set such that \(\overline{Q} = [0,T] \times \overline{\varOmega }\).

The above observation motivates the following extension of the concept of reachability. We say that Q is a partition of the domain \((0,T) \times \varOmega \) if

  • \(Q \subset (0,T) \times \varOmega \) is an open set;

  • \(\overline{Q} = [0,T] \times \overline{\varOmega }.\)

We say that a family of partitions \({\mathcal {Q}}\) is a closed partition set if it is closed with respect to the Hausdorff complementary topology. More specifically, any sequence \(\{ Q_n \}_{n=1}^\infty \subset {\mathcal {Q}}\) contains a subsequence such that

$$\begin{aligned} Q^c_{n(k)} {\mathop {\rightarrow }\limits ^{(H)}} Q^c \ \quad \hbox {as} \ k \rightarrow \infty \ \hbox {for some}\ Q \in {\mathcal {Q}}, \end{aligned}$$

where the symbol \({\mathop {\rightarrow }\limits ^{(H)}}\) denotes convergence in the Hausdorff metric and \(Q^c_n\) denotes the complement \(Q^c_n \equiv ([0,T] \times \overline{\varOmega } ) \setminus Q_n\).

An example of a closed partition set related to the convex integration method is

$$\begin{aligned} \begin{aligned} {\mathcal {Q}} = \Big \{&Q \subset (0,T) \times \varOmega \ \Big | \\&\ Q = ((0,T) \times \varOmega ) \setminus (\cup _{i = 1}^M H_i), \ H_i - \hbox {a hyperplane in}\ R^{N + 1} \Big \}, \end{aligned} \end{aligned}$$

where M is a given positive integer. Note that this is indeed a closed partition set since \((0,T)\times \varOmega \) is a bounded subset of \(R^{N+1}\).

The following property is well known, cf. [11]:

$$\begin{aligned} \begin{array}{c} Q^c_{n} {\mathop {\rightarrow }\limits ^{(H)}} Q^c \ \hbox {and}\ K \subset Q \ \hbox {is a compact set} \end{array} \Longrightarrow \begin{array}{c} \hbox {there exists}\ n(K) \ \hbox {such that}\ \\ K \subset Q_n \ \hbox {for all}\ n \ge n(K) \end{array}. \end{aligned}$$
(28)

Next, we introduce the following generalization of \((W)-\)convergence. Let \({\mathcal {Q}}\) be a closed partition set. We say that a sequence of data \(\left\{ \varrho _{0,n}, {{\mathbf {m}}}_{0,n}, E_{0,n} \right\} _{n=1}^\infty \) \((W[{\mathcal {Q}}])\)-converges to \([\varrho _0, {{\mathbf {m}}}_0, E_0]\),

$$\begin{aligned}{}[ \varrho _{0,n}, {{\mathbf {m}}}_{0,n}, E_{0,n} ] {\mathop {\rightarrow }\limits ^{(W[{\mathcal {Q}}])}} [\varrho _0, {{\mathbf {m}}}_0, E_0], \end{aligned}$$

if

  • $$\begin{aligned} \varrho _{0,n}> 0,\quad \ s(\varrho _{0,n}, {{\mathbf {m}}}_{0,n}, E_{0,n}) \ge s_0 > -\infty ; \end{aligned}$$
  • $$\begin{aligned}{}[ \varrho _{0,n}, {{\mathbf {m}}}_{0,n}, E_{0,n} ] \rightarrow [\varrho _0, {{\mathbf {m}}}_0, E_0] \ \quad \hbox {in}\ L^1(\varOmega ; R^{N+2}); \end{aligned}$$
  • the initial data \([\varrho _{0,n}, {{\mathbf {m}}}_{0,n}, E_{0,n}]\) give rise to a sequence of admissible weak solutions \([\varrho _n, {{\mathbf {m}}}_n, E_n]\) satisfying

    $$\begin{aligned} \int _0^T \int _{\varOmega } \left( \frac{{{\mathbf {m}}}_n \otimes {{\mathbf {m}}}_n}{\varrho _n} - \frac{1}{N} \frac{ |{{\mathbf {m}}}_n|^2}{\varrho _n} {\mathbb {I}} \right) : \nabla _x^2 \varphi \ {\mathrm{d}} {x} \ {\mathrm{d}} t \rightarrow 0 \ \quad \hbox {as}\ n \rightarrow \infty \ \hbox {for any}\ \varphi \in C^\infty _c(Q) \nonumber \\ \end{aligned}$$
    (29)

    for some \(Q \in {\mathcal {Q}}\).

Definition 5.1

Let \({\mathcal {Q}}\) be a closed partition set. We say that a trio \([\varrho _0, {{\mathbf {m}}}_0, E_0]\) is \({\mathcal {Q}}-\)reachable if there exists a sequence of initial data \(\left\{ \varrho _{0,n}, {{\mathbf {m}}}_{0,n}, E_{0,n} \right\} _{n=1}^\infty \) such that

$$\begin{aligned}{}[ \varrho _{0,n}, {{\mathbf {m}}}_{0,n}, E_{0,n} ] {\mathop {\rightarrow }\limits ^{(W[{\mathcal {Q}}])}} [\varrho _0, {{\mathbf {m}}}_0, E_0]. \end{aligned}$$

Obviously any reachable data in the sense of Definition 1.2 are \({\mathcal {Q}}\)-reachable so the set of \({\mathcal {Q}}-\)reachable data is always larger for any closed partition set.

In order to adapt the arguments used in Sects. 3 and 4 we need the following result.

Lemma 5.2

Let \({\mathcal {Q}}\) be a closed partition set.

Then the set of \({\mathcal {Q}}-\)reachable data is closed in \(L^1((0,T) \times \varOmega ; R^{N+2})\).

Proof

Let \([\varrho _{0,n}, {{\mathbf {m}}}_{0,n}, E_{0,n}]\) be a sequence of \({\mathcal {Q}}-\)reachable data such that

$$\begin{aligned}{}[\varrho _{0,n}, {{\mathbf {m}}}_{0,n}, E_{0,n}] \rightarrow [\varrho _0, {{\mathbf {m}}}_0, E_0] \ \quad \hbox {in}\ L^1((0,T) \times \varOmega ; R^{N + 2}). \end{aligned}$$

Let \([\varrho _{0,n,m}, {{\mathbf {m}}}_{0,n,m}, E_{0,n,m}]\) be the corresponding generating sequences,

$$\begin{aligned}{}[\varrho _{0,n,m}, {{\mathbf {m}}}_{0,n,m}, E_{0,n,m}] {\mathop {\rightarrow }\limits ^{(W[{\mathcal {Q}}])}} [\varrho _{0,n}, {{\mathbf {m}}}_{0,n}, E_{0,n}] \ \quad \hbox {as}\ m \rightarrow \infty . \end{aligned}$$

In particular, the corresponding solutions \(\varrho _{m,n}\), \({{\mathbf {m}}}_{m,n}\) satisfy

$$\begin{aligned} \begin{aligned} \int _0^T \int _{\varOmega } \left( \frac{{{\mathbf {m}}}_{n,m} \otimes {{\mathbf {m}}}_{n,m}}{\varrho _{n,m}} - \frac{1}{N} \frac{ |{{\mathbf {m}}}_{n,m}|^2}{\varrho _{n,m}} {\mathbb {I}} \right) : \nabla _x^2 \varphi \ {\mathrm{d}} {x} \ {\mathrm{d}} t \rightarrow 0 \\ \quad \hbox {as}\ m \rightarrow \infty \ \hbox {for any}\ \varphi \in C^\infty _c(Q_n) \end{aligned} \end{aligned}$$

for some \(Q_n \in {\mathcal {Q}}\) and any fixed n.

As the partition set \({\mathcal {Q}}\) is closed, there exists a partition \(Q \in {\mathcal {Q}}\) such that

$$\begin{aligned} K \subset Q \ \hbox {a compact set} \Rightarrow \hbox {there exist}\ n(K) \ \hbox {such that}\ K \subset Q_n \ \hbox {for all}\ n \ge n(K) \end{aligned}$$
(30)

at least for a suitable subsequence (not relabeled).

Let

$$\begin{aligned} \left\{ \varphi ^k \right\} _{k = 1}^\infty , \ \varphi \in C^\infty _c(Q) \end{aligned}$$

be a countable dense subset of the Sobolev space

$$\begin{aligned} W^{\ell ,2}_0 (Q) \equiv \overline{ C^\infty _c(Q) }^{\Vert \cdot \Vert _{\ell ,2} } \hookrightarrow C^2(\overline{Q}) \quad \ \hbox {as soon as} \ \ell > \frac{N + 5}{2}. \end{aligned}$$
(31)

Using property eq. (30), we can find a subsequence

$$\begin{aligned} \begin{aligned} {[}\varrho _{0,n(j), m(j)}, {{\mathbf {m}}}_{0, n(j), m(j)}, E_{0, n(j), m(j)} ]&\rightarrow [\varrho _0, {{\mathbf {m}}}_0, E_0] \ \quad \hbox {as}\ j \\&\rightarrow \infty \ \hbox {in}\ L^1((0,T) \times \varOmega ; R^{N+2}) \end{aligned} \end{aligned}$$

satisfying

$$\begin{aligned} \begin{aligned} \int _0^T \int _{\varOmega } \left( \frac{{{\mathbf {m}}}_{n(j),m(j)} \otimes {{\mathbf {m}}}_{n(j),m(j)}}{\varrho _{n(j),m(j)}} - \frac{1}{N} \frac{ |{{\mathbf {m}}}_{n(j),m(j)}|^2}{\varrho _{n(j),m(j)}} {\mathbb {I}} \right) : \nabla _x^2 \varphi ^k \ {\mathrm{d}} {x} \ {\mathrm{d}} t \rightarrow 0 \\ \hbox {as}\ j \rightarrow \infty \ \hbox {for any}\ k. \end{aligned} \end{aligned}$$
(32)

By virtue of (31), any \(\nabla ^2_x \varphi \) can be uniformly approximated by \(\nabla ^2_x \varphi ^k\), and, as the energies \(E_{n(j), m(j)}\) are bounded, we conclude that (32) holds for any \(\varphi \in C^\infty _c(Q)\). We may infer that

$$\begin{aligned}{}[\varrho _{0,n(j), m(j)}, {{\mathbf {m}}}_{0, n(j), m(j)}, E_{0, n(j), m(j)} ] {\mathop {\rightarrow }\limits ^{(W[{\mathcal {Q}}])}} [\varrho _0, {{\mathbf {m}}}_0, E_0] \ \quad \hbox {as}\ j \rightarrow \infty ; \end{aligned}$$

whence \([\varrho _0, {{\mathbf {m}}}_0, E_0]\) is \({\mathcal {Q}}-\)reachable. \(\square \)

Finally, we observe that any regular initial data \([\varrho _0, {{\mathbf {m}}}_0, E_0]\) satisfy (21). To see this, we first observe, similarly to Sect. 3, that

$$\begin{aligned} \int _0^T \int _{\varOmega } \left[ \left( \frac{{{\mathbf {m}}} \otimes {{\mathbf {m}}} }{\varrho } \right) : \nabla _x^2 \varphi - \frac{1}{N} \frac{ |{{\mathbf {m}}}|^2 }{\varrho } \varDelta _x\varphi \right] \ {\mathrm{d}} {x} \ {\mathrm{d}} t = 0 \ \quad \hbox {for any} \ \varphi \in C^\infty _c(Q). \end{aligned}$$

As \(\varrho \), \({{\mathbf {m}}}\) are smooth, we get

$$\begin{aligned} \mathrm{div}_x\mathrm{div}_x\left( \frac{{{\mathbf {m}}} \otimes {{\mathbf {m}}} }{\varrho } \right) - \frac{1}{N} \varDelta _x\frac{ |{{\mathbf {m}}}|^2 }{\varrho } = 0 \ \quad \hbox {in}\ Q. \end{aligned}$$

However, as \(\overline{Q} = [0,T] \times \overline{\varOmega }\) and the second derivatives of \(\rho \), \({{\mathbf {m}}}\) are continuous, this implies

$$\begin{aligned} \mathrm{div}_x\mathrm{div}_x\left( \frac{{{\mathbf {m}}} \otimes {{\mathbf {m}}} }{\varrho } \right) - \frac{1}{N} \varDelta _x\frac{ |{{\mathbf {m}}}|^2 }{\varrho } = 0 \ \quad \hbox {in}\ (0,T) \times \varOmega . \end{aligned}$$

In particular, we obtain (21) for the initial data.

We have shown the following extension of Theorem 1.3.

Theorem 5.3

Let \(s_0 \in R\) be given and \(\varOmega \subset R^N\), \(N=2,3\) be a bounded smooth domain. Let \({\mathcal {Q}}\) be a closed partition set in \((0,T) \times \varOmega \).

Then the complement of the set of \({\mathcal {Q}}-\)reachable data is an open dense set in \(\texttt {L}^1_{+,s_0} (\varOmega ; R^{N + 2})\).

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].