## 1 Motivation

The Aw–Rascle model (AR) is a second order macroscopic model of vehicular traffic developed originally in one-dimensional framework by Aw and Rascle [3], and independently by Zhang [23]. In contrast to the famous first order prototype of Lighthill and Whitham [22], it is a system of two conservation laws describing the conservation of mass and conservation of linear momentum much in the spirit of the compressible Euler system for gases. However, unlike in the fluid models, the second equation is not associated with the actual velocity of motion u, but with the preferred velocity w:

\begin{aligned} \begin{aligned} \partial _t \varrho + \partial _x(\varrho u)&= 0, \\ \partial _t (\varrho w) + \partial _x (\varrho w u )&= 0, \\ {w}&= {u} + P(\varrho ). \end{aligned} \end{aligned}
(1.1)

The two velocities differ by the velocity offset denoted by $$P(\varrho )>0$$, which, in this particular case, depends only on the density. The relation $$u=w-P(\varrho )$$ means that the actual velocity of motion is always smaller than the preferred velocity depending on the congestion of the cars ahead. The one-dimensional AR system has been derived in [2] from the particle model called Follow-the-Leader model with particular form of the offset function $$P(\varrho )=\varrho ^\gamma$$.

Later on, in [6], a singular offset function was considered

\begin{aligned} \begin{aligned} P(\varrho )=p_\varepsilon (\varrho )=\varepsilon \left( \frac{1}{\varrho }-\frac{1}{{{\bar{\varrho }}}} \right) ^{-\gamma } \end{aligned} \end{aligned}
(1.2)

with the maximal density constraint $${{\bar{\varrho }}}>0$$. This form of the offset function causes that the density $$\varrho$$ stays always below its critical value $${{\bar{\varrho }}}$$, provided it was so initially. As demonstrated in [6] the limit $$\varepsilon \rightarrow 0$$ in the system (1.1) leads to a very interesting constrained pressureles gas dynamics system.

Despite some earlier criticism of the fluid approximations of traffic [13], the modelling modifications proposed in [3, 6, 23] turned out to be a big success. It is therefore natural to seek the generalisation of this model to multi-dimensional setting. However, for example for multi lane traffic, the underlying dynamics of the microscopic system is much more complex than just following the car ahead. In [19] a two-dimensional macroscopic model was derived from the microscopic Follow-the-Leader model with driving direction:

\begin{aligned} \partial _t \varrho + \textrm{div}_x(\varrho {\varvec{u}})&= 0, \end{aligned}
(1.3)
\begin{aligned} \partial _t (\varrho {\varvec{w}}) + \textrm{div}_x(\varrho {\varvec{w}} \otimes {\varvec{u}})&= 0, \end{aligned}
(1.4)
\begin{aligned} {\varvec{w}}&= {\varvec{u}} + {\varvec{P}}(\varrho ), \end{aligned}
(1.5)

with a vector of offset functions in each of directions

\begin{aligned} {\varvec{P}}(\varrho )=[P_1(\varrho ),P_2(\varrho )]. \end{aligned}

This system treats the traffic lanes as a continuum, avoiding the need to prescribe heuristically the dynamics of the flow of vehicles across lanes. The numerical examples from [19] show that it indeed reproduces typical traffic flow situations, including overtaking scenarios.

Another multi-dimansional extension of the AR model was recently studied by Aceves-Sánchez et al. [1] in the context of the pedestrian modelling. The model includes a a singular diffusion term enforcing capacity constraints in the crowd density and inducing a steering behaviour. For single population, it can be written in the same elegant form of two conservation laws (1.3)- (1.5), but the offset function $${\varvec{P}}(\varrho )$$ is now given as a gradient of a singular scalar function

\begin{aligned} {\varvec{P}}(\varrho )=\nabla _xp_\varepsilon (\varrho ), \end{aligned}
(1.6)

with $$p_\varepsilon (\varrho )$$ given by (1.2). Replacement of a vector of scalar functions by the gradient of a single function accounts for including certain non-local effects in the interactions between the individuals at the microscopic level. Therefore, this approach seems to be more suitable for pedestrians, rather than for multi-lane traffic flow. Another generalisation of the AR model including the non-local interactions was recently studied in [12].

The existence of measure-valued solutions to the multi-dimensional AR system with offset function (1.6), called the dissipative Aw-Rascle system (DAR), and their weak-strong uniqueness was recently proved in [9]. The existence of strong solutions and passage to the limit $$\varepsilon$$ are so far restricted to the one space dimension [10], see also [11, 21].

It is worth to mention that the similar one-dimensional system and its singular limit were considered before, in the context of lubrication model for interacting rigid spheres. The macroscopic model derived in [20] assumes that the balls of radius $$r=1$$ move in one dimensional space domain filled with the lubricant with viscosity coefficient $$\varepsilon$$. The corresponding system of equations reads:

\begin{aligned} \begin{aligned}&\partial _t \varrho + \partial _x(\varrho u) = 0,\\&\partial _t(\varrho u)+\partial _x(\varrho u^2)-\partial _x\left( \mu _\varepsilon (\varrho )\partial _x u \right) =0, \end{aligned} \end{aligned}
(1.7)

where $$\mu _\varepsilon =\frac{\varepsilon }{1-\varrho }$$. Note that (1.7) is in fact the compressible, pressureless Navier–Stokes system, studied for example in [18]. By a simple formal calculation one can convert system (1.7) into the one-dimensional DAR system (1.1) with

\begin{aligned} P(\varrho )=\frac{\mu _\varepsilon (\varrho )}{\varrho ^2}\partial _x \varrho . \end{aligned}
(1.8)

As far as we know, extension of this lubrication model to the multi-dimensional case, in particular, the form of the stress and maximal constraint of the density, are not known, and it is not clear if the multi-dimensional DAR system is the right setting. Indeed, performing a similar formal calculation for the multi-dimensional system (1.3)–(1.5) with $${\varvec{P}}(\varrho )=\nabla p(\varrho )$$, we see that it is equivalent to a dissipate pressureless compressible system with degenerate density-dependent shear viscosity and a lower order drift term:

\begin{aligned} \begin{aligned}&\partial _t \varrho + \textrm{div}_x(\varrho {\varvec{u}}) = 0,\\&\partial _t (\varrho {\varvec{u}}) + \textrm{div}_x(\varrho {\varvec{u}} \otimes {\varvec{u}}) = \nabla _x (\varrho Q^\prime (\varrho )\textrm{div}_x{\varvec{u}}) + {\mathcal {L}}[\nabla _x Q(\varrho ), \nabla _x {\varvec{u}}], \end{aligned} \end{aligned}
(1.9)

where $$Q^\prime (\varrho ) = \varrho p^\prime (\varrho )$$ and

\begin{aligned} {\mathcal {L}}[\nabla _x Q(\varrho ), \nabla _x {\varvec{u}}] = \nabla _x(\nabla _x Q(\varrho ) \cdot {\varvec{u}})- \textrm{div}_x(\nabla _x Q(\varrho ) \otimes {\varvec{u}}). \end{aligned}

We can easily check that this drift term is indeed of lower order:

\begin{aligned} \left( {\mathcal {L}}[\nabla _x Q(\varrho ), \nabla _x {\varvec{u}}]\right) _j= \sum _{i=1}^{3} \left( \partial _{x_i}Q(\varrho ) \partial _{x_j}u_{i} - \partial _{x_j} Q(\varrho ) \partial _{x_i} u_i \right) \text { for } j=1,2,3, \end{aligned}

and that it disappears for the one-dimensional case, reducing system (1.9) to the lubrication model (1.7).

In this paper we use the above model as a motivation to further study the role of dissipation in the compressible viscous fluid models. We consider a d-dimensional DAR model with a combination of the offset functions from [19] and [1]. More precisely, we take:

For simplicity, we consider the periodic boundary conditions – the physical domain is identified with the $$d-$$dimensional flat torus

\begin{aligned} {\mathbb {T}}^d= \left( [-1,1]|_{\{ -1; 1 \} } \right) ^d,\ d=2,3. \end{aligned}
(1.13)

The goal of the paper is to show that, similarly to the compressible Euler system, the DAR system is basically ill–posed in the class of weak (distributional) solutions. To this end, we adapt the general approach developed in [16] based on the method of convex integration. This method was introduced by DeLellis and Székelyhidi [15], primarily to prove the existence of infinitely many wild solutions to the incompressible Euler system. Subsequently, it was extended by Chiodaroli [14] to the compressible Euler system, and more recently by Buckmaster and Vicol for the incompressible Navier–Stokes equations [7]. It is not yet known if convex integration technique could be further extended to the compressible Navier–Stokes equations. Note, however, that weak inviscid limit of compressible Navier–Stokes system with degenerate viscosities has been recently used in [8] to generate infinitely many global-in-time admissible weak solutions to the isentropic Euler system. The fact that the convex integration technique works for system (1.10)–(1.12), which in one-dimensional setting coincides with the compressible Navier-Stokes system (1.7), is therefore a very interesting observation. Turning to the multi-dimensional setting and its analogous formulation (1.9), we could conjecture that the convex integration technique provides ill-posedness in the class of weak solutions for certain viscous compressible models with degenerate viscosity coefficients possessing the “two-velocity structure”. Similar structure has been used in the past to prove the existence of solutions to compressible Navier–Stokes equations with density-dependent viscosity [5], and in [4] to consider stochastically perturbed transport terms in the compressible Navier–Stokes system with constant viscosity coefficients.

The paper is organised as follows. In Sect. 2 we state our first main result, Theorem 2.3, about ill posedness of the Aw–Rascle system (1.10)–(1.12) with respect to the initial-final data. The solutions obtained in this section connect arbitrary initial and terminal states, however, they may violate the energy inequality. The ill posedness in the class of admissible weak solutions satisfying this inequality is shown in Sect. 3, the final result is stated in Theorem 3.2. The paper is concluded with a discussion of other boundary conditions.

## 2 Ill Posedness with Respect to the Initial–Final Data

In this section we formulate and prove our first main result: that any initial density–velocity data $$(\varrho _0, {\varvec{u}}_0) = (\varrho (0, \cdot ), {\varvec{u}}(0, \cdot ))$$ can connect to arbitrary terminal state $${(\varrho _T, {\varvec{u}}_T) = (\varrho (T, \cdot ), {\varvec{u}}(T, \cdot ))}$$ via a weak solution to problem (1.10)–(1.13). More specifically, we consider

\begin{aligned} \varrho _0, \varrho _T \in C^2({\mathbb {T}}^d),\ \inf _{{\mathbb {T}}^d} \varrho _0> 0,\ \inf _{{\mathbb {T}}^d} \varrho _T > 0, \ \int _{{\mathbb {T}}^d} \varrho _0 \ \,\textrm{d} {x} = \int _{{\mathbb {T}}^d} \varrho _T \ \,\textrm{d} {x} \end{aligned}
(2.1)

together with

\begin{aligned}{} & {} {\varvec{u}}_0, {\varvec{u}}_T \in C^3({\mathbb {T}}^d; R^d),\ \int _{{\mathbb {T}}^d} \varrho _T {\varvec{u}}_T \ \,\textrm{d} {x} - \int _{{\mathbb {T}}^d} \varrho _0 {\varvec{u}}_0 \ \,\textrm{d} {x}\nonumber \\ {}{} & {} \quad = \int _{{\mathbb {T}}^d} \varrho _0 {\varvec{h}} (\varrho _0) \ \,\textrm{d} {x} - \int _{{\mathbb {T}}^d} \varrho _T {\varvec{h}}(\varrho _T) \ \,\textrm{d} {x}. \end{aligned}
(2.2)

Note that the integral equalities in (2.1), (2.2) represent necessary compatibility conditions as the quantities

\begin{aligned} \int _{{\mathbb {T}}^d} \varrho (t, \cdot ) \ \,\textrm{d} {x},\ \int _{{\mathbb {T}}^d} \varrho {\varvec{w}} (t, \cdot ) \ \,\textrm{d} {x} \end{aligned}

are conserved even in the class of weak solutions.

### Remark 2.1

We suppose that both the initial and the end state density is strictly positive. This is essential for our method to work. Possible “vacuum states” would create non-trivial technical problems that could be possibly overcome as long as the vacuum set is independent of time, cf. Chiodaroli [14].

Here and hereafter, we adopt the standard definition of weak solution via the integral identities.

### Definition 2.2

We say that $$(\varrho ,{\varvec{u}})$$ is a weak solution to (1.10) – (1.12), endowed with the periodic boundary conditions (1.13) and with the initial data $$(\varrho _0, {\varvec{u}}_0) = (\varrho (0, \cdot ), {\varvec{u}}(0, \cdot ))$$, if

\begin{aligned} \int _0^T \int _{{\mathbb {T}}^d} \left[ \varrho \partial _t \varphi + \varrho {\varvec{u}}\cdot \nabla _x\varphi \right] \ \,\textrm{d} {x} \,\textrm{d} t&= 0 \nonumber \\ \text{ for } \text{ any } \ \varphi&\in C^1_c((0,T) \times {\mathbb {T}}^d); \nonumber \\ \int _0^T \int _{{\mathbb {T}}^d} \left[ \varrho {\varvec{w}} \cdot \partial _t \varvec{\varphi }+ \varrho {\varvec{w}} \otimes {\varvec{u}}: \nabla _x\varvec{\varphi }\right] \ \,\textrm{d} {x} \,\textrm{d} t&= 0 \nonumber \\ \text{ for } \text{ any }\ \varvec{\varphi }&\in C^1_c((0,T) \times {\mathbb {T}}^d; R^d). \end{aligned}
(2.3)

We claim the following result.

### Theorem 2.3

( Ill posedness with respect to the data) Let $$d=2,3$$. Suppose that

\begin{aligned} {\varvec{h}} \in C^2 ((0, \infty ); R^d),\ p \in C^2((0,\infty )). \end{aligned}
(2.4)

Let $$(\varrho _0, {\varvec{u}}_0)$$, $$(\varrho _T, {\varvec{u}}_T)$$ satisfy (2.1), (2.2).

Then the system (1.10) – (1.12), endowed with the periodic boundary conditions (1.13) admits infinitely many weak solutions in the class

\begin{aligned} \varrho \in C^2([0,T] \times {\mathbb {T}}^d), {\varvec{u}}\in L^\infty ((0,T) \times {\mathbb {T}}^d; R^d) \end{aligned}

such that

\begin{aligned} \varrho (0, \cdot ) = \varrho _0,\ \varrho (T, \cdot ) = \varrho _T,\ (\varrho {\varvec{u}}) (0, \cdot ) = \varrho _0 {\varvec{u}}_0,\ (\varrho {\varvec{u}}) (T, \cdot ) = \varrho _T {\varvec{u}}_T. \end{aligned}
(2.5)

### Remark 2.4

It will become clear in the course of the proof (see formula (2.12) below) that hypothesis (2.4) can be relaxed to

\begin{aligned} {\varvec{h}} \in C^2(I; R^d),\ {p \in C^2(I)}, \end{aligned}

where $$I \subset (0, \infty )$$ is an open interval containing the convex closure of the range of $$\varrho _0$$, $$\varrho _T$$. In particular, we can choose $${\varvec{h}}(\varrho ),p(\varrho )$$ to be singular as in the form (1.2) proposed by the authors of [6].

### Remark 2.5

Our weak solution from Definition 2.2 satisfies in particular that

\begin{aligned} \varrho {\varvec{w}} \in C_{\textrm{weak}}([0,T]; L^q({\mathbb {T}}^d; R^d)) \ \text{ for } \text{ any }\ 1 \le q < \infty ; \end{aligned}

hence

\begin{aligned} \varrho {\varvec{u}} \in C_{\textrm{weak}}([0,T]; L^q({\mathbb {T}}^d; R^d)) \ \text{ for } \text{ any }\ 1 \le q < \infty , \end{aligned}

and (2.5) makes sense.

In the remaining part of the paper, we develop an abstract framework that enables to prove Theorem 2.3 along with other results stated below.

### 2.1 Momentum Decomposition

Write

\begin{aligned} \varrho {\varvec{u}}= {\varvec{v}} + {\varvec{V}} + \nabla _x\Phi , \end{aligned}
(2.6)

where

\begin{aligned} \textrm{div}_x{\varvec{v}} = 0,\ \int _{{\mathbb {T}}^d} {\varvec{v}} \ \,\textrm{d} {x} = 0,\ {\varvec{V}} = {\varvec{V}}(t) \in R^d. \end{aligned}
(2.7)

Note that (2.6) and (2.7) is the standard Helmholtz decomposition, see [17] for its basic properties. Accordingly, the equation of continuity (1.10) reads

\begin{aligned} \partial _t \varrho + \Delta _x\Phi = 0. \end{aligned}
(2.8)

#### 2.1.1 Density Profile

The next step is adjusting a suitable density profile,

\begin{aligned} \varrho \in C^2([0,T] \times {\mathbb {T}}^d),\ \varrho (0, \cdot ) = \varrho _0, \varrho > 0, \varrho (T, \cdot ) = \varrho _T \end{aligned}
(2.9)

where $$\varrho _0$$, $$\varrho _T$$ are the desired initial and terminal states. In accordance with (2.8), this should be done in such a way that

\begin{aligned} \partial _t \varrho (0, \cdot ) + \Delta _x\Phi _{0} = 0,\ \partial _t \varrho (T, \cdot ) + \Delta _x\Phi _{T} = 0, \end{aligned}
(2.10)

where $$\Phi _0$$, $$\Phi _T$$ are the values of the acoustic potential determined by the Helmholtz decomposition of the initial data, and terminal data

\begin{aligned} \varrho _0 {\varvec{u}}_0 = {\varvec{v}}_0 + {\varvec{V}}_0 + \nabla _x\Phi _0, \ \varrho _T {\varvec{u}}_T = {\varvec{v}}_T + {\varvec{V}}_T + \nabla _x\Phi _T, \end{aligned}
(2.11)

respectively.

Consider the functions

\begin{aligned} H&\in C^\infty [0,T],\ 0 \le H \le 1, H (0) = 1, H (T) = 0, H' (0) = H' (T) = 0, \\ Z_0^\delta&\in C^\infty _c [0,T),\ Z_0^\delta (0) = 0,\ (Z_0^\delta )'(0) = -1, \\ Z_T^\delta&\in C^\infty _c (0,T],\ Z_T^\delta (T) = 0,\ (Z_T^\delta )'(T) = -1, \\ |Z_0^\delta |,\ |Z_T^\delta |&< \delta , \ \delta > 0. \end{aligned}

The desired density profile can be taken as

\begin{aligned} \varrho (t,x) = H(t) \varrho _0 (x) + \varrho _T (x) (1 - H(t) ) + Z_0^\delta (t) \Delta _x\Phi _0(x) + Z_T^\delta (t) \Delta _x\Phi _T(x). \nonumber \\ \end{aligned}
(2.12)

Indeed it is easy to check that (2.9), (2.10) hold while the acoustic potential $$\Phi$$ is uniquely determined by (2.8). Moreover, if $$\delta > 0$$ is chosen small enough, we get

\begin{aligned} \inf _{(0,T) \times {\mathbb {T}}^d} \varrho > 0. \end{aligned}

### 2.2 Transformed Problem I

With $$\varrho$$, $$\Phi$$ fixed in the preceding part, the problem (1.10)–(1.12) reduces to

\begin{aligned} \partial _t ({\varvec{v}} + {\varvec{V}})&+ \textrm{div}_x\left( \frac{ \left( {\varvec{v}} + {\varvec{V}} + \nabla _x\Phi \right) \otimes \left( {\varvec{v}} + {\varvec{V}} + \nabla _x\Phi \right) }{\varrho } + \partial _t \left( \Phi + P(\varrho ) \right) {\mathbb {I}} \right) \nonumber \\&= - \partial _t (\varrho {\varvec{h}}(\varrho )) - \textrm{div}_x\Big ( \left( {\varvec{h}}(\varrho ) + \nabla _xp(\varrho ) \right) \otimes \left( {\varvec{V}} + \nabla _x\Phi \right) \Big ) \nonumber \\&\quad - \nabla _x({\varvec{h}}(\varrho ) + \nabla _xp(\varrho ) ) \cdot {\varvec{v}} \end{aligned}
(2.13)
\begin{aligned} \textrm{div}_x{\varvec{v}}&= 0, \end{aligned}
(2.14)

where $$\nabla _xP(\varrho ) = \varrho \nabla _xp(\varrho )$$. System (2.13), (2.14) still contains two unknowns – $${\varvec{v}}$$ and $${\varvec{V}}$$ that should satisfy the associated initial and terminal conditions

\begin{aligned} {\varvec{v}}(0, \cdot ) = {\varvec{v}}_0,\ {{{\varvec{V}}(0) = {\varvec{V}}_0}},\ {\varvec{v}}(T, \cdot ) = {\varvec{v}}_T,\ {{{\varvec{V}}(T)}} = {\varvec{V}}_T. \end{aligned}

### 2.3 Fixing $${\varvec{V}}$$

In addition to (2.10), the density profile should give rise to the desired momentum average $${\varvec{V}}$$. In accordance with the momentum equation (1.11), we get

\begin{aligned} {\varvec{V}}(t) = {\varvec{V}}_0 - \frac{1}{|{\mathbb {T}}^d|} \int _0^t \int _{{\mathbb {T}}^d} \partial _t (\varrho {\varvec{h}}(\varrho )) (s, \cdot ) \ \,\textrm{d} {x} \textrm{d}s \end{aligned}
(2.15)

so that

\begin{aligned} { \frac{\textrm{d}}{\textrm{d} t}} {\varvec{V}} = - \frac{1}{|{\mathbb {T}}^d|} \int _{{\mathbb {T}}^d} \partial _t (\varrho {\varvec{h}}(\varrho )) (t, \cdot ) \ \,\textrm{d} {x}. \end{aligned}

Moreover, in accordance with (2.2), (2.5), we have

\begin{aligned} \int _{{\mathbb {T}}^d} \varrho _T {\varvec{u}}_T \ \,\textrm{d} {x}&=|{\mathbb {T}}^d| {\varvec{V}}_T = |{\mathbb {T}}^d| {\varvec{V}}(T) = |{\mathbb {T}}^d| {\varvec{V}}_0 - \int _0^T \int _{{\mathbb {T}}^d} \partial _t (\varrho {\varvec{h}}(\varrho )) (s, \cdot ) \ \,\textrm{d} {x} \textrm{d}s \nonumber \\&= \int _{{\mathbb {T}}^d} \varrho _0 {\varvec{u}}_0 \ \,\textrm{d} {x} - \int _{{\mathbb {T}}^d} \varrho _T {\varvec{h}} (\varrho _T) \ \,\textrm{d} {x} + \int _{{\mathbb {T}}^d} \varrho _0 {\varvec{h}}(\varrho _0) \ \,\textrm{d} {x}. \end{aligned}
(2.16)

Consequently, equation (2.13) reduces to

\begin{aligned} \partial _t {\varvec{v}}&+ \textrm{div}_x\left( \frac{ \left( {\varvec{v}} + {\varvec{V}} + \nabla _x\Phi \right) \otimes \left( {\varvec{v}} + {\varvec{V}} + \nabla _x\Phi \right) }{\varrho } + \partial _t \left( \Phi + P(\varrho ) \right) {\mathbb {I}} \right) \nonumber \\&= \left( \frac{1}{|{\mathbb {T}}^d|} \int _{{\mathbb {T}}^d} \partial _t (\varrho {\varvec{h}}(\varrho )) \ \,\textrm{d} {x} - \partial _t (\varrho {\varvec{h}}(\varrho )) \right) - \textrm{div}_x\Big ( \left( {\varvec{h}}(\varrho ) + \nabla _xp(\varrho ) \right) \otimes \left( {\varvec{V}} + \nabla _x\Phi \right) \Big ) \nonumber \\&\quad - \nabla _x({\varvec{h}}(\varrho ) + \nabla _xp(\varrho ) ) \cdot {\varvec{v}} \end{aligned}
(2.17)

Similarly to the acoustic potential $$\Phi$$, the function $${\varvec{V}}$$ is now determined through the given density profile $$\varrho$$ via (2.15).

### 2.4 Elliptic Problem I

To rewrite (2.17) in the form considered in [16], we consider a symmetric traceless tensor

\begin{aligned} {\mathbb {M}} = \nabla _x{\varvec{U}} + \nabla _x{\varvec{U}}^t - \frac{2}{d} \textrm{div}_x{\varvec{U}} {\mathbb {I}}, \end{aligned}
(2.18)

where $${\varvec{U}}$$ is the unique zero–mean solution of the elliptic problem

\begin{aligned} \textrm{div}_x\left( \nabla _x{\varvec{U}} + \nabla _x{\varvec{U}}^t - \frac{2}{d} \textrm{div}_x{\varvec{U}} {\mathbb {I}} \right)&= \textrm{div}_x\Big ( \left( {\varvec{h}}(\varrho ) + \nabla _xp(\varrho ) \right) \otimes \left( {\varvec{V}} + \nabla _x\Phi \right) \Big ) \nonumber \\&\quad -\left( \frac{1}{|{\mathbb {T}}^d|} \int _{{\mathbb {T}}^d} \partial _t (\varrho {\varvec{h}}(\varrho )) \ \,\textrm{d} {x} - \partial _t (\varrho {\varvec{h}}(\varrho )) \right) . \end{aligned}
(2.19)

Indeed the operator on the left–hand side of (2.19) is the Lamé operator. Its kernel on the space of periodic functions consists of constants.

Consequently, problem (2.13), (2.14) can be rewritten in the form

\begin{aligned} \partial _t {\varvec{v}}&+ \textrm{div}_x\left( \frac{ \left( {\varvec{v}} + {\varvec{V}} + \nabla _x\Phi \right) \otimes \left( {\varvec{v}} + {\varvec{V}} + \nabla _x\Phi \right) }{\varrho } + \partial _t \left( \Phi + P(\varrho ) \right) {\mathbb {I}} + {\mathbb {M}} \right) \nonumber \\&= - \nabla _x({\varvec{h}}(\varrho ) + \nabla _xp(\varrho ) ) \cdot {\varvec{v}}, \end{aligned}
(2.20)
\begin{aligned} \textrm{div}_x{\varvec{v}}&= 0, \end{aligned}
(2.21)

with $${\varvec{v}}(0, \cdot ) = {\varvec{v}}_0$$, $${\varvec{v}}(T, \cdot ) = {\varvec{v}}_T$$.

### 2.5 Elliptic Problem II

Similarly to the preceding step, we set

\begin{aligned} {\mathbb {N}} = \nabla _x{\varvec{R}} + \nabla _x{\varvec{R}}^t - \frac{2}{d} \textrm{div}_x{\varvec{R}} {\mathbb {I}}, \end{aligned}
(2.22)

with $${\varvec{R}}$$ solving

\begin{aligned} \textrm{div}_x\left( \nabla _x{\varvec{R}} + \nabla _x{\varvec{R}}^t - \frac{2}{d} \textrm{div}_x{\varvec{R}} {\mathbb {I}} \right) = \nabla _x({\varvec{h}}(\varrho ) + \nabla _xp(\varrho ) ) \cdot {\varvec{v}}. \end{aligned}
(2.23)

Note carefully that $${\mathbb {N}} = {\mathbb {N}}[{\varvec{v}}]$$ depends on the unknown $${\varvec{v}}$$.

Consequently, we may rewrite (2.20), (2.21) in the form

\begin{aligned} \partial _t {\varvec{v}} + \textrm{div}_x\left( \frac{ \left( {\varvec{v}} + {\varvec{V}} + \nabla _x\Phi \right) \otimes \left( {\varvec{v}} + {\varvec{V}} + \nabla _x\Phi \right) }{\varrho } + \partial _t \left( \Phi + P(\varrho ) \right) {\mathbb {I}} + {\mathbb {M}} + {\mathbb {N}}[{\varvec{v}}] \right)&= 0, \end{aligned}
(2.24)
\begin{aligned} \textrm{div}_x{\varvec{v}}&= 0. \end{aligned}
(2.25)

Finally, let us consider the energy associated to the system,

\begin{aligned} e = \frac{1}{2} \frac{|{\varvec{v}} + {\varvec{V}} + \nabla _x\Phi |^2 }{\varrho }. \end{aligned}
(2.26)

Introducing the notation

\begin{aligned} {\varvec{m}} \odot {\varvec{m}} = {\varvec{m}} \otimes {\varvec{m}} - \frac{1}{d} |{\varvec{m}}|^2 {\mathbb {I}}, \end{aligned}

we may rewrite (2.24), (2.25) as an “abstract over-determined (pressureless) Euler system”:

\begin{aligned} \partial _t {\varvec{v}} + \textrm{div}_x\left( \frac{ \left( {\varvec{v}} + {\varvec{V}} + \nabla _x\Phi \right) \odot \left( {\varvec{v}} + {\varvec{V}} + \nabla _x\Phi \right) }{\varrho } + {\mathbb {M}} + {\mathbb {N}}[{\varvec{v}}] \right)&= 0, \end{aligned}
(2.27)
\begin{aligned} \textrm{div}_x{\varvec{v}}&= 0, \end{aligned}
(2.28)
\begin{aligned} \frac{1}{2} \frac{|{\varvec{v}} + {\varvec{V}} + \nabla _x\Phi |^2 }{\varrho } = e&= \Lambda - \frac{d}{2} \partial _t \Big ( \Phi + P(\varrho ) \Big ), \end{aligned}
(2.29)
\begin{aligned} {\varvec{v}}(0, \cdot ) = {\varvec{v}}_0,\ {\varvec{v}}(T, \cdot ) = {\varvec{v}}_T, \end{aligned}
(2.30)

where $$\Lambda = \Lambda (t)$$ is an arbitrary spatially homogeneous function to be adjusted below. Although problem (2.27)-(2.30) is ill posed in the class of strong solutions, it admits infinitely many weak solutions, as we shall see below.

### 2.7 Convex Integration

Motivated by [16, Section 13.2.2], we introduce the class of subsolutions to problem (2.27)– (2.30):

\begin{aligned} X_0&= \left\{ {\varvec{v}} \in C_{\textrm{weak}}([0,T]; L^2({\mathbb {T}}^d; R^d) \cap L^\infty ((0,T) \times {\mathbb {T}}^d; R^d ) \ \Big |\ {\varvec{v}}(0, \cdot ) = {\varvec{v}}_0,\ {\varvec{v}}(T, \cdot ) = {\varvec{v}}_T, \right. \nonumber \\&\partial _t {\varvec{v}} + \textrm{div}_x{\mathbb {F}} = 0 \ \text{ in }\ {\mathcal {D}}'((0,T) \times {\mathbb {T}}^d; R^d) \ \text{ for } \text{ some }\ {\mathbb {F}} \in L^\infty ((0,T) \times {\mathbb {T}}^d; R^{d \times d}_{0,\textrm{sym}} ), \nonumber \\&{\varvec{v}} \in C((0,T) \times {\mathbb {T}}^d; R^d),\ {\mathbb {F}} \in C((0,T) \times {\mathbb {T}}^d; R^{d \times d}_{0,\textrm{sym}} ), \nonumber \\&\sup _{\tau< t \le T, \ x \in {\mathbb {T}}^d} \frac{d}{2} \lambda _{\textrm{max}} \left[ \frac{({\varvec{v}} + {\varvec{V}} + \nabla _x\Phi ) \otimes ({\varvec{v}} + {\varvec{V}} + \nabla _x\Phi )}{\varrho } - {\mathbb {F}} + {\mathbb {M}} + {\mathbb {N}}[{\varvec{v}}] \right] - e< 0 \nonumber \\&\text{ for } \text{ any } \ 0< \tau < T \Big \}. \end{aligned}
(2.31)

The notion of subsolution is inspired by the seminal paper of De Lellis and Székelyhidi [15], however, the definition must be modified considerably to accommodate the present problem. In particular, it contains the non–local term $${\mathbb {N}}[{\varvec{v}}]$$. The general idea of the method of convex integration is obtaining the solution of the problem in the closure of $$X_0$$.

In accordance with (2.29), the energy e in (2.31) is given as

\begin{aligned} e = \Lambda - \frac{d}{2} \partial _t \left( \Phi + P(\varrho ) \right) . \end{aligned}

The symbol $$\lambda _{\textrm{max}}[{\mathbb {A}}]$$ denotes the maximal eigenvalue of a symmetric matrix $${\mathbb {A}}$$. We recall the algebraic inequality

\begin{aligned} \frac{1}{2} |{\varvec{w}}|^2 \le d \lambda _{\textrm{max}} [{\varvec{w}} \otimes {\varvec{w}} - {\mathbb {B}} ],\ {\mathbb {B}} \in R^{d \times d}_{0, \textrm{sym}}. \end{aligned}
(2.32)

As proved in [16, Theorem 13.2.1], problem (2.27)–(2.30) admits infinitely many weak solution if the following holds:

• The set $$X_0$$ of subsolutions is non–empty;

• The set $$X_0$$ is bounded in $$L^\infty ((0,T) \times {\mathbb {T}}^d; R^d)$$;

• The mapping

\begin{aligned} {\varvec{v}} \mapsto {\mathbb {N}}[{\varvec{v}}] \end{aligned}

enjoys the following weak continuity property:

\begin{aligned} {\varvec{v}}_n \rightarrow {\varvec{v}} \ \text{ in }\ C_{\textrm{weak}}([0,\tau ]; L^2({\mathbb {T}}^d; R^d)) \ {}&\text{ and } \text{ weakly-(*) } \text{ in }\ L^\infty ((0,\tau ) \times {\mathbb {T}}^d; R^d)) \nonumber \\&\Rightarrow \nonumber \\ {\mathbb {N}}[{\varvec{v}}_n] \rightarrow {\mathbb {N}}[{\varvec{v}}] \ {}&\text{ in }\ C([0,\tau ] \times {\mathbb {T}}^d; R^{d \times d}) \end{aligned}
(2.33)

for any $$0< \tau \le T$$.

To see that $$X_0$$ is non–empty, it is enough to consider

\begin{aligned} {\varvec{v}} = (1 - t/T) {\varvec{v}}_0 + t/T {\varvec{v}}_T \end{aligned}

the obviously satisfies the initial–terminal conditions, $$\textrm{div}_x{\varvec{v}} = 0$$, and

\begin{aligned} \partial _t {\varvec{v}} = \frac{1}{T} ({\varvec{v}}_T - {\varvec{v}}_0 ). \end{aligned}

Since

\begin{aligned} \int _{{\mathbb {T}}^d} ({\varvec{v}}_T - {\varvec{v}}_0 ) \ \,\textrm{d} {x} = 0, \end{aligned}

it is easy to find (smooth) $${\mathbb {F}} \in L^\infty ((0,T) \times {\mathbb {T}}^d; R^{d \times d}_{0, \textrm{sym}})$$ such that

\begin{aligned} \partial _t {\varvec{v}} + \textrm{div}_x{\mathbb {F}} = 0. \end{aligned}

Finally, we fix $$\Lambda > 0$$ large enough yielding $${\varvec{v}} \in X_0$$ – the set of subsolutions is non–empty. Moreover, with $$\Lambda$$ fixed, we may use inequality (2.32) to concluded that $$X_0$$ is bounded in $$L^\infty ((0,T) \times {\mathbb {T}}^d; R^d)$$. The continuity property (2.33) follows easily from (2.22), (2.23) and the standard elliptic $$L^p$$–theory.

We have proved Theorem 2.3. $$\square$$

## 3 Satisfaction of the Energy Inequality

The AR system (1.10) – (1.12) admits a natural energy functional

\begin{aligned} E(\varrho , {\varvec{u}}) = \frac{1}{2} \varrho \left| {\varvec{u}}+ {\varvec{h}}(\varrho ) + \nabla _xp(\varrho ) \right| ^2. \end{aligned}

Given the periodic boundary conditions, the total energy of smooth solutions is conserved,

\begin{aligned} \int _{{\mathbb {T}}^d} E(\varrho , {\varvec{u}})(t, \cdot ) \ \,\textrm{d} {x} = \int _{{\mathbb {T}}^d} E(\varrho _0, {\varvec{u}}_0) \ \,\textrm{d} {x} \ \text{ for } \text{ any }\ t \in [0,T]. \end{aligned}

Admissible weak solutions should satisfy at least the energy inequality

\begin{aligned} \frac{\textrm{d}}{\,\textrm{d} t } \int _{{\mathbb {T}}^d} E(\varrho , {\varvec{u}}) \ \,\textrm{d} {x} \le 0,\quad \int _{{\mathbb {T}}^d} E(\varrho , {\varvec{u}}) (t, \cdot ) \ \,\textrm{d} {x} \le \int _{{\mathbb {T}}^d} E(\varrho _0, {\varvec{u}}_0 ) \ \,\textrm{d} {x}. \end{aligned}
(3.1)

### Remark 3.1

The first inequality in (3.1) is satisfied in the sense of distributions, and the second one guarantees that there is no initial energy jump. Equivalently, we can include both inequalities in a single weak formulation

\begin{aligned} - \int _0^T \int _{{\mathbb {T}}^d} E(\varrho , {\varvec{u}}) \partial _t \psi \ \,\textrm{d} {x} \,\textrm{d} t \le \int _{{\mathbb {T}}^d} E(\varrho _0, {\varvec{u}}_0 ) \ \,\textrm{d} {x} \end{aligned}

satisfied for any $$\psi \in C^1_c[0,T)$$, $$\psi \ge 0$$, $$\psi (0) = 1$$.

The solutions obtained in Theorem 2.3 connect arbitrary initial and terminal states, in particular, they may violate at least one of the inequalities in (3.1).

To obtain the existence of infinitely many admissible solutions, we change slightly the ansatz in Theorem (2.3) choosing

\begin{aligned} \varrho _0 = \varrho _T,\ {\varvec{u}}_0 = {\varvec{u}}_T = 0. \end{aligned}

Keeping the notation of Sect. 2 we therefore obtain

\begin{aligned} \varrho {\varvec{u}}= {\varvec{v}}, \ \Phi = 0,\ {\varvec{V}} = 0, \end{aligned}

while system (2.27)–(2.29) reduces to

\begin{aligned} \partial _t {\varvec{v}} + \textrm{div}_x\left( \frac{ {\varvec{v}} \odot {\varvec{v}} }{\varrho } + {\mathbb {M}} + {\mathbb {N}}[{\varvec{v}}] \right)&= 0, \end{aligned}
(3.2)
\begin{aligned} \textrm{div}_x{\varvec{v}}&= 0, \end{aligned}
(3.3)
\begin{aligned} \frac{1}{2} \frac{|{\varvec{v}} |^2 }{\varrho } = e&= \Lambda . \end{aligned}
(3.4)

Seeing that $${\varvec{v}} = \varrho {\varvec{u}}$$ with $$\varrho$$ independent of time, we have to fix $$\Lambda$$ in (3.4) so that

\begin{aligned} \frac{\textrm{d}}{\,\textrm{d} t } \int _{{\mathbb {T}}^d} \frac{1}{2} \varrho \left| {\varvec{u}}+ {\varvec{h}}(\varrho ) + \nabla _xp(\varrho ) \right| ^2 \ \,\textrm{d} {x} \le 0. \end{aligned}
(3.5)

We have

\begin{aligned} \frac{1}{2} \varrho \left| {\varvec{u}}+ {\varvec{h}}(\varrho ) + \nabla _xp(\varrho ) \right| ^2 = \frac{1}{2} \frac{|{\varvec{v}}|^2}{\varrho } + \varrho {\varvec{u}}\cdot {\varvec{h}}(\varrho ) + \varrho {\varvec{u}}\cdot \nabla _x\varrho + \frac{1}{2} \varrho |{\varvec{h}}(\varrho ) + \nabla _xp(\varrho ) |^2. \end{aligned}

As $$\varrho = \varrho _0(x)$$ is independent of t, we easily compute

\begin{aligned} \frac{\textrm{d}}{\,\textrm{d} t } \int _{{\mathbb {T}}^d} E(\varrho , {\varvec{u}}) \ \,\textrm{d} {x} = \frac{|{\mathbb {T}}^d|}{2} \Lambda '(t) + \frac{\textrm{d}}{\,\textrm{d} t } \int _{{\mathbb {T}}^d} \varrho {\varvec{u}}\cdot {\varvec{h}}(\varrho ) \ \,\textrm{d} {x}, \end{aligned}

where we have used

\begin{aligned} \int _{{\mathbb {T}}^d} \varrho {\varvec{u}}\cdot \nabla _xp(\varrho ) \ \,\textrm{d} {x} = 0. \end{aligned}

Finally, using the momentum equation (1.11) we compute

\begin{aligned} \frac{\textrm{d}}{\,\textrm{d} t } \int _{{\mathbb {T}}^d} \varrho {\varvec{u}}\cdot {\varvec{h}}(\varrho ) \ \,\textrm{d} {x} = \int _{{\mathbb {T}}^d} \varrho ({\varvec{u}}+ {\varvec{h}}(\varrho ) + \nabla _xp(\varrho ) ) \otimes {\varvec{u}}: \nabla _x{\varvec{h}}(\varrho ) \ \,\textrm{d} {x}. \end{aligned}

Consequently, we may fix $$\Lambda = \Lambda (t)$$ in such a way that

\begin{aligned} \frac{\textrm{d}}{\,\textrm{d} t } \int _{{\mathbb {T}}^d} E(\varrho , {\varvec{u}}) \ \,\textrm{d} {x} \le 0. \end{aligned}

Thus, fixing $$\Lambda$$ and applying Theorem 2.3, we obtain infinitely many solutions of the Aw–Rascle system with a non–increasing total energy profile. This, however, does not exclude the possibility that the energy experiences initial jump, specifically,

\begin{aligned} \liminf _{t \rightarrow 0 +} \int _{{\mathbb {T}}^d} E(\varrho (t), {\varvec{u}}(t)) \ \,\textrm{d} {x} > \int _{{\mathbb {T}}^d} E(\varrho _0, {\varvec{u}}_0) \ \,\textrm{d} {x}. \end{aligned}

To solve the problem of initial energy jump, we use [16, Theorem 13.6.1]. Specifically, there exists a sequence of times $$\tau _n \searrow 0$$ such that the problem (3.2)–(3.4) admits infinitely many weak solutions on the interval $$[\tau _n, T]$$, with the initial data

\begin{aligned} (\varrho (\tau _n, \cdot ), {\varvec{u}}(\tau _n, \cdot )) = (\varrho _0, {\varvec{u}}(\tau _n, \cdot )) \end{aligned}

such that

\begin{aligned} \liminf _{t \rightarrow \tau _n +} \int _{{\mathbb {T}}^d} E(\varrho (t), {\varvec{u}}(t)) \ \,\textrm{d} {x} > \int _{{\mathbb {T}}^d} E(\varrho (\tau _n, \cdot ), {\varvec{u}}(\tau _n, \cdot ) ) \ \,\textrm{d} {x}. \end{aligned}

We have shown the following result.

### Theorem 3.2

(Ill posedness in the class of admissible solutions) Let $$d=2,3$$. Suppose that

\begin{aligned} {\varvec{h}} \in C^2 ((0, \infty ); R^d),\ p \in C^2((0,\infty )). \end{aligned}
(3.6)

Let $$\varrho _0 \in C^2({\mathbb {T}}^d)$$, $$\inf _{{\mathbb {T}}^d} \varrho _0 > 0$$ be given. Then there exists an initial velocity $${\varvec{u}}_0 \in L^\infty ({\mathbb {T}}^d; R^d)$$ such that the system (1.10) – (1.12), endowed with the periodic boundary conditions (1.13) admits infinitely many weak solutions in the class

\begin{aligned} \varrho \in C^2([0,T] \times {\mathbb {T}}^d), {\varvec{u}}\in L^\infty ((0,T) \times {\mathbb {T}}^d; R^d) \end{aligned}

satisfying

\begin{aligned} \varrho (0, \cdot ) = \varrho (T, \cdot ) = \varrho _0,\ (\varrho {\varvec{u}}) (T, \cdot ) = 0, \end{aligned}
(3.7)

together with the energy inequality

\begin{aligned} \frac{\textrm{d}}{\,\textrm{d} t } \int _{{\mathbb {T}}^d} E(\varrho , {\varvec{u}}) \ \,\textrm{d} {x} \le 0,\ \int _{{\mathbb {T}}^d} E(\varrho , {\varvec{u}}) (t, \cdot ) \ \,\textrm{d} {x} \le \int _{{\mathbb {T}}^d} E(\varrho _0, {\varvec{u}}_0 ) \ \,\textrm{d} {x}. \end{aligned}

### Remark 3.3

Hypothesis (3.6) can be relaxed exactly as in Remark 2.4.

We conclude the paper with two remarks concerning other choices of boundary data.

### Remark 3.4

The periodic boundary data can be replaced by more physically realistic boundary conditions, namely

\begin{aligned} {\varvec{u}}\cdot {\varvec{n}}|_{\partial \Omega } = 0, \end{aligned}
(3.8)

where $$\Omega \subset R^d$$ is a bounded domain with smooth boundary. Accordingly, the weak formulation of the equation (1.11) reads

\begin{aligned} \int _0^T \int _{\Omega } \Big ( \varrho {\varvec{w}} \cdot \partial _t \varvec{\varphi }+ \varrho {\varvec{w}} \otimes {\varvec{u}}: \nabla _x\varvec{\varphi }\Big ) \ \,\textrm{d} {x} \,\textrm{d} t = - \int _{\Omega } \varrho _0 {\varvec{w}}_0 \cdot \varvec{\varphi }(0, \cdot ) \ \,\textrm{d} {x}, \end{aligned}
(3.9)

for any test function $$\varvec{\varphi }$$ in the class

\begin{aligned} \varvec{\varphi }\in C^1_c([0,T) \times \overline{\Omega }; R^d),\ \varvec{\varphi }\cdot {\varvec{n}}|_{\partial \Omega } = 0. \end{aligned}
(3.10)

For the proofs from the previous sections to be adaptable to this boundary conditions, we have to impose a geometric restriction on the shape of the domain $$\Omega$$, specifically, $$\Omega$$ is not rotationally symmetric with respect to some axis.

### Remark 3.5

We can also consider the case of general boundary conditions on a regular bounded domain $$\Omega$$, namely,

\begin{aligned} \varrho {\varvec{u}}\cdot {\varvec{n}}|_{\partial \Omega } = {{\varrho _B v_B}},\ \varrho |_{\partial \Omega } = \varrho _B,\ \nabla _x\varrho \cdot {\varvec{n}}|_{\partial \Omega } = D_N \varrho _B, \end{aligned}
(3.11)

where $$D_N$$ is the Dirichlet-to-Neumann operator.

To accommodate (3.11), we consider a weaker formulation of (1.11), namely

\begin{aligned} \int _0^T \int _{\Omega } \Big ( \varrho {\varvec{w}} \cdot \partial _t \varvec{\varphi }+ \varrho {\varvec{w}} \otimes {\varvec{u}}: \nabla _x\varvec{\varphi }\Big ) \ \,\textrm{d} {x} \,\textrm{d} t = - \int _{\Omega } \varrho _0 {\varvec{w}}_0 \cdot \varvec{\varphi }(0, \cdot ) \ \,\textrm{d} {x} \end{aligned}
(3.12)

for any test function $$\varvec{\varphi }$$ in the class

\begin{aligned} \varvec{\varphi }\in C^1_c([0,T) \times {\Omega }; R^d). \end{aligned}
(3.13)

In the case of general boundary conditions, we can obtain the existence of infinitely many solutions for given data. The related energy inequality must be modified accordingly to discuss admissible solutions for certain data.