Abstract
We prove nonuniqueness of weak solutions to multidimensional generalisation of the AwRascle model of vehicular traffic. Our generalisation includes the velocity offset in a form of gradient of density function, which results in a dissipation effect, similar to viscous dissipation in the compressible viscous fluid models. We show that despite this dissipation, the extension of the method of convex integration can be applied to generate infinitely many weak solutions connecting arbitrary initial and final states. We also show that for certain choice of data, ill posedness holds in the class of admissible weak solutions.
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1 Motivation
The Aw–Rascle model (AR) is a second order macroscopic model of vehicular traffic developed originally in onedimensional 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:
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 \(u=wP(\varrho )\) means that the actual velocity of motion is always smaller than the preferred velocity depending on the congestion of the cars ahead. The onedimensional AR system has been derived in [2] from the particle model called FollowtheLeader model with particular form of the offset function \(P(\varrho )=\varrho ^\gamma \).
Later on, in [6], a singular offset function was considered
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 multidimensional 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 twodimensional macroscopic model was derived from the microscopic FollowtheLeader model with driving direction:
with a vector of offset functions in each of directions
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 multidimansional extension of the AR model was recently studied by AcevesSánchez 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
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 nonlocal effects in the interactions between the individuals at the microscopic level. Therefore, this approach seems to be more suitable for pedestrians, rather than for multilane traffic flow. Another generalisation of the AR model including the nonlocal interactions was recently studied in [12].
The existence of measurevalued solutions to the multidimensional AR system with offset function (1.6), called the dissipative AwRascle system (DAR), and their weakstrong 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 onedimensional 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:
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 onedimensional DAR system (1.1) with
As far as we know, extension of this lubrication model to the multidimensional case, in particular, the form of the stress and maximal constraint of the density, are not known, and it is not clear if the multidimensional DAR system is the right setting. Indeed, performing a similar formal calculation for the multidimensional 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 densitydependent shear viscosity and a lower order drift term:
where \( Q^\prime (\varrho ) = \varrho p^\prime (\varrho )\) and
We can easily check that this drift term is indeed of lower order:
and that it disappears for the onedimensional 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 ddimensional 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
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 globalintime admissible weak solutions to the isentropic Euler system. The fact that the convex integration technique works for system (1.10)–(1.12), which in onedimensional setting coincides with the compressible NavierStokes system (1.7), is therefore a very interesting observation. Turning to the multidimensional setting and its analogous formulation (1.9), we could conjecture that the convex integration technique provides illposedness in the class of weak solutions for certain viscous compressible models with degenerate viscosity coefficients possessing the “twovelocity structure”. Similar structure has been used in the past to prove the existence of solutions to compressible Navier–Stokes equations with densitydependent 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 initialfinal 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
together with
Note that the integral equalities in (2.1), (2.2) represent necessary compatibility conditions as the quantities
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 nontrivial 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
We claim the following result.
Theorem 2.3
( Ill posedness with respect to the data) Let \(d=2,3\). Suppose that
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
such that
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
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
hence
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
where
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
2.1.1 Density Profile
The next step is adjusting a suitable density profile,
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
where \(\Phi _0\), \(\Phi _T\) are the values of the acoustic potential determined by the Helmholtz decomposition of the initial data, and terminal data
respectively.
Consider the functions
The desired density profile can be taken as
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
2.2 Transformed Problem I
With \(\varrho \), \(\Phi \) fixed in the preceding part, the problem (1.10)–(1.12) reduces to
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
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
so that
Moreover, in accordance with (2.2), (2.5), we have
Consequently, equation (2.13) reduces to
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
where \({\varvec{U}}\) is the unique zero–mean solution of the elliptic problem
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
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
with \({\varvec{R}}\) solving
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
2.6 Adjusting the Energy
Finally, let us consider the energy associated to the system,
Introducing the notation
we may rewrite (2.24), (2.25) as an “abstract overdetermined (pressureless) Euler system”:
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):
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
The symbol \(\lambda _{\textrm{max}}[{\mathbb {A}}]\) denotes the maximal eigenvalue of a symmetric matrix \({\mathbb {A}}\). We recall the algebraic inequality
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
the obviously satisfies the initial–terminal conditions, \(\textrm{div}_x{\varvec{v}} = 0\), and
Since
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
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
Given the periodic boundary conditions, the total energy of smooth solutions is conserved,
Admissible weak solutions should satisfy at least the energy inequality
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
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
Keeping the notation of Sect. 2 we therefore obtain
while system (2.27)–(2.29) reduces to
Seeing that \({\varvec{v}} = \varrho {\varvec{u}}\) with \(\varrho \) independent of time, we have to fix \(\Lambda \) in (3.4) so that
We have
As \(\varrho = \varrho _0(x)\) is independent of t, we easily compute
where we have used
Finally, using the momentum equation (1.11) we compute
Consequently, we may fix \(\Lambda = \Lambda (t)\) in such a way that
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,
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
such that
We have shown the following result.
Theorem 3.2
(Ill posedness in the class of admissible solutions) Let \(d=2,3\). Suppose that
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
satisfying
together with the energy inequality
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
where \(\Omega \subset R^d\) is a bounded domain with smooth boundary. Accordingly, the weak formulation of the equation (1.11) reads
for any test function \(\varvec{\varphi }\) in the class
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,
where \(D_N\) is the DirichlettoNeumann operator.
To accommodate (3.11), we consider a weaker formulation of (1.11), namely
for any test function \(\varvec{\varphi }\) in the class
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.
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Funding
The work of E.F. was partially supported by the Czech Sciences Foundation (GAČR), Grant Agreement 21–02411 S. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840. The work of N.C. and E.Z. was supported by the EPSRC Early Career Fellowship no. EP/V000586/1.
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Chaudhuri, N., Feireisl, E. & Zatorska, E. Nonuniqueness of Weak Solutions to the Dissipative Aw–Rascle Model. Appl Math Optim 90, 19 (2024). https://doi.org/10.1007/s0024502410158x
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DOI: https://doi.org/10.1007/s0024502410158x