A Space-time Nonlocal Traffic Flow Model: Relaxation Representation and Local Limit^††thanks: This rsearch is supported in part by US NSF DMS-1937254, DMS-2012562, and CNS-2038984.

We consider the following nonlocal conservation law modeling traffic flow

	$\displaystyle\partial_{t}\rho(t,x)+\partial_{x}(\rho(t,x)v(q(t,x)))=0,\quad x\in\mathbb{R},\,t>0,$		(1.1)
	$\displaystyle\mbox{where}\qquad q(t,x)=\int_{0}^{\infty}\rho(t-\gamma s,x+s)w(s)\,ds.$		(1.2)

Here, the quantity $\rho(t,x)\in[0,1]$ represents the traffic density, where $\rho=0$ indicates an empty road ahead and $\rho=1$ models bumper-to-bumper traffic jam. The nonlocal quantity $q(t,x)$ is a weighted average of $\rho(t^{\ast},x^{\ast})$ along the space-time path

$$t^{\ast}=t-\gamma s,\quad x^{\ast}=x+s,\qquad\mbox{for}~{}s\in[0,\infty),$$

with an averaging kernel $w=w(s)$. The vehicle velocity $v=v(q(t,x))$ depends on the nonlocal traffic density $q(t,x)$ through a decreasing function $v(\cdot)$. The model (1.1)-(1.2) is the evolution associated to a past-time condition $\rho(t,x)$ given on the half plane $t\leq 0$.

The model (1.1)-(1.2) takes inspiration from the classical Lighthill-Whitham-Richards (LWR) model [35, 36]

$\displaystyle\partial_{t}\rho(t,x)+\partial_{x}(\rho(t,x)v(\rho(t,x)))=0,\quad x\in\mathbb{R},\,t>0,$

(1.3)

in which the vehicle velocity $v=v(\rho(t,x))$ depends only on the local traffic density $\rho(t,x)$. The LWR model (1.3) is a scalar conservation law with the flux function $f(\rho)\doteq\rho v(\rho)$. In the instance of inter-vehicle communication [18], the flux may have a nonlocal dependence on traffic density in order to capture each vehicle’s reaction to downstream traffic conditions. It is useful to incorporate time delays of this traffic density information in the distance [40, 31]. In (1.1)-(1.2), we incorporate both nonlocal fluxes and time delays via velocities that depend on a weighted space-time average of the traffic density, assuming that the traffic density information travels at a constant speed $\gamma^{-1}$.

If the choice of rescaled weights $w_{\varepsilon}(s)=\varepsilon^{-1}w(s/\varepsilon)$ is made, then formally the equations (1.1)-(1.2) converge to the local equation (1.3) as $\varepsilon\to 0$. The main goal of this paper is to demonstrate this in a rigorous manner via convergence of solutions.

There has recently been much research interest in nonlocal effects in phenomena described by conversation laws; there is a wide variety of applications but a dearth of analytical understanding. Some application areas from which nonlocal conservation laws arise are traffic flows [34, 33, 8, 30, 29, 10, 9, 11], sedimentation [1], pedestrian traffic [16, 5], material flow on conveyor belts [27, 38], and the numerical approximation of local conservation laws [21, 20, 19, 23].

For several traffic flow models, the nonlocal mechanism is introduced in the flux term. One such model that recovers the LWR model (1.3) when the effect is localized was proposed in [2, 26]:

	$\displaystyle\partial_{t}\rho(t,x)+\partial_{x}(\rho(t,x)v(q(t,x)))=0,\quad x\in\mathbb{R},\,t>0,$		(1.4)
	$\displaystyle\mbox{where}\qquad q(t,x)=\int_{0}^{\infty}\rho(t,x+s)w(s)\,ds.$		(1.5)

Various analytical aspects of this model have been investigated, including the existence and uniqueness of solutions [2, 26, 4], existence and stability of traveling wave solutions [37, 39], development of numerical schemes [26, 6, 25], and stability analysis of the model in the case where the domain (road) is a closed ring [28]. Convergence of solutions of (1.4)-(1.5) to its local limit (which is the LWR model (1.3)) was established in [4, 3] by way of an a priori BV estimate and an entropy estimate, both of which were obtained via reformulation of the nonlocal model as a $2\times 2$ relaxation system in the case of exponential weight kernels. This is not the only mechanism that has been used to investigate the nonlocal-to-local limit; see the works of [13, 32, 12, 14, 15, 24].

We conduct an analogous study of the nonlocal-to-local limit for the model (1.1)-(1.2) with suitable choices of the functions $w,v$ and the past-time condition. To fix ideas, we make the following assumptions on $w,v$:

The average density $q$ is taken along a space-time curve that requires traffic density data for all past times $t\leq 0$. Therefore, the model (1.1)-(1.2) shall be equipped with a past-time condition on the lower half-plane, i.e.,

$\displaystyle\rho(t,x)=\rho_{-}(t,x),\quad(t,x)\in(-\infty,0]\times\mathbb{R},$

(1.7)

where $\rho_{-}\in\mathbf{L}^{\infty}((-\infty,0]\times\mathbb{R})$ is a given function.

Our first main result is the existence of Lipschitz solutions to the past-time value problem (1.1)-(1.2)-(1.7) with Lipschitz past-time data $\rho_{-}$.

Formally, as the time-delay parameter $\gamma$ approaches zero, the system (1.1)-(1.2) approaches the nonlocal-in-space system (1.4)-(1.5). This is also true in a qualitative sense; each of the key estimates for (1.1)-(1.2) remain valid as $\gamma\to 0$, as the bounding constants neither vanish nor blow up. Analogous statements of all of our results hold for (1.4)-(1.5), see [4], and can be formally recovered from our results by taking $\gamma\to 0$. However, quantitatively stronger results hold for (1.4)-(1.5). For example, the main estimates in Proposition 3.1 and Theorem 1.3 concern $\mathbf{L}^{1}$ estimates in space and time, whereas the analogous results for (1.4)-(1.5) hold for $\mathbf{L}^{1}$ in space and $\mathbf{C}^{0}$ in time; see again [4].

The proof of Theorem 1.1 makes up Section 2. We use a fixed point argument combined with the method of characteristics, which is heavily inspired by the proof of [4] for the existence of solutions to the nonlocal-in-space model.

In certain modelling applications, it might only be possible to gather the traffic data at a certain initial time. In such a case, a natural choice of past-time data via the following extension of initial data:

$\displaystyle\rho_{-}(t,x)$

$\displaystyle=\rho_{0}(x),\qquad(t,x)\in(-\infty,0]\times\mathbb{R},$

(1.10)

for a given function $\rho_{0}:\mathbb{R}\to\mathbb{R}$. We can then treat (1.1)-(1.2)-(1.7) as an initial-value problem, since the quantity $q$ depends only on $t\in(0,\infty)$. To be precise, with (1.10), the equation (1.2) becomes

$$q(t,x)=\int_{0}^{\infty}\rho_{0}\Big{(}x+\frac{t}{\gamma}+s\Big{)}w\Big{(}s+\frac{t}{\gamma}\Big{)}ds+\int_{0}^{\frac{t}{\gamma}}\rho(t-\gamma s,x+s)w(s)ds.$$

(1.11)

Under this consideration, the equations (1.1)-(1.2)-(1.7) where the past-time data is given by the equation (1.10) are equivalent to the Cauchy problem (1.1)-(1.11) with the initial condition $\rho(0,x)=\rho_{0}(x)$.

For the particular choice (1.10) of past-time data, we establish the well-posedness of the Cauchy problem in the setting of weak solutions, which is our second main result.

The key tool used to prove Theorem 1.2 is the $\mathbf{L}^{1}$-stability of Lipschitz solutions; once that is established in Proposition 3.1, the existence and uniqueness of weak solutions follows by using Theorem 1.1 and an approximation argument.

With the well-posedness of the problem (1.1)-(1.11) in hand, we analyze the nonlocal-to-local limit. This limit is realized in the following way: consider the rescaled kernels $w_{\varepsilon}(s)=\varepsilon^{-1}w(s/\varepsilon)$. Taking $\varepsilon\to 0$, the kernel $w_{\varepsilon}$ converges to a Dirac delta function, and so – formally – solutions of the nonlocal model (1.1)-(1.11) converge to the entropy admissible solution of the local model (1.3). We make the choice of exponential kernel function for $w$, defined as

$\displaystyle w(s)=e^{-s},\qquad w_{\varepsilon}(s)=\varepsilon^{-1}w(s/\varepsilon)=\varepsilon^{-1}e^{-s/\varepsilon},\quad s\in[0,\infty).$

(1.14)

With $w$ defined as in (1.14), the model (1.1)-(1.11) (and more generally (1.1)-(1.2)) can be rewritten as a relaxation system:

	$\displaystyle\partial_{t}\rho+\partial_{x}(\rho v(q))$	$\displaystyle=0,$		(1.15)
	$\displaystyle\partial_{t}q-\gamma^{-1}\partial_{x}q$	$\displaystyle=(\gamma\varepsilon)^{-1}(\rho-q).$		(1.16)

Utilizing the special features of this relaxation system formulation (1.15)-(1.16), a uniform global BV bound on $\rho$ that is independent of the relaxation parameter $\varepsilon$ can be proved, which serves as a key estimate for the compactness theory and guarantees the existence of a limit of the solutions.

The choice (1.14) is the same as the one made in [4] to analyze the nonlocal-to-local limit for the nonlocal-in-space model (1.4)-(1.5). Our methods closely follow theirs, but the relaxation system (1.15)-(1.16) is a genuine system of conservation laws in the original $(t,x)$-coordinate system, and we additionally take this into account. We remark that, in the case of $\gamma=0$, the condition (1.8) holds whenever $w^{\prime}(s)\leq 0~{}\forall s\in[0,+\infty)$, and (1.17) becomes $\min_{\rho\in[0,1]}|v^{\prime}(\rho)|\geq\left\lVert v^{\prime\prime}\right\rVert_{\infty}$. These conditions on the functions $w,v$ are the same as the ones proposed in [4] for the nonlocal-in-space model (1.4)-(1.5).

Finally, we show that any limit solution of the space-time nonlocal model (1.1)-(1.11) when $\varepsilon\to 0$ is the unique weak entropy solution of (1.3).

This paper is organized as follows. First, we establish the existence of Lipschitz solutions from Lipschitz past-time data in Section 2 (Theorem 1.1). In Section 3 we establish the $\mathbf{L}^{1}$ stability estimate for Lipschitz solutions and prove Theorem 1.2. Section 4 is devoted to the uniform BV bound estimate of solutions based on the model’s relaxation system formulation (Theorem 1.3), which guarantees the existence of local limit solutions. Section 5 provides the proof of entropy admissibility of the local limit solution and completes the nonlocal-to-local limit theorem (Theorem 1.4).

To begin, we make precise the conditions on the past-time data. First, the initial values of $\rho$ and $q$ corresponding to a past-time condition $\rho_{-}$ are denoted throughout the paper as

$$\rho_{0}(x)\doteq\rho_{-}(0,x),\quad q_{0}(x)\doteq\int_{0}^{\infty}\rho_{-}(-\gamma s,x+s)w(s)\,ds,\qquad x\in\mathbb{R}.$$

(2.1)

Second, for a given constant $L>0$ we introduce the following notation for a class of functions for past-time data $\rho_{-}$ with $\rho_{0}$ and $q_{0}$ given by (2.1) correspondingly.

$$\begin{split}&\mathcal{X}_{\mathrm{Lip},L}\\ &\doteq\left\{\begin{gathered}\rho_{-}\in\mathbf{L}^{\infty}((-\infty,0]\times\mathbb{R}):\,\inf_{(t,x)\in(-\infty,0]\times\mathbb{R}}\rho_{-}(t,x)>0,\;\sup_{(t,x)\in(-\infty,0]\times\mathbb{R}}\rho_{-}(t,x)<1,\\ \qquad\inf_{x\in\mathbb{R}}\rho_{0}(x)(1+\gamma v(q_{0}(x)))>0,\quad\sup_{x\in\mathbb{R}}\rho_{0}(x)(1+\gamma v(q_{0}(x)))<1,\\ \qquad\sup_{(t,x)\in(-\infty,0]\times\mathbb{R}}|(\partial_{x}-\gamma\partial_{t})\rho_{-}(t,x)|\leq L,\quad\sup_{x\in\mathbb{R}}|\partial_{x}(\rho_{0}(x)(1+\gamma v(q_{0}(x))))|\leq L\end{gathered}\right\}.\end{split}$$

(2.2)

Now we define

$$g(\rho)\doteq\rho(1+\gamma v(\rho)),\quad\rho\in[0,1].$$

(2.3)

Under the Assumption 1, we have $g(0)=0$ and $g(1)=1$. Moreover, it holds that $g^{\prime}(\rho)>0$ for $\rho\in[0,1]$ provided $\gamma\left\lVert v^{\prime}\right\rVert_{\infty}<1$. In this case the function $g$ is monotone and we let $g^{-1}$ denote the inverse function of $g$. We define

	$\displaystyle\rho_{\mathrm{min}}\doteq\min\left\{\inf_{(t,x)\in(-\infty,0]\times\mathbb{R}}\rho_{-}(t,x),~{}g^{-1}\left(\inf_{x\in\mathbb{R}}\rho_{0}(x)(1+\gamma v(q_{0}(x))\right)\right\},$		(2.4)
	$\displaystyle\rho_{\mathrm{max}}\doteq\max\left\{\sup_{(t,x)\in(-\infty,0]\times\mathbb{R}}\rho_{-}(t,x),~{}g^{-1}\left(\sup_{x\in\mathbb{R}}\rho_{0}(x)(1+\gamma v(q_{0}(x))\right)\right\},$		(2.5)

where $\rho_{0}$ and $q_{0}$ are defined in (2.1). It is clear that $0<\rho_{\mathrm{min}}\leq\rho_{\mathrm{max}}<1$ for any $\rho_{-}\in\mathcal{X}_{\mathrm{Lip},L}$.

The essential idea in the proof of Theorem 1.1 is to reformulate the model as a fixed point problem and apply the contraction mapping theorem. We first define the fixed point mapping on a proper domain with a finite time horizon, and then show it is contractive through a priori $\mathbf{L}^{\infty}$ and Lipschitz estimates. The fixed point solution is shown to be a Lipschitz solution to the model and it can be extended to all times $t>0$.

First let us fix a time horizon $[0,T]$ where $T>0$, and suppose $\rho_{\mathrm{min}},\rho_{\mathrm{max}}$, are as defined in (2.4)-(2.5). For any $0\leq\rho_{a}<\rho_{\mathrm{min}}$ and $\rho_{\mathrm{max}}<\rho_{b}\leq 1$, we define the domain

$$\mathcal{D}_{T,L,\rho_{a},\rho_{b}}\doteq\left\{\rho\in\mathbf{L}^{\infty}([0,T]\times\mathbb{R})\,:\,\begin{gathered}\rho_{a}\leq\rho(t,x)\leq\rho_{b},\;(t,x)\in[0,T]\times\mathbb{R},\\ |(\partial_{x}-\gamma\partial_{t})\rho(t,x)|\leq 3L,\;(t,x)\in(0,T)\times\mathbb{R},\\ \rho(0,x)=\rho_{0}(x),\;x\in\mathbb{R}\end{gathered}\right\}.$$

Then we introduce a directional derivative operator

$$\partial_{y}\doteq\partial_{x}-\gamma\partial_{t},$$

where the direction is taken along the line integral paths in (1.2), and an auxiliary variable

$$z\doteq\rho(1+\gamma v(q)).$$

With the above definitions, the past-time value problem (1.1)-(1.2)-(1.7) can be reformulated as a system to be solved on $[0,T]\times\mathbb{R}$.

	$\displaystyle q(t,x)=\int_{0}^{t/\gamma}\rho(t-\gamma s,x+s)w(s)\,ds+\int_{t/\gamma}^{\infty}\rho_{-}(t-\gamma s,x+s)w(s)\,ds,$		(2.6)
	$\displaystyle z(t,x)=\rho(t,x)(1+\gamma v(q(t,x))),$		(2.7)
	$\displaystyle\partial_{t}z(t,x)+\partial_{y}\left(\frac{v(q(t,x))}{1+\gamma v(q(t,x))}z(t,x)\right)=0.$		(2.8)

This representation motivates the following step-by-step definition of a mapping $\Gamma:\,\mathcal{D}_{T,L,\rho_{a},\rho_{b}}\to\mathbf{L}^{\infty}([0,T]\times\mathbb{R})$.

1. 1.

   With a given $\rho_{-}\in\mathcal{X}_{\mathrm{Lip},L}$ and for any $\rho\in\mathcal{D}_{T,L,\rho_{a},\rho_{b}}$, we define $q(t,x;\rho,\rho_{-})$ for all $(t,x)\in[0,T]\times\mathbb{R}$ as in (2.6).

2. 2.

   We define $z(t,x;\rho,\rho_{-})$ for all $(t,x)\in[0,T]\times\mathbb{R}$ as the solution to the linear Cauchy problem (2.8) with the above $q(t,x;\rho,\rho_{-})$ and the initial condition

   $$z(0,x;\rho_{-})=\rho_{0}(x)(1+\gamma v(q_{0}(x))).$$

3. 3.

   With $z(t,x;\rho,\rho_{-})$ and $q(t,x;\rho,\rho_{-})$ defined above, we define $\tilde{\rho}(t,x;\rho,\rho_{-})$ as

   $\displaystyle\tilde{\rho}(t,x;\rho,\rho_{-})=\frac{z(t,x;\rho,\rho_{-})}{1+\gamma v(q(t,x;\rho,\rho_{-}))},\quad(t,x)\in[0,T]\times\mathbb{R}.$

Finally we define the mapping $\Gamma$ by

$$\Gamma[\rho](t,x)\doteq\tilde{\rho}(t,x;\rho,\rho_{-}),\quad(t,x)\in[0,T]\times\mathbb{R}.$$

The outline of the proof of Theorem 1.1 is to establish the following facts:

• •

  For any $\rho\in\mathcal{D}_{T,L,\rho_{a},\rho_{b}}$, $\Gamma[\rho]\in\mathcal{D}_{T,L,\rho_{a},\rho_{b}}$;

• •

  $\Gamma$ is a contraction mapping on $\mathcal{D}_{T,L,\rho_{a},\rho_{b}}$ in the $\mathbf{L}^{\infty}$ norm;

• •

  The contraction mapping theorem gives the unique fixed point $\rho\in\mathcal{D}_{T,L,\rho_{a},\rho_{b}}$, i.e. $\Gamma[\rho]=\rho$;

• •

  The fixed point solution is Lipschitz and it solves the system (2.6)-(2.7)-(2.8) for $t\in[0,T]$;

• •

  By continuation, the constructed solution for $t\in[0,T]$ can be extended to $t\in[0,\infty)$ and so it solves the past-time value problem (1.1)-(1.2)-(1.7).

We remark here that the map $\Gamma$ as constructed requires no relation between $\rho$ and $\rho_{-}$ at $t=0$ to hold. However, the condition $\rho(0,x)=\rho_{0}(x)$ is imposed so that quantities such as $q(t,x)$ are Lipschitz with appropriate constant.

The proof consists of six steps. In the proof, we omit the notations $\rho$ and $\rho_{-}$ in $q(t,x;\rho,\rho_{-})$, $z(t,x;\rho,\rho_{-})$ and $\tilde{\rho}(t,x;\rho,\rho_{-})$ for simplicity, but keep in mind that they both depend on $\rho$ and $\rho_{-}$. In addition, we use the equation (1.2) for $q$ to simplify the calculation, but keep in mind that the nonlocal integral for $q$ involves $\rho_{-}$ and its precise form is (2.6).

To give a $\mathbf{L}^{\infty}$ bound on $\tilde{\rho}$, we note that

$$g(\rho_{a})<g(\rho_{\mathrm{min}})\leq z(0,x)\leq g(\rho_{\mathrm{max}})<g(\rho_{b}),\quad x\in\mathbb{R}.$$

By integrating (2.3) and using the uniform bound

$\displaystyle\left\lVert\frac{-v^{\prime}(q)}{(1+\gamma v(q))^{2}}\partial_{y}q\right\rVert_{\infty}$

$\displaystyle\leq 3\left\lVert v^{\prime}\right\rVert_{\infty}L,$

we obtain that

$\displaystyle g(\rho_{a})\leq z(t,x)\leq g(\rho_{b}),\quad(t,x)\in[0,T]\times\mathbb{R},$

when $T$ is sufficiently small. This together with (2.11) gives

$\displaystyle\rho_{a}\leq\tilde{\rho}(t,x)\leq\rho_{b},\quad(t,x)\in[0,T]\times\mathbb{R}.$

(2.14)

Now let us give a bound on $\partial_{y}\tilde{\rho}$. Taking the directional derivative $\partial_{y}$ of the equation (2.9), we obtain

	$\displaystyle\partial_{t}(\partial_{y}z)+$	$\displaystyle\frac{v(q)}{1+\gamma v(q)}\partial_{y}(\partial_{y}z)=\partial_{y}z\frac{-2v^{\prime}(q)}{(1+\gamma v(q))^{2}}\partial_{y}q$
		$\displaystyle+z\left[\frac{2\gamma(v^{\prime}(q))^{2}-v^{\prime\prime}(q)(1+\gamma v(q))}{(1+\gamma v(q))^{3}}(\partial_{y}q)^{2}+\frac{-v^{\prime}(q)}{(1+\gamma v(q))^{2}}\partial_{yy}q\right].$		(2.15)

At time $t=0$, by the equation (2.9) we write

$\displaystyle\partial_{y}z(0,x)=(1+\gamma v(q(0,x)))\partial_{x}z(0,x)+z(0,x)\frac{\gamma v^{\prime}(q(0,x))}{1+\gamma v(q(0,x))}\partial_{y}q(0,x),$

using that $\rho_{-}\in\mathcal{X}_{\mathrm{Lip},L}$ we have

$\displaystyle|\partial_{y}z(0,x)|\leq\left(1+\gamma v_{\mathrm{max}}+\gamma\left\lVert v^{\prime}\right\rVert_{\infty}\right)L\leq\frac{4}{3}L,\quad x\in\mathbb{R}.$

We integrate the equation (2.3) along the characteristic curves defined in (2.10). With the uniform bounds

	$\displaystyle\left\lVert\frac{-2v^{\prime}(q)}{(1+\gamma v(q))^{2}}\partial_{y}q\right\rVert_{\infty}$	$\displaystyle\leq 6\left\lVert v^{\prime}\right\rVert_{\infty}L,$
	$\displaystyle\left\lVert z\frac{2\gamma(v^{\prime}(q))^{2}-v^{\prime\prime}(q)(1+\gamma v(q))}{(1+\gamma v(q))^{3}}(\partial_{y}q)^{2}\right\rVert_{\infty}$	$\displaystyle\leq\left(2\gamma\left\lVert v^{\prime}\right\rVert_{\infty}^{2}+\left\lVert v^{\prime\prime}\right\rVert_{\infty}\right)9L^{2}\cdot g(\rho_{b}),$
	$\displaystyle\left\lVert z\frac{-v^{\prime}(q)}{(1+\gamma v(q))^{2}}\partial_{yy}q\right\rVert_{\infty}$	$\displaystyle\leq 6w(0)\left\lVert v^{\prime}\right\rVert_{\infty}L\cdot g(\rho_{b}),$

we deduce from a comparison argument that

$\displaystyle\sup\nolimits_{x\in\mathbb{R}}|\partial_{y}z(t,x)|\leq Z(t),\quad t\in[0,T],$

(2.16)

where $Z(t)$ is the solution to the linear ODE

$\displaystyle\dot{Z}=aZ+b,\quad Z(0)=\frac{4}{3}L,$

(2.17)

with constant coefficients given by

$\displaystyle a\doteq\frac{6\left\lVert v^{\prime}\right\rVert_{\infty}L}{1-\gamma v_{\mathrm{max}}},\qquad b\doteq\frac{9\left(2\gamma\left\lVert v^{\prime}\right\rVert_{\infty}^{2}+\left\lVert v^{\prime\prime}\right\rVert_{\infty}\right)L^{2}+6w(0)\left\lVert v^{\prime}\right\rVert_{\infty}L}{1-\gamma v_{\mathrm{max}}}.$

By choosing $T$ sufficiently small, we obtain that $|\partial_{y}z(t,x)|\leq Z(T)\leq 2L$ for $(t,x)\in[0,T]\times\mathbb{R}$. Then the identity

	$\displaystyle\partial_{y}\tilde{\rho}(t,x)$
	$\displaystyle=\frac{(1+\gamma v(q(t,x)))\partial_{y}z(t,x)-\gamma z(t,x)v^{\prime}(q(t,x))\partial_{y}q(t,x)}{(1+\gamma v(q(t,x)))^{2}},\quad(t,x)\in[0,T]\times\mathbb{R},$

implies that

$\displaystyle|\partial_{y}\tilde{\rho}(t,x)|\leq 2L+3\gamma\left\lVert v^{\prime}\right\rVert_{\infty}L\leq 3L,\quad(t,x)\in[0,T]\times\mathbb{R}.$

(2.18)

The equality $\tilde{\rho}(0,x)=\rho_{0}(x)$ is clear from the definition. Using the obtained $\mathbf{L}^{\infty}$ and directional Lipschitz bounds (2.14)-(2.18), we conclude that there exist $L,T>0$, depending only on $\gamma,v,L_{0},\rho_{\mathrm{min}},\rho_{\mathrm{max}}$, such that $\Gamma$ maps $\mathcal{D}_{T,L,\rho_{a},\rho_{b}}$ to itself.