On critical thresholds for hyperbolic balance law systems

TL;DR

This paper reviews the development of critical threshold phenomena in hyperbolic balance law systems, focusing on Euler-Poisson-alignment and hyperbolic relaxation systems, highlighting their mathematical and modeling significance.

Contribution

It provides a comprehensive review of the theoretical progress on critical thresholds in specific hyperbolic PDE systems, emphasizing nonlocal models.

Findings

Analysis of critical thresholds in Euler-Poisson-alignment systems

Insights into hyperbolic relaxation systems behavior

Importance of nonlocal PDE systems in modeling

Abstract

We review the theoretical development in the study of critical thresholds for hyperbolic balance laws. The emphasis is on two classes of systems: Euler-Poisson-alignment (EPA) systems and hyperbolic relaxation systems. We start with an introduction to the `Critical Threshold Phenomena' and study some nonlocal PDE systems, which are important from modeling point of view.

Equations

$$\begin{array}{rl}& {\rho }_t+(\rho u{)}_x=0,\\ & u_t+u{u}_x=-{\varphi }_x,\\ & -{\varphi }_{xx}=\rho ,\end{array}$$

$$\frac{d(\rho (t,X))}{dt}=-\rho g,$$

$$\frac{d(g(t,X))}{dt}=-g^2+\rho .$$

$$\frac{d}{dt}(g/\rho )=1,\frac{d}{dt}(1/\rho )=\frac{g}{\rho }.$$

$$\frac{1}{\rho (t,X(t))}=\frac{t^2}{2}+t\frac{g(0)}{\rho (0)}+\frac{1}{\rho (0)}.$$

$${\rho }_t+(\rho u{)}_x=0,$$

$$u_t+u{u}_x=-k{\varphi }_x+{\int }_{\mathbb{R}}\psi (x-y)(u(t,y)-u(t,x))\rho (t,y)dy,$$

$$-{\varphi }_{xx}=\rho -c,$$

$$c={\int }_{-\frac{1}{2}}^{\frac{1}{2}}{\rho }_0(y)dy.$$

$$0\le {\psi }_{min}\le \psi \le {\psi }_{max}.$$

$$G_t+(Gu{)}_x=k(\rho -c),$$

$${\rho }_t+(\rho u{)}_x=0,$$

$${\psi }_{max}-{\psi }_{min}<\frac{e^{\frac{{\tan }^{-1}\hat{z}}{\hat{z}}}\left(1-e^{-\frac{\pi }{\tilde{z}}-\frac{\pi }{\hat{z}}}\right)}{2\left(1+e^{-\frac{\pi }{\tilde{z}}}\right)}\lambda ,$$

$$\big{(}G_{0}(x),\rho_{0}(x)\big{)}\in\Sigma_{1},\quad\forall\,x\in\mathbb{T},$$

$$\big{(}G_{0}(x),\rho_{0}(x)\big{)}\in\Sigma_{2},\quad\forall\,x\in\mathbb{T},$$

$${\psi }_{max}-{\psi }_{min}<\frac{e^{\frac{{\tan }^{-1}\hat{z}}{\hat{z}}}}{2\left(1+e^{-\frac{\pi }{\tilde{z}}}\right)}\lambda ,$$

$$\big{(}G_{0}(x),\rho_{0}(x)\big{)}\in\Sigma_{3},\quad\forall\,x\in\mathbb{T},$$

$$\hat{z}:=\sqrt{{\left(\frac{\lambda }{{\psi }_{max}}\right)}^2-1}\text{and}\tilde{z}:=\sqrt{{\left(\frac{\lambda }{{\psi }_{min}}\right)}^2-1}.$$

$$G^{\prime }=-G(G-\psi \ast \rho )+k(\rho -c),$$

$${\rho }^{\prime }=-\rho (G-\psi \ast \rho ).$$

$$4(\Vert \psi {\Vert }_{L^1}-2\gamma )<\frac{e^{\frac{{\tan }^{-1}\hat{z}}{\hat{z}}}\left(1-e^{-\frac{\pi }{\tilde{z}}-\frac{\pi }{\hat{z}}}\right)}{2\left(1+e^{-\frac{\pi }{\tilde{z}}}\right)}\lambda ,$$

$$\big{(}G_{0}(x),\rho_{0}(x)\big{)}\in\Sigma_{L}^{1},\quad\forall\,x\in\mathbb{T},$$

$$\big{(}G_{0}(x),\rho_{0}(x)\big{)}\in\Sigma_{L}^{2},\quad\forall\,x\in\mathbb{T},$$

$$4(||\psi |{|}_{L^1}-2\gamma )<\frac{e^{\frac{{\tan }^{-1}\hat{z}}{\hat{z}}}}{2\left(1+e^{-\frac{\pi }{\tilde{z}}}\right)}\lambda ,$$

$$\big{(}G_{0}(x),\rho_{0}(x)\big{)}\in\Sigma_{L}^{3},\quad\forall\,x\in\mathbb{T},$$

$$\hat{z}:=\sqrt{{\left(\frac{\lambda }{2(\Vert \psi {\Vert }_{L^1}-\gamma )}\right)}^2-1}\text{and}\tilde{z}:=\sqrt{{\left(\frac{\lambda }{2\gamma }\right)}^2-1}.$$

$$\underset{x}{\inf }G(0,x)>0,$$

$${\rho }_t+(\rho \text{v}{)}_x=0,$$

$${\rho }_t+(\rho \text{v}{)}_x=0,x\in \mathbb{R},t>0,$$

$$u_t+u{u}_x=\rho (\text{v}-u),$$

$$\begin{array}{rl}& Q\in W^{1,1}(\mathbb{R}),Q(x)=Q(-x)\text{ (symmetric)},\\ & {\int }_{\mathbb{R}}Q(x)dx=1,\text{and }Q\text{ is decreasing away from origin}.\end{array}$$

$${\rho }_x,u_x\in C([0,\infty );H^s(\mathbb{R})),s\ge 1$$

$$\rho (t,x)\le \frac{\sup {\rho }_0|f(u_0)-u_0|}{e^{{\int }_{u_0(x)}^{u(t,x)}\frac{d\xi }{f(\xi )-\xi }}|f(u(t,x))-u(t,x)|}.$$

Full text

1. Introduction

For first order hyperbolic conservation laws, it is generic that the solutions lose smoothness even if the initial data is smooth, [8, 14]. However, addition of source terms can balance this ‘breaking’ and result in global-in-time smooth solutions for a large class of initial data. For the question of global behavior of strong solutions, the choice of the initial data and/or damping forces is decisive. The classical stability analysis can fail either for large perturbations of some wave patterns or when the steady state solution may be only conditionally stable due to the weak dissipation in the system, see for example [11, 21]. On the other hand, the notion of critical threshold (CT) has been shown to be powerful in describing the conditional stability for underlying physical problems, and the associated phenomena does reflect the delicate balance among various forcing mechanisms, [1, 2, 7, 9].

An example to illustrate this is the pressureless Euler-Poisson (EP) system that consists of the continuity equation, Burgers’ equation with an electric source through a potential, and the Poisson equation for the potential,

[TABLE]

with smooth initial data $(\rho_{0}\geq 0,u_{0})\in C^{1}(\mathbb{R})\times C^{1}(\mathbb{R})$. Taking the spatial derivative of the second equation and setting $g(t,X):=u_{x}(t,X)$ for $\frac{dX}{dt}=u(t,X)$, we can obtain an ODE system along the characteristics,

[TABLE]

We have omitted the parameter for the ODE, that is a consequence of the method of characteristics, to avoid excess notation. For global well-posedness of (1.1), the issue now has reduced to ensuring the all-time-boundedness of $\rho,g$ as solutions of the above ODE system. If we assume the initial density to be nowhere zero, then the first ODE implies that the density remains positive for all times. Using simple calculations, one has,

[TABLE]

Hence,

[TABLE]

From this expression, we can conclude that $\rho(t)$ is finite for all time if and only if $g(0)\geq-\sqrt{2\rho(0)}$. Therefore, for (1.1), there is global classical solution if and only if for all $x\in\mathbb{R}$, we have $(u_{0x}(x),\rho_{0}(x))\in\{(g,\rho):g\geq-\sqrt{2\rho},\rho\geq 0\}$. The question of global-in-time existence vs finite-time-breakdown has now been reduced to whether the initial data crosses a certain threshold, which here, is the curve, $\{(g,\rho):g=-\sqrt{2\rho},\rho\geq 0\}$ (on the phase space of $(u_{0x},\rho_{0})$). This is the CT curve. It divides the initial data into two sets: the subcritical region (the good part) and the supercritical region (the bad part).

The stimulating interplay of old and new ideas continue to lead to increased understanding as well as an ever-larger set of techniques with CT analysis for a larger array of problems. The results on CT phenomena reviewed here are obtained in a series of papers [3, 4, 5, 6], which provide an account on recent developments of CT theory for a class of hyperbolic balance laws. The corresponding methods in obtaining these results will be briefly illustrated.

2. Euler-Poisson alignment dynamics

The Euler-Poisson-alignment (EPA) system models phenomena where matter can be regarded as consisting of a continuum of moving particles, such as flow of charge or flocking of birds, see for example [9, 20]. It consists of the following system of equations,

[TABLE]

where $x\in\mathbb{T}$ (periodic torus) or $x\in\mathbb{R}$, and $t>0$. The initial data is $\rho_{0}(x),u_{0}(x)$. Here, $k$ is the forcing coefficient, the sign of which signifies the type of forcing between particles. $c$ is the background term. The equation (2.1b) has two types of forces: the alignment force modeled through the influence function $\psi$, and the electric force modeled through the potential $\phi$. The alignment force affects the collective motion of particles as to how they react to other particles around them. $\psi$ is nonnegative and symmetric. It models the strength and extent of the interaction.

$\phi$ is the potential that models electric or gravitational forces as and according when $k>0$ or $k<0$ respectively. The system (2.1), with $\psi\equiv 0$ is called the Euler-Poisson (EP) system (illustrated in Section 1) and has been a topic of intensive study by various researchers due to its ability to model numerous physical phenomena, [12, 17, 18].

When $k=0$ in (2.1), it is called the Euler-alignment (EA) system. EA systems have been analyzed with different types of $\psi$. In [19], the author obtained global existence results for $\psi\in L^{1}$. In [6], we relaxed the hypothesis and used a different, more elementary technique to arrive at the result. The gist of the result is mentioned in Theorem 2.3.

EPA systems with background ($c>0$) have to be studied on a periodic spatial domain and not on $\mathbb{R}$. This is owing to assumptions required for local well-posedness. Therefore, we let the spatial variable space be $\mathbb{T}=[-1/2,1/2)$, the periodic torus. Local existence requires the assumption: $\int_{-1/2}^{1/2}\rho(t,y)-c\,dy=0$. This equality holds for all time if it holds initially. This is because mass is conserved by (2.1a). In view of this, we set,

[TABLE]

Our first set of results is for $\psi\in L^{\infty}(\mathbb{T})$ having,

[TABLE]

With the above assumption, (2.1) can be reformulated into a simpler system. Setting $G:=u_{x}+\psi\ast\rho$ with $G_{0}(x):=u_{0x}(x)+(\psi\ast\rho_{0})(x)$, we obtain the following.

[TABLE]

with initial data $(G_{0},\rho_{0})\in H^{s}(\mathbb{T})\times(H^{s}(\mathbb{T})\cap L^{1}_{+}(\mathbb{T}))$, for $s>1/2$. The local existence for such system is known, [7]. In particular, if initial data is smooth, then a smooth solution exists for some finite time.

Theorem 2.1** (Bounded alignment force).**

Consider (2.3) with repulsive electric force $k>0$ and bounded alignment influence $\psi$ satisfying (2.2). Set $\lambda:=2\sqrt{\frac{k}{c}}$. Suppose the initial data $(G_{0},\rho_{0})$ is smooth and lies in the space mentioned above. Then there exist sets $\Sigma_{1},\Sigma_{2},\Sigma_{3}$ such that,

1. (1)

Weak alignment* ($\psi_{max}<\lambda$): under the admissible condition*

[TABLE]

if the initial data lie in the subcritical region $\Sigma_{1}$, namely

[TABLE]

then (2.3) admits global-in-time classical solutions. 2. (2)

Strong alignment* ($\psi_{min}\geq\lambda$): if the initial data lie in the subcritical region $\Sigma_{2}$, namely*

[TABLE]

then (2.3) admits global-in-time classical solutions. 3. (3)

Medium alignment* ($\psi_{min}<\lambda\leq\psi_{max}$): under the admissible condition*

[TABLE]

if the initial data lie in the subcritical region $\Sigma_{3}$, namely

[TABLE]

then (2.3) admits global-in-time classical solutions.

Here, the parameters $\hat{z}$ and $\tilde{z}$ are defined as

[TABLE]

*Note that $\hat{z}$, $\tilde{z}$ could be real, purely imaginary, as well as infinity. *

The weak and medium alignment situations require an additional structural inequality (2.4) and (2.5) respectively, so that a subcritical region can be obtained through our techniques. These inequalities only depend on the parameters of the EPA system, $k,c,\psi$. These conditions arise due to the presence of oscillatory solutions and as a consequence of our method in handling these to arrive at the thresholds. In the strong alignment case, the solutions decay exponentially to the equilibrium solution without any oscillations, obviating the requirement of any additional condition.

We also prove the corresponding finite-time-breakdown result but do not include it here. We obtain regions $\Delta_{1},\Delta_{2},\Delta_{3}$ which are the supercritical regions for the weak, strong and medium alignment cases respectively.

Similar to the example in the introduction, the fundamental step in analyzing EPA systems for thresholds is to derive an ODE system along the common characteristic path, $\left\{(t,x):\frac{dx}{dt}=u(t,x(t)),\,x(0)=\alpha\right\}$. $\alpha\in\mathbb{T}$ is the parameter which is fixed for a single characteristic path. The resulting ODE system is analyzed for each path. The global well-posedness of unknowns in (2.3) is obtained by combining the all-time-existence of the unknowns in (2.7) for all $\alpha\in\mathbb{T}$. For (2.3), the resulting ODE system is,

[TABLE]

Next, we move on to another important situation of that of the weakly singular kernel, that is, $\psi\in L^{1}(\mathbb{T})$. Here, we do not have the (2.2) type of bounds which are essential in the threshold analysis.

Therefore, we have to modify our technique. Here, we need to improve the bounds on $\psi\ast\rho$ to obtain valid thresholds. Following is the global existence result.

Theorem 2.2** (Weakly singular alignment force).**

Let $\psi\in L^{1}(\mathbb{T})$. Set $\lambda:=2\sqrt{\frac{k}{c}}$. Suppose the initial data to (2.3) satisfies the hypothesis of Theorem 2.1. Then there exists sets $\Sigma_{L}^{1},\Sigma_{L}^{2},\Sigma_{L}^{3}$ such that,

1. (1)

Weak alignment* ($\|\psi\|_{L^{1}}-\gamma<\tfrac{\lambda}{2}$): under the admissible condition*

[TABLE]

if the initial data lie in the subcritical region $\Sigma_{L}^{1}$, namely

[TABLE]

then $(G,\rho)$ remain bounded in all time. 2. (2)

Strong alignment* ($\gamma\geq\tfrac{\lambda}{2}$): if the initial data lie in the subcritical region $\Sigma_{L}^{2}$, namely*

[TABLE]

then $(G,\rho)$ remain bounded in all time. 3. (3)

Medium alignment* ($\gamma<\frac{\lambda}{2}\leq||\psi||_{1}-\gamma$): under the admissible condition*

[TABLE]

if the initial data lie in the subcritical region $\Sigma_{L}^{3}$, namely

[TABLE]

then $(G,\rho)$ remain bounded in all time.

Consequently, (2.3) has a global smooth solution. Here, $\gamma=\int_{1/2}^{1}\psi^{*}(x)\,dx$, where $\psi^{*}:(0,1]\to\mathbb{R}$ is the decreasing rearrangement of $\psi$ on $\mathbb{T}$. The parameters $\hat{z}$ and $\tilde{z}$ are defined as

[TABLE]

Figure 2 illustrates the shape of $\Sigma_{L}^{1}$ and $\Sigma_{L}^{2}$. The steady-state solution $(G,\rho)=(c\|\psi\|_{L^{1}},c)\in\Sigma_{L}^{i}$. Hence, the region $\Sigma_{L}^{i}$ contains initial data around the steady state and we obtain a nontrivial subcritical region.

The admissible conditions (2.8) and (2.9) are similar to (2.4) and (2.5) respectively, in the sense that both imply that nonlocality of $\psi$ is not too strong. When $\psi$ is bounded, $\psi_{max}-\psi_{min}$ is its oscillation. It is zero if and only if $\psi$ is constant. Correspondingly, when $\psi$ is unbounded but integrable, $\psi_{max}-\psi_{min}$ is replaced by $4(\|\psi\|_{L^{1}}-2\gamma)$. Note that $\|\psi\|_{L^{1}}-2\gamma\geq 0$, and the equality holds if and only if $\psi$ is a constant. Therefore, all of (2.8), (2.9), (2.4), (2.5) impose an upper bound on how much $\psi$ is offset from a constant.

The following result is for the EA system with weakly singular influence function.

Theorem 2.3**.**

Let $\psi\in L^{1}(X)$ ($X=\mathbb{R}$ or $\mathbb{T}$) be non-negative. Consider (2.3) with $k=0$. If

[TABLE]

then there exists global-in-time $C^{1}$ solution to the system (2.3). Moreover, $\rho,G$ have uniform bounds in terms of $\psi,\rho_{0},G_{0}$.

3. Nonlocal Euler system with relaxation

Several traffic-flow and fluid-flow models are modeled through a density that follows the continuity equation with a nonlocal flux, see for example [10, 15]. A general equation is,

[TABLE]

for some nonlocal v. We augment this model with a velocity obeying Burgers’ equation with some source terms. The source terms are such that the particles modeled move towards an equilibrium state, imparting some order and regularity to the system. We consider the following pressureless Euler-like model,

[TABLE]

with initial data $(\rho_{0}\geq 0,u_{0})$. The quantity v determines the steady state velocity. Motivated by physical assumptions, we set $\text{v}=Q\ast u$, with,

[TABLE]

Evidently, $Q\ast u$ is a weighted average of the velocity, with maximum weight at the point itself and decreasing symetrically as one moves away. To our knowledge, the well-posedness of (3.1) was not known. Due to the nonlocal flux, it cannot be concluded from existing literature for hyperbolic balance laws. Inspired by Kato and Majda, see [13, 16], we use energy methods to prove local existence/uniqueness for a general multidimensional system, the one dimensional form of which is (3.1), in a relatively more general space. In particular, we allow for solutions that need not decay at infinity, though they are bounded. We do not state the result here but instead, focus on the one dimensional threshold result.

Theorem 3.1**.**

Consider (3.1) with $\text{v}=Q\ast u$ and $Q$ satisfying (3.2). Suppose the initial density is nonnegative and the initial data $\rho_{0},u_{0}\in L^{\infty}(\mathbb{R})$, and $\rho_{0x},u_{0x}\in H^{s}(\mathbb{R})$, $s\geq 1$.

• •

[Subcritical region] A unique solution $\rho,u\in C([0,\infty);L^{\infty}(\mathbb{R}))$ and

[TABLE]

exists if $u_{0x}(x)+\rho_{0}(x)\geq 0$ for all $x\in\mathbb{R}$.

• •

[Supercritical region] If $\exists x_{\ast}\in\mathbb{R}$ for which $u_{0x}(x_{\ast})<-\rho_{0}(x_{\ast})$, then $u_{x}\to-\infty$ in finite time.

Another interesting case is when v is local in (3.1), that is, $\text{v}=f(\rho,u)$. Let $\Theta=\{(\rho,u):u=f(\rho,u)\}$. Then (3.1) is strictly hyperbolic as long as $(\rho,u)\in\Theta^{c}$. However, this cannot be guaranteed a priori and (3.1) could degenerate from strict hyperbolic to weak hyperbolic at certain time. It turns out that for $f_{\rho}=0$ ($f$ only depending on the velocity), strict hyperbolicity can be guaranteed if the initial data lies in a certain set. The threshold analysis and results depend heavily on whether (3.1) is strictly or weakly hyperbolic.

Theorem 3.2**.**

Consider the system (3.1) with $\text{v}=f(u)$ and initial conditions $(\rho_{0}\geq 0,u_{0})\in C_{b}^{1}(\mathbb{R})\times C_{b}^{1}(\mathbb{R})$ with $\inf|f(u_{0})-u_{0}|>0$. If $f_{u}\leq 0$ for solution $u$ under consideration, then,

1. (1)

Bounds on $u$ and $\rho$:* $u(t,\cdot)$ is uniformly bounded and satisfies $|f(u(t,\cdot))-u(t,\cdot)|>0$ for $t>0$. And*

[TABLE] 2. (2)

Global solution:* If*

[TABLE]

then there exists a global classical solution $\rho,u\in C^{1}((0,\infty)\times\mathbb{R})$. Moreover, $\rho,u_{x}$ are uniformly bounded with

[TABLE]

where $M=\max\{\sup\rho_{0},\sup(u_{0x}+\rho_{0})\}$. 3. (3)

Finite time breakdown:* If $\exists x_{0}\in\mathbb{R}$ such that*

[TABLE]

then $\lim_{t\to t_{c}}|\rho_{x}|=\infty$ or $\lim_{t\to t_{c}}|u_{x}|=\infty$ for some $t_{c}>0$.

The condition $\inf|f(u_{0})-u_{0}|>0$ ensures strict hyperbolicity by ensuring $\inf|f(u(t,\cdot))-u(t,\cdot)|>0$ for all $t>0$. If this does not hold, then we have Theorem 3.3. Also, the upper bound on $\rho$ can be infinitely large as the system gets ‘closer’ to weakly hyperbolic. This indeed exhibits the borderline behaviour of density between strictly and weakly hyperbolic systems. In strictly hyperbolic systems (with non-erratic source terms) density is bounded for all times, even when shock forms, whereas in weakly hyperbolic systems, density becomes unbounded when shock forms, as is the case in (2.1).

Theorem 3.3**.**

Let $f$ be a smooth function depending on $u$ only, i.e., $f_{\rho}=0$. Consider the system (3.1) subject to initial conditions, $(\rho_{0}\geq 0,u_{0})\in C^{1}_{b}(\mathbb{R})\times C^{2}_{b}(\mathbb{R})$. If $f_{u}\leq 0$ for the solution $u$ of consideration, then

1. (1)

Global Solution:* If*

[TABLE]

then there exists a global solution

[TABLE]

*Moreover, $u,\rho,u_{x}$ are uniformly bounded. Also, we have the following, *

[TABLE]

where $C_{1}=C_{1}(||\rho_{0}||_{C^{1}},||u||_{C^{2}})$ and $C_{2}=C_{2}(||u||_{C^{2}},||\rho_{0}||_{\infty})$.

We would like to point out some key differences in Theorems 3.2 and 3.3. Firstly, $\rho,u$ are bounded for all times in the former which is not true for the latter wherein there might be density concentration, that is, $\rho\to\infty$ in finite time. Secondly, the space in which the solutions lie is different due to an extra degree of smoothness needed for velocity which arises in proving the local existence in the case of pressureless Eulerian systems.

The following result is for a general $f$, that is, it is a function of both density and velocity. Here, we cannot guarantee a priori strict hyperbolicity and it is imperative that we consider (3.1) as weakly hyperbolic. As a result, we need more conditions for global existence.

Theorem 3.4**.**

Let $f=f(\rho,u)$ be a smooth function of its variables. Consider the system (3.1) with initial conditions $(\rho_{0}\geq 0,u_{0})\in C_{b}^{1}(\mathbb{R})\times C_{b}^{2}(\mathbb{R})$. If $f_{u}\leq 0$ for the solutions under consideration, then $u,\rho,u_{x}$ are uniformly bounded. If in addition to $u_{0x}+\rho_{0}\geq 0$,

• •

$(\rho f)_{\rho\rho}\geq 0$, $f_{uu}\leq 0$ along with

[TABLE]

OR

• •

$(\rho f)_{\rho\rho}\leq 0$, $f_{uu}\geq 0$ along with

[TABLE]

then there exists a global solution $\rho,u\in C^{1}((0,\infty)\times\mathbb{R})$. In addition,

[TABLE]

The breakdown result is same for Theorems 3.3 and 3.4.

Theorem 3.5** (Finite time breakdown).**

Consider (3.1) with $\text{v}=f(\rho,u)$. If there exists an $x_{\ast}$ such that $u_{0x}(x_{\ast})+\rho_{0}(x_{\ast})<0$, then $\lim_{t\to t_{c}^{-}}|u_{x}(t,x_{c})|=-\infty$ for some $x_{c}$ and $t_{c}>0$.

A key step is to identify a quantity $e:=u_{x}+\rho$, that simplifies (3.1). $e$ is quite analogous to $G$ in (2.3). This transformation results in,

[TABLE]

If $f_{u}\leq 0$, we can bound $e,\rho$ in tandem, that is, both quantities are all-time-bounded or break down together. Theorems 3.3 and 3.4 can be proved thereafter.

Under the assumptions of Theorem 3.2, where the system is strictly hyperbolic, we find the two Riemann invariants and analyze them. From (3.1), one of the Riemann invariants is simply $u$. The other invariant $R$ can be evaluated using conventional techniques. Bounds on $R_{x}$ ensure bounds on $\rho_{x}$. The asymptotic bound on $\rho$ in Theorem 3.2 is obtained by bounding $R$.

Acknowledgments

This work was supported in part by the National Science Foundation under Grant DMS1812666.