# Failure of uniqueness for scalar conservation laws

Abraham Sylla, Shyam Sundar Ghoshal, Parasuram Venkatesh

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Abraham Sylla, Shyam Sundar Ghoshal, Parasuram Venkatesh. Failure of uniqueness for scalar con- servation laws. 2026. hal-05457589

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# Failure of uniqueness for scalar conservation laws

Shyam Sundar Ghoshal*

Abraham Sylla†

Parasuram Venkatesh*

## Abstract

In this article, we develop what are, to the best of our knowledge, the first *negative* results for scalar conservation laws. We begin with explicit examples where bounded initial data leads to $L^\infty$ blow-up despite flux regularity. More strikingly, we demonstrate that Kružkov's entropy equalities alone fail to ensure uniqueness in this regime by constructing infinitely many entropy solutions to a single Cauchy problem with bounded initial datum, each continuous in time with respect to the $L^1$ norm. Thus, we demonstrate that the $L^\infty$ assumption is essential for the doubling of variables argument, and hence for the uniqueness of entropy solutions to scalar conservation laws. On the positive side, we develop a novel theory for scalar conservation laws with spatial heterogeneity by adapting the front tracking method. We recover uniqueness by imposing a Lax-type condition in addition to the entropy inequality, motivated by the properties of our front tracking approximations. Unbounded Kružkov solutions do not necessarily satisfy the weak formulation; we show that global weak solutions may not even *exist* in a natural class for some Cauchy problems of this form, even when Kružkov entropy solutions exist. Finally, we construct an explicit example of global *ill-posedness* with bounded initial datum.

**Keywords:** scalar conservation laws, front tracking, generalised characteristics, heterogeneous flux, blow-up, entropy solutions.

**MSC2020:** 35L65, 35L67 (Primary), 35A02, 35B44 (Secondary)

## Contents

<table><tr><td><b>1 Introduction</b></td><td><b>2</b></td></tr><tr><td>1.1 Structure of the paper</td><td>4</td></tr><tr><td>1.2 Preliminaries</td><td>4</td></tr><tr><td>1.3 Structural assumptions</td><td>6</td></tr><tr><td>1.4 Summary of results</td><td>7</td></tr><tr><td><b>2 Well-posedness</b></td><td><b>7</b></td></tr><tr><td>2.1 $L^\infty$ blow-up of entropy solutions</td><td>8</td></tr><tr><td>2.2 Non-uniqueness</td><td>9</td></tr><tr><td>2.2.1 Entropy-preserving solution</td><td>10</td></tr><tr><td>2.2.2 Non-uniqueness with one discontinuity</td><td>11</td></tr><tr><td>2.2.3 Non-uniqueness from bounded initial data</td><td>12</td></tr><tr><td>2.3 Non-existence of weak solutions</td><td>14</td></tr><tr><td>2.4 A sufficient condition for blow-up of entropy solutions</td><td>16</td></tr><tr><td>2.5 A sufficient condition for non-uniqueness of entropy solutions</td><td>18</td></tr></table>

<footnote>*Centre for Applicable Mathematics, Tata Institute of Fundamental Research</footnote>

<footnote>†LAMFA CNRS UMR 7352, Université de Picardie Jules Verne</footnote>
In several space dimensions, the right-hand side of (4) is replaced by the spatial divergence of $f_u$; note that the two definitions coincide when $n = 1$. However, this condition is not sharp. The equation (2) has also been studied under the coercive genuinely non-linear assumptions in [7], relying on the correspondence between scalar conservation laws and Hamilton-Jacobi equations in one space dimension. For instance, if $f(x, u) = (2 + \sin(x))u^2$, then $f$ does not satisfy (4), but the associated Cauchy problem is still well-posed since $f$ is 'genuinely non-linear', i.e. $f_{uu}$ is non-vanishing.

In this paper we construct a counter-example demonstrating that the condition cannot be arbitrarily relaxed. That is, we construct a smooth flux violating (4) such that the Cauchy problem may be ill-posed globally in time, even with smooth and bounded initial data. This indicates that the existence of solutions is sensitive to the structure of the flux, even with strong regularity assumptions.

The non-uniqueness of entropy solutions to non-linear partial differential equations is an active research program. Most notably for the equations of fluid dynamics, the 'convex integration' method pioneered by De Lellis and Székelyhidi [12] has proven fruitful in the construction of multiple admissible solutions to Cauchy problems with irregular initial data. For the compressible Euler equations, this regularity assumption can be relaxed, and the non-uniqueness is exhibited after the formation of singularities [6]. Our constructions follow a similar pattern; in particular even for smooth initial data, there may be infinitely many global-in-time entropy solutions to the Cauchy problem. A secondary active question is the search for a criterion that picks out a unique solution among the pathological ones. We provide such a criterion in the form of a Lax-type condition on interfaces generating the pathologies, returning a unique solution as the limit of front-tracking approximations.

Kružkov's proof for the uniqueness of entropy solutions does not require the assumption (4), it merely requires that the flux be continuously differentiable in all its variables with $f_x$ Lipschitz in $u$ on compact sets [18, Theorem 1]. However, the comparison principle used in the argument crucially relies on uniform bounds of the solutions being compared. Thus, it only demonstrates the uniqueness of $L^\infty$ solutions. We construct examples of fluxes satisfying all assumptions of the uniqueness theorem but without global $L^\infty$ a priori estimates. That is, the $L^\infty$ norm of entropy solutions may blow up in finite time, and the comparison principle is valid only up to this time. The blow-up goes hand-in-hand with non-uniqueness of entropy solutions; we construct explicit examples of Cauchy problems that admit infinitely many entropy solutions. The initial data may also be chosen such that $u_0 \in L^\infty$. Although we work with an explicit example for illustration, our construction is generic, and applies to a large number of fluxes.

Both constructions are essentially one-dimensional, but can be carried out in arbitrary spatial dimensions by a simple extension. Thus, we direct our attention to fluxes of the specific form

$$f(x, u) = g(x)h(u), \qquad (5)$$

where $g$ is locally Lipschitz and $h \in C^2(\mathbb{R})$ is strictly convex with $h'(0) = h(0) = 0$, and thus $h(u) \ge 0$ with equality holding if and only if $u = 0$. In the sequel, we make a slight abuse of notation by writing $f(x, u(x, t))$ as $f \circ u(x, t)$. The precise structural assumptions are outlined in section 1.3. If $g$ in (5) takes on both positive and negative values, then the flux $f$ is not genuinely non-linear everywhere and no known well-posedness theory applies. Thus, we develop a new framework based on the earlier work [24], inspired by the front tracking method of Dafermos [8].
Cauchy problems with such fluxes have entropy solutions that are obtained as limits of front tracking approximations. However, entropy solutions in the sense of (3) may not be unique. We introduce a further constraint to recover uniqueness – in particular, it is satisfied by limits of front tracking approximations, which therefore converge to a unique limit. In the final section, we show that outside the class of fluxes with form (5), the pathologies can worsen, such that global entropy solutions may not even exist.

Fluxes of this form satisfy neither the Kružkov assumptions [18] nor the coercivity conditions of Colombo, Perrollaz, and Sylla in [7]. This has important consequences for well-posedness. In particular, although entropy solutions are unique, they may not exist as bounded functions, though our structural constraints outlined in sections 1.3 can ensure that they are at least locally integrable.

Time-regularity of entropy solutions for multiplicative flux equations was analysed by Felix Otto in [20], though well-posedness was not discussed. More generally, the study of conservation laws with spatial heterogeneity has been conducted in various forms. Motivated by relativistic applications, LeFloch and Ben-Artzi [2] analysed conservation laws on Riemannian or Lorentzian manifolds, where spatial heterogeneities naturally arise. Lengeler and Müller [19] further proved the $L^1$ contraction property on closed manifolds. Convergence of a numerical scheme was demonstrated by Abraham Sylla [23], discretising the heterogeneity and applying the theory of discontinuous flux [1]. Ghoshal and Venkatesh [14] demonstrated studied the asymptotic emergence and $L^2$ stability of simple shock for spatially heterogeneous conservation laws. Anne-Laure Dalibard [11] developed a pure $L^1$ theory for such conservation laws, based on the kinetic formulation [21]. A front tracking approximation scheme for such equations was developed by Parasuram Venkatesh [24], which we leverage here to construct entropy solutions of the Cauchy problem (2).

## 1.1 Structure of the paper

The rest of this paper is structured as follows. In the following section 1.2 we collate some results for scalar conservation laws in one spatial dimension that are crucial for our constructions. In section 1.3 we list the structural assumptions we impose on the flux (5) in order to develop the well-posedness theory. Section 1.4 colloquially summarises our results for the ease of reference. Next, in section 2 we demonstrate the possibility of $L^\infty$ blow up in finite time, and show how Kružkov’s entropy formulation can still make sense, though the entropic solution may not satisfy (2) in the weak sense, unless it is uniformly bounded. We also prove the non-uniqueness of entropy solutions, first with an explicit example and then more generically in Section 2.9. In order to recover a unique solution, we develop an approximation theory for the Cauchy problem, through which we also prove the existence of solutions in the large. Section 3 concerns the front tracking method we use to prove the existence of Kružkov entropy solutions; we also prove that the approximations converge to a unique limit by imposing a Lax-type condition that allows us to recover uniqueness. In section 4.1, we illustrate the construction of some explicit solutions for equations whose fluxes do not satisfy all the structural assumptions. Finally, in section 4.2, we construct a Cauchy problem for a scalar conservation law for which no global solution exists.

## 1.2 Preliminaries

Here we recall some preliminary concepts that we will use in the sequel. Dafermos’ classical text [10] is a comprehensive reference for all the results here. Differentiable solutions of (2)
exist locally in time for smooth initial data [17], but generally break down in finite time. Hence the concept of entropy is introduced – solutions satisfying a family of inequalities of the form

$$ \eta(u)_t + Q(x, u)_x + \eta'(u)f_x(x, u) - Q_x(x, u) \le 0, \qquad (6) $$

in the sense of distributions over $C_c^\infty(\mathbb{R} \times [0, \infty))$ for all pairs of functions $\eta, Q$ such that the ‘entropy’ $\eta$ is a convex function and the ‘entropy flux’ $Q(x, \cdot)$ is an antiderivative of $\eta'(\cdot)f_u(x, \cdot)$ for each fixed $x$. However, this conception of entropy is too strong when solutions are unbounded, as we shall see later. The characteristic equations associated to (2), valid for Lipschitz solutions, are given by

$$ \begin{aligned} \dot{q}(t) &= f_u(q(t), p(t)), \\ \dot{p}(t) &= -f_x(q(t), p(t)). \end{aligned} \qquad (7) $$

If $u$ is a weak entropic solution of (2) which is continuous on either side of a curve $\gamma(t)$, then $\gamma$ satisfies the Rankine-Hugoniot condition

$$ \dot{\gamma}(t) = \frac{f(\gamma(t), u(\gamma(t)+, t)) - f(\gamma(t), u(\gamma(t)-, t))}{u(\gamma(t)+, t) - u(\gamma(t)-, t)}. \qquad (8) $$

Let us recall Dafermos’ theory of generalised characteristics [9] for genuinely non-linear scalar conservation laws, which allows for the use of characteristics (7) even when the solution develops discontinuities [13].

Given an entropy solution $u(x, t)$ and flux $f$ that is strictly convex we have that from every point $(x, t)$ with $t > 0$, we can define a unique forward characteristic $y_f : [t, \infty) \to \mathbb{R}$ and a non-empty set of backward characteristics $y_b : [0, t] \to \mathbb{R}$, i.e. Lipschitz curves with $y(t) = x$ solving the differential inclusion

$$ \dot{y}(s) \in [f_u(y(s), u(y(s)+, s)), f_u(y(s), u(y(s)-, s))] \qquad (9) $$

on their respective domains, where $u(x \pm, t)$ respectively denote the left and right traces in space of $u$ at $(x, t)$. If the flux is uniformly convex in the conserved variable, then these traces always exist for positive times due to the regularising effect of the non-linearity [22]. Furthermore, the interval in (9) is non-empty for all $(x, t)$ with $t > 0$ since $f_u$ is monotone increasing in the second argument and the traces satisfy the Oleinik inequality $u(x-, t) \ge u(x+, t)$ at positive times. Note that along the trajectories of (7),

$$ \begin{aligned} \frac{d}{dt} f(y(t), z(t)) &= f_x(y(t), z(t))\dot{y}(t) + f_u(y(t), z(t))\dot{z}(t) \\ &= f_x(y(t), z(t))f_u(y(t), z(t)) - f_u(y(t), z(t))f_x(y(t), z(t)) \qquad (10) \\ &= 0. \end{aligned} $$

The differential inclusion is more restrictive than it appears prima facie. If $y : [t_0, T] \to \mathbb{R}$ is a Lipschitz solution to (9) for some $t_0 \ge 0$, then for almost all $t \in [t_0, T]$ the curve either satisfies the differential equation or the Rankine-Hugoniot condition, i.e. we have that

$$ \dot{y}(t) = \begin{cases} f_u(y(t), u(y(t), t)) & \text{if } u(y(t)-, t) = u(y(t)+, t), \\ \frac{f(y(t), u(y(t)-, t)) - f(y(t), u(y(t)+, t))}{u(y(t)-, t) - u(y(t)+, t)} & \text{if } u(y(t)-, t) > u(y(t)+, t). \end{cases} $$
The extremal backward characteristics from any point $(x, t)$ with $t > 0$ are 'genuine', i.e. $y(s)$ for $s \in (0, t)$ are points of continuity for $u$, and $u(y(s), s) = z(s)$.

With these generalised characteristics, a front tracking scheme can be constructed to approximate solutions of Cauchy problems with respect to certain conservation laws with heterogeneous flux [24]. The basic idea is to approximate $\operatorname{sgn}(u_0(x))f(x, u_0(x))$ by a piece-wise constant function, and solve the 'generalised Riemann problem' at discontinuities by either a Rankine-Hugoniot shock curve or an approximate rarefaction fan, and solving forward by entropic shocks whenever fronts interact.

## 1.3 Structural assumptions

Let us explicitly state our assumptions on the multiplicative (5) flux $f(x, u) = g(x)h(u)$.

**Regularity:** $h \in C^2(\mathbb{R})$, $g \in C(\mathbb{R})$. (R)

In addition, if $a, b \in \mathbb{R}$ such that for all $x \in [a, b]$, $g(x) \neq 0$, then $g \in C^2([a, b])$.

**Stationarity:** $h(0) = h'(0) = 0$. (S)

**Strict Convexity:** $h'$ is a strictly increasing function. (SC)

**Log Lipschitz Growth:** $\log \circ h \in \operatorname{Lip}((-1, 1)^c)$. (LL)

Throughout this paper, we will consider convexity in terms of (SC), and mutatis mutandis for concavity. Note that (LL) ensure the finite speed of propagation property on compact sets in space for flux-bounded solutions: if $f \circ u(x, t)$ is uniformly bounded, then so is $f_u \circ u(x, t)$, since for non-zero $u$ we have that

$$f_u \circ u(x, t) = g(x)h'(u) = g(x)h(u) \frac{h'(u)}{h(u)}.$$

Note that $h(u)$ in the denominator is non-zero if $u \neq 0$ by (S) and (SC). In addition to these, we need assumptions to control how the flux behaves when $g$ vanishes. Let $G = \{x \in \mathbb{R} : g(x) = 0\}$. In order to construct entropy solutions, we require that $G$ consist of disjoint (possibly degenerate) closed intervals $[a, b]$ (possibly with $a = -\infty$ or $b = +\infty$) such that for each such interval, $G \setminus [a, b]$ is still closed, i.e. the zeroes of $g$ can only accumulate on a complete interval where $g$ vanishes. We can make this a precise assumption as follows: the set of zeroes $G$ of $g$ must satisfy a discreteness condition.

**Boundary Regularity of G:** the boundary $\partial G$ has no accumulation point. (B)

At these points, we require that $g$ have a finite order of vanishing, that is, if $\bar{x} \in \partial G$, then for some positive exponent $\eta \ge 1$ and constant $K$, we must have that $|g(x)| \ge K|x - \bar{x}|^\eta$ in some neighbourhood of $\bar{x}$. More precisely,

**Order of Vanishing:** $\exists \eta \ge 1, K > 0 : \forall \bar{x} \in \partial G, \exists \delta > 0 : |g(x)| \ge K|x - \bar{x}|^\eta, \quad (V)$
$x \in (\bar{x} - \delta, \bar{x} + \delta) \cap G^c$.

Let $\eta \ge 1$ be the smallest exponent which suffices for all points of $\partial G$, then we impose the following growth assumption on $h$.

**Compensating Growth:** $\exists C, \varepsilon, M > 0 : h(u) \ge C|u|^{\eta+\varepsilon}$ for $|u| > M$. (CG)

The assumption (CG) ensures that $u \in L_{\text{loc}}^1(\mathbb{R} \times [0, \infty))$ whenever $f \circ u$ is uniformly bounded.
## 1.4 Summary of results

Large parts of the paper involve explicit constructions of counter-examples. For the readers' convenience, we colloquially summarise our findings here. We remark that our negative results hold in arbitrary spatial dimensions by trivial extension, while the positive results are limited to one spatial dimension.

1. $L^\infty$ blow-up: in section 2, we demonstrate that solutions of (1) may not remain bounded globally in time. Theorem 2.2 makes precise the conditions under which such blow-up generically occurs.

2. Non-uniqueness: following the possibility of unbounded entropy solutions to (1), in section 2.2 we demonstrate that they may not be unique. Thus, we show that Krukov's 'doubling-of-variables' argument crucially relies on $L^\infty$ bounds and fails without them.

3. Well-posedness: in section 3, we show that the equation (2) is well-posed for fluxes satisfying the structural assumptions outlined in section 1.3, i.e. entropy solutions exist for locally integrable initial data, and we recover uniqueness and stability despite the negative results of section 2.2 by imposing a further Lax-type condition in addition to the entropy inequality.

4. Ill-posedness: finally, we demonstrate that there exist conservation laws (1) for which no global entropy solution may exist, by constructing an explicit example in section 4.2 where the entropy solution blows up everywhere in space at a finite positive time.

All together, our results show that the theory of scalar conservation laws with spatially heterogeneous flux is not a simple extension of the homogeneous theory. To the best of our knowledge, these are the first negative results for scalar conservation laws with smooth fluxes; regularity of the flux itself plays almost no role here.

# 2 Well-posedness

The notion of entropy solution for us will be an adaptation of the standard one used for scalar conservation laws, i.e. $u \in L^1_{\text{loc}}(\mathbb{R} \times [0, T))$ is a solution of the Cauchy problem (2) in $\Omega_T = \mathbb{R} \times [0, T]$ if $f \circ u$ is uniformly bounded in $\Omega_T$, and for all $\varphi \in C_c^\infty(\mathbb{R} \times [0, T), [0, \infty))$ and $k \in \mathbb{R}$:

$$ \iint_{\mathbb{R}_+^2} |u(x,t) - k| \varphi_t(x,t) + \operatorname{sgn}(u(x,t) - k) [f \circ u(x,t) - f(x,k)] \varphi_x(x,t)dx \\ \geq \iint_{\mathbb{R}_+^2} \operatorname{sgn}(u(x,t) - k) f_x(x,k) \varphi(x,t)dxdt - \iint_{\mathbb{R}} |u_0(x) - k| \varphi(x,0)dx. \quad (11) $$

Equivalently, we can characterise entropy solutions as those satisfying the above inequality for positive test functions supported in the open upper half plane and continuously approaching the initial value $u_0$ in $L^1_{\text{loc}}$ as $t \to 0$. We remark that it is sufficient for $u$ and $(t, x) \mapsto f(x, u(t, x))$ to be merely $L^1_{\text{loc}}$ in order to make sense of this definition. However, entropy solutions may not satisfy the integral or 'weak' formulation of (2), unless $u \in L^\infty(\mathbb{R})$, in which case the integral formulation follows directly from (11) by considering
$k = \pm \|u\|_{L^{\infty}}$. We also show that global weak solutions may not even exist in the natural class for some Cauchy problems.

Entropy solutions also satisfy the trace property in domains of genuine non-linearity, i.e. where $g \neq 0$. In fact, the solutions are locally special functions of bounded variation at all but countably many time levels, following the argument in [22].

**Lemma 2.1.** If $x \in G^c$ and $t > 0$, then for any entropy solution $u$, the left and right spatial traces $u(x \pm t)$ exist and satisfy the inequality $u(x \pm t) \ge u(x \pm t)$ if $g(x) > 0$ and $u(x \pm t) \le u(x \pm t)$ if $g(x) < 0$.

*Proof.* Let $t > 0$, $x \in G^c$ and assume without loss of generality that $g(x) > 0$. Since $G$ is a closed set, there is a maximal open interval $(a, b) \subseteq G^c$ such that $x \in (a, b)$. There exists some $d > 0$ be such that $[x - d, x + d] \subseteq (a, b)$. Set $M = \|f \circ u\|_{L^{\infty}}$, then by (LL), we have that $f_u \circ u$ is also uniformly bounded, say by $R > 0$. Now, there exists some $\varepsilon > 0$ small enough such that $t > \varepsilon$ and $d > R\varepsilon$. Consider the trapezoidal domain

$$T_x = \{(y, s) : s \in [t - \varepsilon, t], y \in [x - d + R(s - t + \varepsilon), x + d - R(s - t + \varepsilon)]\}.$$

Since $f \circ u$ is bounded, and $[x - d, x + d]$ is a compact subset of $G^c$, we have that $u$ is uniformly bounded in $T_x$ as well. If we let $u_0(x) = u(x, t - \varepsilon)$ for $x \in [x - d, x + d]$ and zero otherwise, then in the domain $T_x$, the bounded measurable function $u$ is an entropy solution of a genuinely non-linear scalar conservation law. Since the entropy solutions of such equations correspond with viscosity solutions of the associated Hamilton-Jacobi equation [5], it follows that $u$ has spatial traces in the domain $T_x \cap [s > t - \varepsilon]$, which includes the point $(x, t)$. Since $(x, t)$ was arbitrarily chosen from $G^c \times (0, \infty)$, we are done. $\square$

Note that if $u \in L^{\infty}$, then the traces of $f \circ u$ on $G \times [0, T]$ are zero. In the next section we demonstrate that working with $L_{\text{loc}}^1$ is necessary for multiplicative fluxes, as the $L^{\infty}$ norm can blow up even for uniformly bounded initial data. For such solutions, $f \circ u$ may have non-zero trace on the degenerate interface.

## 2.1 $L^{\infty}$ blow-up of entropy solutions

For particular fluxes, we can show explicitly that the Cauchy problem corresponding to (2) is not globally well-posed in $L^{\infty}$, though it still is in $L_{\text{loc}}^1$. Let $f$ be given by

$$f(x, u) = xu^2. \qquad (12)$$

Although $f$ as defined satisfies the assumptions of Theorem 2.8 proved later, it is instructive to first visualise the blow-up phenomenon explicitly. This flux is convex for $x > 0$ and concave for $x > 0$, but degenerate for $x = 0$. The characteristic equations thus reduce to

$$\dot{q}(t) = 2q(t)p(t), \qquad (13)$$

$$\dot{p}(t) = -p(t)^2. \qquad (14)$$

We can integrate this system by first solving in $p$, and then using the Hamiltonian structure of the system. More precisely, if $p(0) = 0$, then, $p \equiv 0$. Otherwise,

$$p(t) = \frac{1}{t + p(0)^{-1}}.$$
We see that $p$ blows up in finite time if $p(0) < 0$. Then, using the fact that $t \mapsto f(q(t), p(t))$ is constant, we deduce that

$$q(t) = q(0) (tp(0) + 1)^2, \qquad (15)$$

which holds for as long as $p$ exists, i.e. for $t < -p(0)^{-1}$. If $q(0) = 0$, however, the trivial solution satisfies (13) globally in time.

With this explicit formula for characteristics, we have the following blow-up result.

**Theorem 2.2.** Fix $u_0 \in C \cap L^\infty(\mathbb{R})$. Assume that $u_0$ is non-increasing on $(-\infty, 0]$, non-decreasing on $[0, \infty)$, and that $u_0(0) < 0$. If $u_0(0) < 0$, then a continuous entropy solution of (2) exists for $t < -u_0(0)^{-1}$, and its $L^\infty$ norm of the solution blows up at $(x, t) = (0, -u_0(0)^{-1})$.

*Proof.* A continuous solution exists as long as distinct characteristics do not intersect. The characteristic emanating from $x = 0$ is just the trivial one. Hence, it is sufficient to show that characteristics emanating from $x > 0$ or $x < 0$ do not intersect with each other, respectively. Let $0 \le q_1(0) < q_2(0)$; then by hypothesis $u_0(0) \le p_1(0) \le p_2(0) \le 0$. For $0 < t < -u_0(0)^{-1}$, by (15) we have that

$$q_1(t) = q_1(0) (tp_1(0) + 1)^2, \qquad (16)$$

$$q_2(t) = q_2(0) (tp_1(0) + 1)^2. \qquad (17)$$

Now, we can divide (16) by (17) to get

$$\frac{q_1(t)}{q_2(t)} = \frac{q_1(0)}{q_2(0)} \left( \frac{tp_1(0) + 1}{tp_2(0) + 1} \right)^2.$$

Since the right-hand side is always less than one, we see that the characteristics, as long as they exist, cannot meet. The same argument applies, *mutatis mutandis*, for characteristics emanating from negative values of $x$. Blow-up of the characteristics as $t$ approaches $-u_0(0)^{-1}$ when $u_0(0) < 0$ follows from (14), continuity, and positive/negative monotonicity. $\square$

If $u_0$ is merely bounded and non-decreasing/non-increasing on $[0, \infty)$ and $(-\infty, 0]$ respectively, similar results hold. At points of discontinuity, the forward characteristics fan out as a rarefaction wave, and are again non-intersecting as long as they exist. Consider

$$u_0(x) = \begin{cases} -1 & \text{if } |x| \le 1, \\ 0 & \text{if } |x| > 1, \end{cases} \qquad (18)$$

which satisfies the prescribed monotonicity conditions. Then we have a blow-up at $t = 1$, and the characteristics can be pictured as in Figure 1

## 2.2 Non-uniqueness

Unexpectedly, the entropy condition (11) is not sufficient to ensure uniqueness. If a uniformly bounded entropy solution exists, then it is still unique in the class of bounded solutions [18, Theorem 1], but we know from Theorem 2.2 that we cannot remain in this class. Let us consider the flux $f(x, u) = xu^2$ as before.

Recall that stationary solutions in $x > 0$ are of the form $f(x, u) = c$ for $u(x) \le 0$ and some real number $c$, and that the positive stationary solution corresponding to $c$ does
<figure><img src="image_2.png" /><figcaption>Figure 1: Characteristics associated with the Cauchy problem for (2) with $f(x, u) = xu^2$ and initial data given by (18). The $L^\infty$ norm of the entropy solution $u$ blows up along $\{x=0, t \ge 1\}$.</figcaption></figure>

not satisfy (11), to rephrase. However, if we add a dissipative term to the other side of the boundary, we can cancel out the trace term at the interface, and this ‘glued’ solution stands in contrast with the one we would obtain by solving as a rarefaction fan.

### 2.2.1 Entropy-preserving solution

First, we construct a discontinuous entropy-preserving stationary solution of (2) for the flux $f(x, u) = xu^2$, that is a function that satisfies the entropy inequality as an equality. The construction generically works for any similarly multiplicative flux. We also remark that instead of $g(x) = x$, we could also take a function $g$ that is uniformly bounded but agrees with the identity function on some compact interval $[-K, K]$ where $K > 0$. Our results are essentially localised to the interface $x = 0$ where the flux flips from convex to concave.

**Lemma 2.3.** *Define*

$$u(x) = \begin{cases} -1/\sqrt{-x} & \text{if } x < 0, \\ 1/\sqrt{x} & \text{if } x > 0. \end{cases}$$

Then $u$ is stationary entropy solution of the equation (2) with flux $f(x, u) = xu^2$; in particular, it satisfies the entropy inequality as an equality.

*Proof.* Since $u$ is time-independent, it is trivially $L^1_{\text{loc}}$-continuous in time. Thus, it is sufficient to prove the entropy inequality for positive test functions $\varphi$ supported in the open upper-half plane. Furthermore, $f(x, u(x)) = \text{sgn}(x)$ for all $x \ne 0$, hence $u$ is a classical solution of (2) away from the interface $\{x=0\}$. Now, let $\varphi \in C_c^\infty(\mathbb{R} \times [0, \infty), [0, \infty))$ and $k \in \mathbb{R}$. Choose some $\varepsilon > 0$ such that $\varepsilon^{-1} > k^2$, so that $|u(x)| > |k|$ for all $|x| \le \varepsilon$. Then, by splitting up the integral in (11) and integrating by parts, we have that

$$ \begin{align*} E &:= \iint_{\mathbb{R}^2_+} |u(x) - k| \varphi_t + \text{sgn}(u(x) - k) \{[f(x, u(x)) - f(x, k)] \varphi_x - f_x(x, k)\varphi\} dxdt \\ &= \int_0^\infty \int_{-\varepsilon}^\varepsilon |u(x) - k| \varphi_t + \text{sgn}(u - k) \{[f(x, u) - f(x, k)] \varphi_x - f_x(x, k)\varphi\} dxdt \\ &\quad - \int_0^\infty [f(\varepsilon, u) - f(\varepsilon, k)] \varphi(\varepsilon, t)dt - \int_0^\infty [f(-\varepsilon, u) - f(-\varepsilon, k)] \varphi(-\varepsilon, t)dt. \end{align*} $$
Now take the limit as $\varepsilon \to 0$. Using that $f \circ u$ is constant, the integrability of integrands, and the fact that $f(\varepsilon, k) \to 0$ as $\varepsilon \to 0$ for any fixed $k \in \mathbb{R}$, we obtain:

$$E = \int_{0}^{\infty} (-f(1, u(1)) - f(-1, u(-1)))\varphi(0, t)dt = 0.$$

Since $k, \varphi$ were arbitrary, this proves our claim that $u$ is an entropy solution that satisfies the inequality (11) as an equality. □

We remark that if we define $u(x) = 0$ for $x < 0$ instead, the same calculation shows that $u$ would still be an admissible solution of (2), albeit one satisfying the entropy condition as a strict inequality. However, in both cases, $u$ does not satisfy the integral formulation of (2).

### 2.2.2 Non-uniqueness with one discontinuity

Before we establish non-uniqueness of entropy solutions of (2) even for $L^\infty$ initial data, we first establish it in a special case involving discontinuities, i.e. shocks. We use this as a stepping stone for the final construction. For the same flux $f(x, u) = xu^2$ as before, consider the initial datum

$$u_0(x) = \begin{cases} -1/\sqrt{-x} & \text{if } x < 0, \\ 0 & \text{if } x > 0. \end{cases} \qquad (19)$$

A similar calculation as in the proof of Lemma 2.3 shows that this datum is a stationary entropy solution of (2).

Later, in Lemma 3.1 we show this for more general stationary solutions. However, it is not the unique entropy solution of the Cauchy problem. Define

$$u(x,t) = \begin{cases} -1/\sqrt{-x} & \text{if } x < 0, \\ 1/\sqrt{x} & \text{if } 0 < x < t^2/4, \\ 0 & \text{if } x > t^2/4. \end{cases} \qquad (20)$$

We can also write this function succinctly as

$$u(x,t) = \begin{cases} \operatorname{sgn}(x)/\sqrt{|x|} & \text{if } x < y(t), \\ 0 & \text{otherwise,} \end{cases}$$

where $y(t) = t^2/4$. This function is illustrated in Figure 2.

We claim that $u$ is also an entropy solution of the same Cauchy problem. For test functions $\varphi$ supported away from $t = 0$, this is clear from Lemma 2.3 and the fact that $u$ satisfies the Rankine-Hugoniot condition along the curve $\gamma(t) = t^2/2$ with a well-signed jump. Since $1/\sqrt{x}$ is also integrable near zero, it follows that $\|u(\cdot, t) - u_0\|_{L^1} \to 0$ as $t \to 0$, and thus $u$ is an entropy solution of the Cauchy problem. In [4], the authors construct a solution of Burgers equation that satisfies the entropy inequality in the upper half plane but fails to achieve the initial value as $t \to 0$. In our example, however, we have $L_{\text{loc}}^1$-continuous non-unique solutions.
<figure><img src="image_3.png" /><figcaption>Figure 2: One of the infinitely many Kružkov entropy solutions to the Cauchy problem for (2) with flux $f(x, u) = xu^2$ and initial data given by (19).</figcaption></figure>

### 2.2.3 Non-uniqueness from bounded initial data

By Kruzkov’s doubling-of-variables argument [18, Theorem 1], we know that uniqueness holds for the Cauchy problem within the class of bounded entropy solutions whenever they exist. However, when the $L^\infty$ norm blows up for bounded initial data, we can exploit it to generate infinitely many entropy solutions on a time horizon that is large enough to allow for blow-up. Consider the initial datum

$$ u_0(x) = \begin{cases} -1/\sqrt{-x} & \text{if } x < -1, \\ 0 & \text{if } x > -1. \end{cases} \qquad (21) $$

Since the discontinuity at $x = -1$ is entropic, and the data is a stationary solution on either side of the discontinuity, we can solve the PDE exactly by solving the Rankine-Hugoniot ordinary differential equation

$$ \begin{aligned} \dot{y}(t) &= \sqrt{-y(t)}, \\ y(0) &= -1, \end{aligned} $$

at least as long as $y < 0$. However, the solution is easily seen to be $y(t) = -(t-2)^2/4$ for $t < 2$, which means that at $t = 2$, the solution has precise profile (19), and hence the Cauchy problem (2) with flux $xu^2$ and $L^\infty$ initial datum (21) also has infinitely many entropy solutions. Figure 3 illustrates the finite-time blow-up.

Note that there is nothing special about the quadratic exponent here, and we could have used any flux of the form $xu^s$ with $s > 1$ to carry out the earlier computations and generate infinitely many entropy solutions. This is essentially due to the non-uniqueness of Cauchy problems for ODEs with merely Hölder continuous right-hand sides. For a flux of the form $xu^s$, the initial datum/stationary solution (19) is simply

$$ u_0(x) = \begin{cases} -1/\sqrt{-x} & \text{if } x < 0, \\ 0 & \text{if } x > 0, \end{cases} $$
<figure><img src="image_4.png" /><figcaption>Figure 3: The entropy solution to the Cauchy problem for (2) with flux $f(x, u) = xu^2$ and initial data given by (21) evolves into the profile (19) at $t = 2$, beyond which there are infinitely many entropic continuations.</figcaption></figure>

and the non-stationary solution analogous to (20) is now

$$u(x, t) = \begin{cases} -1/\sqrt{-x} & \text{if } x < 0, \\ 1/\sqrt{x} & \text{if } 0 < x < y(t), \\ 0 & \text{if } x > y(t), \end{cases}$$

where

$$y(t) = \left( \frac{s-1}{s} t \right)^{\frac{s}{s-1}}.$$

Hence, even mild super-linearity can induce non-uniqueness. Finally, we remark that this phenomenon can also be extended to several space dimensions by a simple extension.

**Corollary 2.3.1.** Consider the multidimensional scalar conservation law

$$u_t + (x_1 u^2)_{x_1} + \sum_{i=2}^{n} f_i(u)_{x_i} = 0 \quad \text{in } \mathbb{R}^n \times [0, \infty),$$

where $n \ge 2$ and $f_i$ are arbitrary $C^2$ functions. For initial data of the form

$$u_0(x_1, \dots, x_n) = \begin{cases} -x_1^{-1/2} & \text{if } x_1 \ge 1, \\ 0 & \text{otherwise,} \end{cases}$$

there exist infinitely many solutions satisfying Kružkov's entropy inequalities.

Since the extension in higher dimensions is trivial, the corollary follows directly from the preceding construction.
## 2.3 Non-existence of weak solutions

Consider again the Cauchy problem for the following conservation law:

$$ 
\begin{aligned}
u_t + (xu^2)_x &= 0, \\
u(x, 0) &= u_0(x).
\end{aligned} 
\qquad (22) 
$$

In this section, we will once again consider initial data of the form

$$ u_0(x) = \begin{cases} 1/\sqrt{x} & \text{if } x > 0, \\ 0 & \text{if } x < 0. \end{cases} \qquad (23) $$

We claim that there are no weak solutions of (22) in the class of locally integrable functions $u$ with $\|f \circ u\|_{L^\infty} = 1$, where $f(x, u) = xu^2$. Note that $\|f \circ u_0\|_{L^\infty} = 1$, and entropy solutions of (2) generally satisfy a flux-maximum principle. The method of proof will be to assume that a weak solution exists, and derive a contradiction. In order to do this, we first need some lemmas on the necessary conditions that any weak solution, if it exists, must satisfy.

**Lemma 2.4** (Trace at the interface). Let $u$ be a weak solution of (22). Then, $f \circ u(x, t)$ has zero trace at the interface $x = 0$, i.e. for all $T > 0$:

$$ \lim_{x \to 0^\pm} \int_0^T |f \circ u(x, t)| dt = 0. $$

*Proof.* Consider the test function $\varphi_\varepsilon(x, t) = \varepsilon\eta_\varepsilon(x)\psi(t)$, where $\psi \in C_c^\infty([0, \infty))$ and $\eta_\varepsilon$ is from a family of approximate identities, with $\eta$ supported in $[-1, 1]$ and $\eta'(x) \operatorname{sgn}(x) \ge 0$. If $u$ is a weak solution, we can use $\varphi_\varepsilon$ as a test function and obtain, for all $\varepsilon > 0$:

$$ \int_0^\infty \psi'(t) \int_{-\varepsilon}^\varepsilon \varepsilon \eta_\varepsilon(x) u(x,t) dx dt = - \int_0^\infty \psi(t) \int_{-\varepsilon}^\varepsilon \varepsilon \eta_\varepsilon'(x) f \circ u(x,t) dx dt. $$

Since $f \circ u$ is non-negative (respectively, non-positive) if $x > 0$ (respectively, $x < 0$), and the term on the left-hand side vanishes as $\varepsilon \to 0$ (since the integrand is absolutely integrable and the domain shrinks in measure to zero), the term on the right-hand side must vanish too. Since $\psi$ was arbitrary, this concludes the proof of the strong trace property. $\square$

The following corollary now follows immediately from Lemma 2.4.

**Corollary 2.4.1.** Let $u$ be a weak solution of (22) with initial data $u_0$ such that $u_0(x) = 0$ for $x < 0$. Then, the function $\tilde{u}$ defined as

$$ \tilde{u}(x, t) = \begin{cases} u(x, t), & \text{if } x \ge 0, \\ 0, & \text{if } x < 0, \end{cases} $$

is also a weak solution of the same Cauchy problem.

Hence, we will assume from now on that, without loss of generality, the weak solution $u$ is supported in the domain $\{x \ge 0\}$. If $u$ agrees with the entropy solution outside a compact domain, we have the following result.
**Lemma 2.5 (Integral evolution).** Let $u$ be a weak solution of (22) with initial data (23) such that $u(x,t) = -x^{-1/2}$ for all $x > K, t \le T$ for some positive constants $K, T$. Then, for almost all $0 \le t \le T$:

$$ \int_0^\infty u(x,t) + \frac{1}{\sqrt{x}} dx = -t. $$

*Proof.* Since the equality is trivially true for $t = 0$, we will only consider the case of $t > 0$. Furthermore, by Corollary 2.4.1, we may assume without loss of generality that $u - u_0$ has compact support in $\mathbb{R} \times [0, T]$. Consider the domain $[0, K+1] \times [0, t]$, where $t < T$. By approximating the characteristic function of this domain, and assuming that $t$ is a Lebesgue point of $u : [0, T] \to \mathbf{L}_{\text{loc}}^1(\mathbb{R})$, we have that

$$ \int_0^t f \circ u(K+1, s) - f \circ u(0+, s) ds + \int_0^{K+1} u(x,t) - \frac{1}{\sqrt{x}} dx = 0. $$

From the strong trace property of Lemma 2.4 and the fact that $u - u_0 = 0$ for $x > K$, the lemma follows, since $f \circ u_0(K+1) = 1$. □

Next, we localise the weak solution in the following lemmas, showing that the far-field condition can be assumed without much loss of generality.

**Lemma 2.6.** Let $\Omega_T = \mathbb{R} \times [0, T]$, and let $u$ be a weak solution of (22) with the initial data (23) such that $\|f \circ u\|_{\mathbf{L}^\infty} < \infty$, and $u$ has well-defined spatial traces in the domain $\{x > 0\}$. Let $\|f \circ u\|_{\mathbf{L}^\infty} = \bar{C}$. Then, there exists $K > 0$ and a (possibly different) weak solution $\tilde{u}$ such that $\tilde{u}(x,t) = -x^{-1/2}$ for all $x > K$.

*Proof.* Without loss of generality, consider the left-continuous representative of $u$. Define the function $Q$ as follows:

$$ Q(x,t) = \begin{cases} \frac{f \circ u(x,t) - 1}{u(x,t) + x^{-1/2}}, & \text{if } u(x,t) \neq -x^{-1/2}, \\ -2x^{1/2}, & \text{otherwise.} \end{cases} $$

Then, if $\gamma(t)$ is the solution of the initial value problem

$$ \begin{aligned} \dot{\gamma}(t) &= Q(\gamma(t), y), \\ \gamma(0) &= x_0 > 0, \end{aligned} $$

we can define a new solution $\tilde{u}$ that satisfies the far-field criterion of Lemma 2.5 on any time interval $[0, T]$ where $\gamma(t) > 0$ for all $T > 0$ as follows:

$$ \tilde{u}(x,t) = \begin{cases} u(x,t) & \text{if } x \le \gamma(t), \\ -x^{-1/2} & \text{if } x > \gamma(t). \end{cases} $$

Since the flux $f(x, u) = xu^2$ satisfies (LL), the curve $\gamma$ is uniformly Lipschitz. Since it is precisely either a Rankine-Hugoniot curve or a genuine characteristic, it is trivial to see that $\tilde{u}$ is a weak solution.

We claim that $\gamma(t) > 0$ for all $t \in [0, T]$. Let $\gamma(t) > 0$ for $t < t^*$ and $\gamma(t^*) = 0$ for some $t^* \in [0, T]$. Then $\tilde{u}$ is a weak solution on $\Omega_{t^*}$. However, $\tilde{u}$ must violate the integral evolution identity of Lemma 2.5, which is a contradiction. Hence, $\gamma(t) > 0$ for all $t \in [0, T]$, and we are done. □
With this lemma, we can restrict ourselves to the case of weak solutions satisfying the far-field criterion of Lemma 2.5. Hence, we have the following theorem on non-existence of weak solutions.

**Theorem 2.7.** Let $u \in L_{\text{loc}}^1(\Omega_T)$ such that $u(x, 0) = u_0(x)$, where $u_0$ is defined as in (23) and $t = 0$ is a Lebesgue point of $u : [0, T] \to L_{\text{loc}}^1(\mathbb{R})$. If $\|f \circ u\|_{L^\infty} \le 1$, then $u$ cannot be a weak solution of the Cauchy problem (22).

*Proof.* Note that $\|f \circ u_0\|_{L^\infty} = 1$ and therefore the inequality condition is really an equality.

Suppose, if possible, that $u$ is a weak solution of (22). By the Lemmas 2.4 and 2.6, we may assume that $u(x, t) - u_0(x)$ is zero for $x \in [0, K]^c$ for some $K$. That is, we may assume without loss of generality that $u$ satisfies the far-field criterion of Lemma 2.5. However, if $\|f \circ u\|_{L^\infty} \le 1$, then for $x > 0$ we have that

$$u(x, t) \geq -\frac{1}{\sqrt{x}} \implies u(x, t) + \frac{1}{\sqrt{x}} \geq 0,$$

but this is precisely the integrand in Lemma 2.5. Since the integral, however, must be negative at all positive times, this is a contradiction. Hence, $u$ cannot be a weak solution of (22) with initial data $u_0$ if $\|f \circ u\|_{L^\infty} \le 1$. $\square$

## 2.4 A sufficient condition for blow-up of entropy solutions

Consider a general flux of the form

$$f(x, u) = g(x)h(u), \qquad (24)$$

where $h, g$ satisfy the structural assumptions outlined in section 1.3. Note that the flux $f(x, u) = xu^2$ considered earlier is a special case of such a flux. The characteristic system (7) reduces to

$$\dot{q}(t) = g(q)h'(p), \qquad (25)$$

$$\dot{p}(t) = -g'(q)h(p). \qquad (26)$$

For fluxes of this form, we have the following result.

**Theorem 2.8.** Consider the Cauchy problem (2) for a flux with multiplicative heterogeneity of the form (24). If, for some zero $x_0$ of $g$ we have that $g'(x_0) \neq 0$, then there exist compactly supported initial data $u_0$ such that the $L^\infty$ norm of the entropy solution $u$ blows up in finite time. In particular, we can also choose $u_0 \in C_c^1(\mathbb{R})$.

*Proof.* Without loss of generality, assume that $x_0 = 0$ and $g'(0) > 0$. Since $g \in C^2$, there exists $K, \beta > 0$ such that for all $x \in [-K, K] : g'(x) \ge \beta$. Note that $f$ is now strictly convex or concave if $K > x > 0$ or $-K < x < 0$ respectively, since $g$ is monotone strictly increasing on $[-K, K]$ with $g' \ge \beta$. Let $m > 0$, and consider initial data of the form

$$\bar{u}_0(x) = \begin{cases} 0 & \text{if } |x| > K, \\ -m & \text{if } |x| \le K. \end{cases}$$

We claim that the entropy solution $\bar{u}$ for this equation blows up in finite time. Firstly, note that at $x = \pm K$, the jumps are of rarefaction-type. Furthermore, from (25) we see that
forward characteristics $q$ emanating from positive $x$ travel at negative speed, and vice versa, while the forward characteristic with $q(0) = 0$ is simply $q(t) = 0$.

Since $g$ is Lipschitz, and also $g' \ge \beta$, from (26) we have that

$$-Lh(p(t)) \le \dot{p}(t) \le -\beta h(p(t)).$$

In particular, $p$ cannot blow up instantaneously if $p(0) = -m$. However, by uniform convexity of $h$ we have that

$$\dot{p}(t) \le \frac{\alpha\beta}{2} p(t)^2,$$

and hence must blow up in finite time. By the conservation of $f$ along characteristic trajectories, this can only happen when $q(t) = 0$. Hence, if we can show that distinct forward characteristics do not intersect except at $x = 0$, we are done. Clearly, no characteristic can meet the trivial zero characteristic except at $x = 0$.

Now suppose $q_1, q_2$ are two distinct characteristics with $0 < q_1(0) < q_2(0) < K$. Then $p_i(0) = -m$ for the corresponding value functions along the trajectories. Suppose, if possible, that $0 < q_1(\tau) = q_2(\tau)$. Now, for $i = 1, 2$, we know that

$$h(p_i(t)) = \frac{g(q_i(0))h(p_i(0))}{g(q_i(t))}. \qquad (27)$$

Since $q_1(0) < q_2(0)$, furthermore, it follows that $\dot{q}_1(\tau) > \dot{q}_2(\tau)$, so that by (25) and the monotonicity of $h'$,

$$\begin{aligned} g(q_1(\tau))h'(p_1(\tau)) &> g(q_2(\tau))h'(p_2(\tau)), \\ \implies h'(p_1(\tau)) &> h'(p_2(\tau)), \\ \implies p_1(\tau) &> p_2(\tau). \end{aligned}$$

Hence, by (27) and the monotonicity of $h$ for negative values, we have that

$$\begin{aligned} \frac{g(q_1(0))h(p_1(0))}{g(q_1(\tau))} &> \frac{g(q_2(0))h(p_2(0))}{g(q_2(\tau))}, \\ \implies g(q_1(0))h(-m) &> g(q_2(0))h(-m), \\ \implies g(q_1(0)) &> g(q_2(0)), \end{aligned}$$

which contradicts the hypothesis that $g' \ge \beta$ since $q_1(0) < q_2(0)$. Hence, forward characteristics from $x \in [0, K)$ do not intersect before they blow up. The same argument holds, mutatis mutandis, for the non-intersection of forward characteristics from $x \in (-K, 0]$.

To complete the proof, we need to show that characteristics emanating from the rarerfaction fans at $\pm K$ do not intersect either. Let $q_1(0) = q_2(0) = K$, without loss of generality, with $-m \le p_1(0) < p_2(0) < 0$. Note that it is again clear that the characteristic with $p(0) = 0$ cannot intersect any other. If $q_1(\tau) = q_2(\tau)$, then we must have that $0 > p_1(\tau) > p_2(\tau)$ once again, and hence $h(p_1(\tau)) < h(p_2(\tau))$, but from (27) we have that

$$\begin{aligned} h(p_1(\tau)) &= \frac{g(q_1(0))h(p_1(0))}{g(q_1(\tau))} \\ &= \frac{g(q_2(0))h(p_2(0))}{g(q_2(\tau))} \frac{h(p_1(0))}{h(p_2(0))} \\ &= h(p_2(\tau)) \frac{h(p_1(0))}{h(p_2(0))}. \end{aligned}$$
Since it was assumed that $0 > p_2(0) > p_1(0)$, it follows that $h(p_1(\tau)) > h(p_2(\tau))$, which is a contradiction. Hence, characteristics of the rarefaction fan(s) are also non-intersecting. Note that the solution $u$ is supported in $[-K, K]$ for all time, and only blows up along $x=0$. If $g'(0) < 0$, then use $m$ instead of $-m$ in the construction of $\bar{u}_0$.

Now, $\bar{u}_0$ as constructed is not continuous. However, from (26) we know that a bounded solution exists locally in time, so for $\varepsilon > 0$ small enough, let $u_0(x) = \bar{u}(x, \varepsilon)$. We have already shown that it is compactly supported in space at all times, and that the characteristics do not intersect, hence $u_0 \in C_c^1(\mathbb{R})$. $\square$

## 2.5 A sufficient condition for non-uniqueness of entropy solutions

$\mathbf{L}^\infty$ blow-up goes along with the non-uniqueness, and hence this is also a sufficient condition for the existence of initial data with infinitely many Kružkov entropy solutions.

**Theorem 2.9.** Consider the Cauchy problem (2) for a flux with multiplicative heterogeneity of the form (24). If, for some zero $x_0$ of $g$ we have that $g'(x_0) \neq 0$, then there exists initial data $u_0 \in \mathbf{L}^\infty(\mathbb{R})$ such that the Cauchy problem has infinitely many Kružkov entropy solutions.

*Proof.* Without loss of generality, suppose $x_0 = 0$ and $g'(0) > 0$. We will imitate the construction for $f(x, u) = xu^2$; since the construction is localised in space, we may assume that $g' \ge \beta > 0$ for some positive constant $\beta$. Let $u_+$ denote the continuous stationary solution of the Cauchy problem of such that $u_r(x) \ge 0$ with $u_r(x) = 0$ for $x < 0$ and $f(x, u_r(x)) = 1$ for $x > 0$. Similarly, let $u_l(x)$ denote the continuous stationary negative solution supported on $(-\infty, 0]$ with $f(x, u_l(x)) = -1$ for $x < 0$. First, we will show that function $u(x, t)$ is an entropy solution of (2), where $u(x, t) = u_l(x) + u_r(x)$, or in other words

$$u(x, t) = \begin{cases} u_l(x) & \text{if } x < 0, \\ u_r(x) & \text{if } x > 0. \end{cases}$$

It is clear that $u$ is a classical solution of (2) away from the interface $\{x=0\}$, and by (V)-(CG) $u$ and $f \circ u$ are also locally integrable. Hence, for any positive $\varphi \in C_c^1(\mathbb{R} \times [0, \infty))$ and $k \in \mathbb{R}$, the same computation as was done in Lemma 2.3 shows that $u$ as defined is an entropy solution of (2). For completeness, we replicate the computation here.

$$ \begin{aligned} E &:= \iint_{\mathbb{R}_+^2} |u(x) - k| \varphi_t + \operatorname{sgn}(u(x) - k) \{[f(x, u(x)) - f(x, k)] \varphi_x - f_x(x, k)\varphi\} dxdt \\ &= \int_0^\infty \int_{-\varepsilon}^\varepsilon |u(x) - k| \varphi_t + \operatorname{sgn}(u - k) \{[f(x, u) - f(x, k)] \varphi_x - f_x(x, k)\varphi\} dxdt \\ &\quad - \int_0^\infty [f(\varepsilon, u) - f(\varepsilon, k)] \varphi(\varepsilon, t)dt - \int_0^\infty [f(-\varepsilon, u) - f(-\varepsilon, k)] \varphi(-\varepsilon, t)dt. \end{aligned} $$

As $\varepsilon \to 0$, by the construction of $u$, we obtain:

$$E = \int_{0}^{\infty} (-f(1, u(1)) - f(-1, u(-1)))\varphi(0, t)dt = 0.$$

Now, consider the initial data $u_0(x) = u_l(x)$. By the blow-up criterion of Theorem 2.8, we can obtain this profile at a positive time from $\mathbf{L}^\infty$ initial data. Hence, it is sufficient to demonstrate that the Cauchy problem with initial data $u_l(x)$ has infinitely many entropy
solutions. Note that $u_l$ itself is already a Kružkov entropy solution. Thus, if $\gamma(t)$ denotes the Rankine-Hugoniot curve connecting the stationary solution $u_r$ with zero to the right with $\gamma(t) > 0$ for $t > 0$ and $\gamma(t) = 0$ for $t \le 0$, we can define an infinite family of entropy solutions to the Cauchy problem $u_\lambda(x, t)$ parametrised by $\lambda \in [0, \infty)$ as follows:

$$ u_{\lambda}(x, t) = \begin{cases} u_l(x) & \text{if } x < 0, \\ u_r(x) & \text{if } 0 \le x < \gamma(t - \lambda), \\ 0 & \text{if } \gamma(t - \lambda) \le x. \end{cases} $$

□

Hence, the sufficient condition for blow-up as established in Theorem 2.8 is also a sufficient condition for the non-uniqueness of entropy solutions. Kružkov's 'doubling of variables' argument [18, Theorem 1] works only in the $L^\infty$ setting.

## 3 Front tracking approximations

Despite the pathologies uncovered in the preceding sections, we can select a unique entropy solution of (2) as the limit of front-tracking approximations, suitably defined. Recall that $G = \{x : g(x) = 0\}$ is a closed set, by the continuity of $g$ and therefore the complement of $G$ is a countable union of disjoint open intervals, say $G^c = \cup(a_i, b_i)$ where the possibility that $a_i = -\infty, b_i = +\infty$ are not excluded. By the assumption (B), we also know that these boundary points cannot accumulate.

$L^\infty$ bounds fail to hold without coercivity, so numerical schemes such as [23] break down (see section 2.4). Hence, we employ a novel front tracking scheme to demonstrate the well-posedness of (2). Consider a flux of the form (5) such that $g$ is supported on an interval $(a, b)$, possibly with $a = -\infty$ or $b = +\infty$. Without loss of generality we assume that $g > 0$ here, the case $g < 0$ can be analysed in similar fashion.

Since the flux is uniformly zero outside the domain $(a, b)$, the solution is stationary regardless of initial datum, and thus we can restrict ourselves to the case of $u_0$ supported in $(a, b)$ without any loss of generality. We can demonstrate the existence of solutions by means of a heterogeneous front-tracking scheme as developed in [24], since $f$ is strictly convex on the open interval and uniformly convex on compact sets within it. The steps are detailed as follows:

1. Approximate $\mathcal{F}(x, u) = \operatorname{sgn}(u)f(x, u)$ by $\delta$-approximate piecewise constant functions $\mathcal{F}^\delta$, which converge to $\mathcal{F}$ in $L^1_{\text{loc}}(\mathbb{R})$ as $\delta \to 0$.

2. For each $\delta$, solve the Riemann problems at the initial time either as Rankine-Hugoniot shocks or $\delta$-fans [24, Definition 1].

3. Whenever interactions occur, the forward solution is a Rankine-Hugoniot shock in all cases [24, Theorem 2.4].

The only additional difficulty in this case is handling the interfaces where $g$ vanishes. Since we will work locally for the front tracking, it is enough to consider the case where $b = +\infty$ and, without loss of generality, $a = 0$. Thus in this case $g$ is a positive function on the positive real numbers. Set

$$ u_0(x) = \begin{cases} 0 & \text{if } x < 0, \\ u_r(x) & \text{if } x \ge 0, \end{cases} \qquad (28) $$
where $u_r$ is a differentiable function such that $x \mapsto f(x, u_r(x))$ is constant, say equal to $f_r$. For any positive number $f_r \in \mathbb{R}$ we can find exactly two such functions satisfying our requirement, one positive and the other negative, by the implicit function theorem. If $f_r = 0$, then $u_r \equiv 0$ is the only such function. These are precisely the stationary solutions corresponding to the flux [24, Lemma 2.1]. We have the following important lemma which assures us that $u_0$ is locally integrable.

**Lemma 3.1.** *When the flux satisfies (V) and (CG), generalised Riemann data of the form (28) are integrable near the interface. More generally, if $x \mapsto f(x, u_0(x))$ is bounded, then $u_0$ is locally integrable.*

*Proof.* Let $f_r > 0$ and consider the negative stationary solution $u_r$ associated with $f_r$. Then, since $f(x, u_r(x)) = f_r$ for all positive $x$, we have that

$$h(u_r(x)) = \left( \frac{f_r}{g(x)} \right).$$

Now, if $p$ is the order of vanishing for $g$ in (V), then near zero $g$ behaves like $x \mapsto x^p$, while by (CG), we have that $|y| \le h((C^{-1}y)^{1/(p+\varepsilon)})$ for $|y|$ large enough. Hence, for some constants $\tilde{C} > 0$ and $\theta \in (0, 1)$, we have that

$$u_r(x) \le \frac{\tilde{C}}{x^{1-\theta}},$$

and thus the singularity at $x = 0$ is integrable. Since $u_0$ is uniformly bounded outside a neighbourhood of 0 if $f$ is, the local integrability of more general functions supported on the positive real numbers trivially follows. $\square$

In section 4.1 we demonstrate that the growth exponent condition on $h$ is sharp with respect to the behaviour of $g$ near its zeroes.

We claim that if $u_0 < 0$, then the stationary solution $u(x, t) = u_0(x)$ is an entropy solution of the Cauchy problem (2) associated with initial datum (28). It is trivial to see that, away from the interface, $u$ is a classical solution of (2). Let $k \in \mathbb{R}$ be given. Now, $u_0$ blows up near $x = 0$; in particular, $u_0(x) \to -\infty$ as $x \to 0$. Hence, there exists $N > 0$ such that $u_0(x) < k$ for all $x \in (0, N)$. Taking $\varepsilon \in (0, N)$ arbitrarily small, and noting that $u(x, t)$ is a classical solution of (2) for $x > 0$, we have that for any positive test function $\varphi$,

$$\begin{aligned} L &= \iint_{\mathbb{R}_+^2} |u-k| \varphi_t + \operatorname{sgn}(u-k) [f(x,u)-f(x,k)] \varphi_x dxdt \\ &= \int_0^\varepsilon \int_0^\infty |u-k| \varphi_t + \operatorname{sgn}(u-k) [f_r-f(x,k)] \varphi_x dxdt \\ &\quad + \int_\varepsilon^\infty \int_0^\infty |u-k| \varphi_t + \operatorname{sgn}(u-k) [f(x,u)-f(x,k)] \varphi_x dxdt. \end{aligned}$$

Since $u(x, t) = u_0(x)$ is a classical solution of (2) for $x \ge \varepsilon$, we can apply integration by parts on the second integral. Now, $u_0(\varepsilon) < k$, hence $\operatorname{sgn}(u-k) < 0$ on this boundary, and thus

$$\begin{aligned} L &= \int_0^\varepsilon \int_0^\infty |u-k| \varphi_t + \operatorname{sgn}(u-k) [f_r - f(x,k)] \varphi_x dxdt \\ &\quad + \int_0^\infty [f_r - f(\varepsilon,k)] \varphi(\varepsilon,t) dt \\ &\quad + \int_\varepsilon^\infty \int_0^\infty \operatorname{sgn}(u-k) f_x(x,k) \varphi dxdt - \int_\varepsilon^\infty |u_0(x)-k| \varphi(x,0) dx. \end{aligned}$$
Now, as $\varepsilon \to 0$, the first integral vanishes by the local integrability of the integrand, and the second integral approaches

$$f_r \int_0^\infty \varphi(0, t)dt,$$

since $f(\varepsilon, k) \to 0$ as $\varepsilon \to 0$ for any fixed $k \in \mathbb{R}$. Since $f_r > 0$, the test function $\varphi$ and real number $k$ were arbitrarily chosen and the integrand at the initial time is non-negative even for $x < 0$, we can conclude that (11) holds; the stationary solution $u(x, t) = u_0(x)$ is an entropy solution of (2) with initial datum (28) when $u_0 < 0$, and the same computation shows that this is no longer true when $u_0 > 0$.

In order to avoid the trouble with positive stationary solutions, we ensure that our initial discretisation $\mathcal{F}^\delta$ is always 0 in a small neighbourhood of the origin, which vanishes as the discretisation parameter $\delta \to 0$. Then, fronts only interact with the interface to produce entropic interfaces. However, note that solutions satisfying the entropy condition (11) need not satisfy the weak formulation of (2); if they happen to be uniformly bounded in addition to satisfying the entropy condition, then we can say that they are weak solutions by considering $k = \pm \|u\|_{L^\infty}$, but not otherwise.

## 3.1 Flux with discrete zero set

Since solutions can be analysed in each maximal open interval, it is sufficient to generalise the previous case where $G$ is a singleton, say $G = \{0\}$, by working within each maximal open domain where $g$ is non-negative. If $f$ satisfies the structural constraint with respect to all the zeroes in $G$, then the Cauchy problem (2) has locally integrable solutions for flux-bounded initial data. More precisely, we have the following result:

**Theorem 3.2.** Let $f$ be given by (5) satisfy our assumptions and structural constraint. Fix $u_0 \in L_{\text{loc}}^1(\mathbb{R})$ such that the function $\theta(x) = \text{sgn}(u_0(x))f(x, u_0(x))$ is locally of bounded variation. Then, the Cauchy problem (2) has an entropy solution in the sense of (11) globally in time.

## 3.2 Uniqueness and stability

We adopt Kružkov's 'doubling of variables' technique in order to prove the uniqueness and stability of solutions. The main difficulty arises in dealing with merely integrable rather than uniformly bounded functions. We say that $u$ satisfies the interface condition on $G \times (0, T)$ if for each maximal interval $(a, b) \subseteq G^c$, we have one of the following:

1. If $g > 0$ in $(a, b)$ then $u(a+, t) < +\infty, u(b-, t) > -\infty$ for $t \in (0, T)$.

2. If $g < 0$ in $(a, b)$ then $u(a+, t) > -\infty, u(b-, t) < +\infty$ for $t \in (0, T)$.

Our front tracking algorithm seems to indicate that the domain of dependence for solutions is localised within each maximal interval where $g$ is non-zero. Hence, limits of front tracking approximations all satisfy this interface condition. Furthermore, non-uniqueness involves characteristics emanating from the degenerate interface at positive times. Thus, we expect that such a Lax-type condition on the solution should allow us to recover uniqueness. The following theorem confirms this intuition.

**Theorem 3.3 (Uniqueness and stability).** Let $u, v$ be entropy solutions of (2) in a domain containing $\Omega_T$ satisfying the interface condition on $G \times (0, T)$ with respective initial values
$u_0, v_0$. Then, for any $a, b \in \mathbb{R}$ such that $a < b$ with $g(a) = g(b) = 0$ and $g(x) \neq 0$ for $x \in (a, b)$, we have that for all $T > 0$:

$$ \int_a^b |u(x, T) - v(x, T)| dx \le \int_a^b |u_0(x) - v_0(x)| dx. \quad (29) $$

*Proof.* Our method of proof is to use the following inequality for positive test functions $\varphi$ supported in the interior of $\Gamma_T := [a, b] \times [0, T]$.

$$ -\iint_{\Gamma_T} |u - v| \varphi_t + \operatorname{sgn}(u - v) [f(x, u) - f(x, v)] \varphi_x dx dt \le 0. \quad (30) $$

Let $S$ denote the compact support of $\varphi$. Since $f \circ u, f \circ v$ are bounded, $u, v \in L^\infty(S)$ and thus (30) follows from the arguments of Kružkov [18, Theorem 1], which only requires regularity of the flux and boundedness of the entropy solutions. Now we choose a special family of test functions $\varphi_\varepsilon$ as follows: let $\eta_\varepsilon$ denote the standard, compactly supported approximations of identity, and let $\delta_\varepsilon$ denote the function

$$ \delta_\varepsilon(x) = \int_{-\infty}^{x} \eta_\varepsilon(z) dz, $$

so that it is strictly increasing, $\delta_\varepsilon(y) = 0$ for $y \le -\varepsilon$, and $\delta_\varepsilon(y) = 1$ for $y \ge \varepsilon$. Then, for $0 < s < r < T$ and $\varepsilon > 0$ small enough, define

$$ \varphi_{\varepsilon}(x, t) = [\delta_{\varepsilon}(t-s) - \delta_{\varepsilon}(t-r)] [\delta_{\varepsilon}(x-(a+2\varepsilon)) - \delta_{\varepsilon}(x-(b-2\varepsilon))], $$

i.e. $\varphi_\varepsilon \in C_c^\infty$ approximates the characteristic function of the rectangular domain $(a, b) \times (s, r)$. We can write this as $\varphi_\varepsilon(x, t) = \tau_\varepsilon(t)\psi_\varepsilon(x)$, and note that $\tau_\varepsilon, \psi_\varepsilon$ approximate the characteristic functions of the intervals $[s, r]$ and $[a, b]$ respectively. Now, for each $\varepsilon > 0$, the entropy solutions $u, v$ are uniformly bounded on the support of $\varphi_\varepsilon$. Since the integrand in (30) consists of uniformly bounded terms for any fixed $\varepsilon > 0$ small enough, we can rewrite the inequality as follows:

$$ \begin{aligned} & \int_{r-\varepsilon}^{r+\varepsilon} \eta_{\varepsilon}(t) \int_{a}^{b} |u-v| \psi_{\varepsilon}(x) dx dt - \int_{s-\varepsilon}^{s+\varepsilon} \int_{a}^{b} |u-v| \psi_{\varepsilon}(x) dx dt \\ & \le \int_{a+\varepsilon}^{a+3\varepsilon} \eta_{\varepsilon}(x) \int_{s-\varepsilon}^{r+\varepsilon} \operatorname{sgn}(u-v) [f(x,u) - f(x,v)] \tau_{\varepsilon}(t) dt dx \\ & \quad - \int_{b-3\varepsilon}^{b-\varepsilon} \eta_{\varepsilon}(x) \int_{s-\varepsilon}^{r+\varepsilon} \operatorname{sgn}(u-v) [f(x,u) - f(x,v)] \tau_{\varepsilon}(t) dt dx. \end{aligned} \quad (31) $$

Now, if $u \ne v$, then

$$ \operatorname{sgn}(u - v)[f(x, u) - f(x, v)] = |u - v| \frac{f(x, u) - f(x, v)}{u - v}, $$

so by the interface condition on $u, v$ and the mean value theorem, the right-hand side of (31) must converge to a non-positive quantity as $\varepsilon \to 0$, since all the other terms in the integrand(s) are positive. Taking $\varepsilon \to 0$ on the left-hand side as well and letting $s \to 0, r \to T$, we obtain (29) as required, which completes the proof. $\square$

Hence, although entropy solutions of (2) may not be unique, the limits of front-tracking approximations are. We remark that Theorem 3.3 also allows us to extend the well-posedness theory to any initial data $u_0$ such that $f \circ u_0$ is merely uniformly bounded.
# 4 Explicit examples

We solve some more initial value problems explicitly, and demonstrate that our sufficient conditions for $L^\infty$ blow-up is sharp.

## 4.1 Heterogeneity with vanishing derivative

To see that the derivative condition in Theorem 2.8 is sharp, consider the flux

$$f(x, u) = x^2 u^2. \qquad (32)$$

The characteristic ODEs for (2) reduce in this case to

$$\dot{q}(t) = 2q(t)^2p(t), \qquad (33)$$

$$\dot{p}(t) = -2q(t)p(t)^2. \qquad (34)$$

If $q(0) = 0$ or $p(0) = 0$, then $q(t) = q(0)$ and $p(t) = p(0)$. This equation (34) is no longer autonomous; the system is now genuinely coupled. However, since $f$ is invariant along trajectories, we know that $q(t)p(t) = q(0)p(0)$. Hence, the equations reduce to

$$\dot{q}(t) = 2q(0)p(0)q(t), \qquad (35)$$

$$\dot{p}(t) = -2q(0)p(0)p(t). \qquad (36)$$

If we denote $C = 2q(0)p(0)$, then the system of equations (35), (36) have explicit solution

$$q(t) = q(0) \exp(Ct), \qquad (37)$$

$$p(t) = p(0) \exp(-Ct), \qquad (38)$$

and thus neither the characteristic trajectory nor the value can blow up in finite time. Note that $f$ as defined in (32) does not satisfy our structural assumption, however, and thus stationary solutions are not integrable. This is easily seen as follows. Let $f = 1$, and consider the positive stationary solution $u$ associated with it. Then (to rephrase)

$$u(x, t) = \sqrt{\frac{1}{x^2}} = \frac{1}{|x|},$$

which is clearly not integrable. Hence, this also shows that the exponent in our structural constraint is sharp. If $u_0 \in L^\infty$, however, we can avoid this issue since we have explicit control on the $L^\infty$ norm. At the front tracking level, we choose our $\delta$-approximate piecewise stationary solution to be zero in a small neighbourhood of the origin, ensuring that we remain in the $L^\infty$ and hence $L^1_{\text{loc}}$ class.

## 4.2 Ill-posedness

If we relax the constraint that our flux be of multiplicative form, the blow-up can be even more pathological. Consider the following spatially heterogeneous flux for (2)

$$f(x, u) = xu^2 + u^4. \qquad (39)$$

The equations (7) then reduce to

$$\dot{q}(t) = 2q(t)p(t) + 4p(t)^3, \qquad (40)$$

$$\dot{p}(t) = -p(t)^2. \qquad (41)$$
Note that (41) is still an autonomous Riccati equation that may blow up in finite time and is explicitly given by $p(t) \equiv 0$ if $p(0) = 0$, and

$$p(t) = \frac{1}{t + p(0)^{-1}}$$

if $p(0) \neq 0$. Consider the constant initial datum $u_0 \equiv -1$. By the method of characteristics, a smooth solution exists locally in time. Given the explicit form of $p(t)$, we can compute $q(t)$ for any value of $q(0)$ by conservation of $f$ along trajectories (40)-(41). In particular, we have that

$$q(t)p(t)^2 + p(t)^4 = q(0)p(0)^2 + p(0)^4,$$

and hence, for $p(0) \neq 0$,

$$ \begin{aligned} q(t) &= p(t)^{-2} (q(0)p(0)^2 + p(0)^4 - p(t)^4) \\ &= \frac{q(0) + 1}{p(t)^2} - p(t)^2. \end{aligned} \qquad (42) $$

Since $p(0) = -1$ for all characteristics, we can write (42) as

$$q(t) = (1 - t)^2 (q(0) + 1) - \frac{1}{(1 - t)^2},$$

and thus $q(t)$ blows up as $t \to 1$ as well. Now, it is clear that for $t < 1$, no two characteristics can intersect, and that the value of $u(x, t)$ for any fixed time level is constant in $x$. Hence, the Cauchy problem corresponding to this flux is ill-posed globally in time – there is no way to continue the solution beyond $t = 1$, even if we look for merely locally integrable functions.

As with the construction of non-unique solutions, this ill-posedness result can be generalised to arbitrary spatial dimensions by a simple extension. In particular, the following Cauchy problem is globally ill-posed, for any arbitrary choice of functions $f_i$:

$$ \begin{aligned} u_t + (x_1 u^2 + u^4)_{x_1} + \sum_{i=2}^{n} f_i(u)_{x_i} &= 0, \qquad (43) \\ u(x, 0) &= -1. \end{aligned} $$

The ill-posedness of (43) is a trivial consequence of the result for (39).

# 5 Acknowledgements

SSG and PV would like to thank the Department of Atomic Energy, Government of India, for their support under project no. 12-R&D-TFR-5.01-0520.

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