Lagrangian Representation for Systems of Conservation Laws: An Overview

Abstract

We present an overview on some recent works in collaboration with S. Bianchini (see Bianchini and Modena in Lagrangian representation for solution to general systems of conservation laws [9] and the Ph.D. thesis Modena in Interaction functionals, Glimm approximations and Lagrangian structure of BV solutions for hyperbolic systems of conservation laws [15]), in which we propose a way to describe BV solutions to hyperbolic systems of conservation laws in one space dimension from a Lagrangian point of view.

1 Introduction

One of the key observations in fluid dynamics is that the fluid flow can be described from two different (and in some sense complementary) points of view: the Lagrangian points of view (in which the trajectory in space–time of each single fluid particle is tracked) and the Eulerian point of view (in which one looks at fluid motion focusing on fixed locations in the space through which the fluid flows as time passes).

From a mathematical perspective, such duality between the Lagrangian and the Eulerian approach can be seen, for instance, in the framework of the continuity equation:

$$\begin{aligned} \begin{aligned}&{\left\{ \begin{array}{ll} \partial _t v(t,x) + \mathrm {div}_x \big (v(t,x) b(t,x) \big ) &{}= 0, \\ v(0,x) &{}= \bar{v}(x), \end{array}\right. } \\ \end{aligned} \end{aligned}$$

(1)

where $v: [0, \infty ) \times \mathbb R^d \rightarrow \mathbb R$ is the unknown and $b: [0, \infty ) \times \mathbb R^d \rightarrow \mathbb R^d$ is a given vector field. It is well known that, under suitable regularity assumptions, the solution to (1) can be written, for any time $t \in [0, \infty )$, as

$$\begin{aligned} v(t, \cdot ) \mathscr {L}^1 = \mathtt X(t)_\sharp \big ( \bar{v} \mathscr {L}^1 \big ), \end{aligned}$$

(2)

where $\mathtt X: [0, \infty ) \times \mathbb R^d \rightarrow \mathbb R^d$ is the flow generated by the ODE

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{\partial \mathtt X}{\partial t}(t,y) &{}= b(t, \mathtt X(t,y)), \\ \mathtt X(0,y) &{}= y, \end{array}\right. } \end{aligned}$$

(3)

$\mathscr {L}^d$ is the Lebesgue measure on $\mathbb {R}^d$ and $\sharp$ denotes the push-forward in the sense of measures.¹

In the framework of the continuity equation, the Lagrangian and the Eulerian approach help each other: For instance, in the smooth setting, one can use the ODE (3) (Lagrangian approach) to solve the PDE (1) (Eulerian approach), while in the non-smooth setting one can use the PDE to solve the ODE (see [13]). The duality between the two approaches can be used not only to prove the existence of solutions, but also to prove their uniqueness and their stability and to investigate further properties of them, like their fine structure, their regularity, and so on. In few words, we could say that two is better than one: what cannot be done using the Lagrangian approach could be hopefully done using the Eulerian one, and vice versa.

For these reasons, it is an interesting question whether systems of conservation laws

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u + \partial _x F(u) &{} = 0, \\ u(0, x) &{} = \bar{u}(x), \end{array}\right. } \quad u = u(t,x) \in \mathbb {R}^n, \ t \ge 0, \ x \in \mathbb {R}, \ \end{aligned}$$

(4)

can be analyzed from a Lagrangian point of view. Here, $F: \mathbb {R}^n \rightarrow \mathbb {R}^n$ is a generic smooth function, which is only assumed to be strictly hyperbolic; i.e. its differential DF(u) has n distinct real eigenvalues in each point of its domain. We restrict our analysis to one space dimension, since this is the setting where a satisfying well-posedness theory for entropic solutions is available.²

For a scalar conservation law with a smooth initial datum, the method of characteristics provides a reasonable Lagrangian approach to the problem. Such method was extended by C. Dafermos (through the notion of generalized characteristics in [12]) to systems whose characteristic fields are either genuinely nonlinear or linearly degenerate,³ and to initial data which are just BV. However, Dafermos’ approach can not be further generalized to systems where the flux F has no convexity properties.

Another Lagrangian approach in the analysis of conservation laws was proposed by T.-P. Liu in [14], where he introduced the notion of wave tracing for the waves present in an approximate solution to the system (4), constructed by means of the Glimm scheme. However, in [14], only approximate solutions (which in some sense are just piecewise constant functions) are considered.

Recently, some papers appeared in which a Lagrangian analysis is developed for the exact (and not approximate) entropic solution to conservation laws with a flux F which does not satisfy any convexity assumption. In particular,

• in [4, 5] (see also for a previous, slightly different approach [10]) S. Bianchini and E. Marconi develop a Lagrangian approach for the solution to the Cauchy problem associated to a scalar conservation law ($n=1$), whose flux $F: \mathbb {R}\rightarrow \mathbb {R}$ is any smooth function and whose initial datum $\bar{u} \in L^\infty (\mathbb {R})$ is any bounded function;

• in [9, 15] S. Bianchini and the author develop a Lagrangian approach for the solution to the Cauchy problem associated to a system of conservation laws ($n \ge 1$), whose flux is any smooth strictly hyperbolic function and whose initial datum $\bar{u} \in BV(\mathbb {R})$ is a function of bounded variation.

In both cases, the starting point is the analysis of BV entropic solutions to scalar conservation laws. The extension to $L^\infty$ initial data (for the scalar equation) [4, 5] or to systems [9, 15] requires, however, several new ideas. The goal of this notes is to present the notion of Lagrangian representation for BV entropic solutions to systems of conservation laws (4), proposed in [9, 15], and to present the main ideas behind the construction of such Lagrangian representation, focusing in particular on the difficulties in extending the scalar BV analysis to the system case.

As a final remark, we would like to stress that both in the scalar case and in the system one, the Lagrangian analysis is done in the same setting in which the well-posedness of the Cauchy problem is already know. We do not want to use Lagrangian methods to prove such well-posedness again. Rather, the aim of our new Lagrangian tools is to analyze in a more precise way the solution u to the Cauchy problem (4), in order to prove further properties of it. As an example, in the scalar case, the Lagrangian approach can be used to prove the concentration of entropies (see the papers by Bianchini and Marconi [4, 5]); in the system case, the Lagrangian tools can be used to study the fine structure of the solution (see the paper by Bianchini and the author [9] and the Ph.D. thesis of the author [15]).

2 Analysis of BV Solutions to Scalar Conservation Laws

The starting point of our analysis is the study of entropic BV solutions to scalar conservation laws

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u + \partial _x F(u) = 0, \\ u(0, x) = \bar{u}(x), \end{array}\right. } \quad \bar{u} \in BV(\mathbb {R}) \text { with compact support,} \ F: \mathbb {R}\rightarrow \mathbb {R}\text { smooth}. \end{aligned}$$

(5)

The first question we have to answer is: What is a good notion of Lagrangian representation of the solution to (5)? A hint in this direction is given by the following observation: By Vol’pert’s rule⁴ the distributional derivative $\partial _x u$ (which is a measure, being $u \in BV$) satisfies the 1D continuity equation

$$\begin{aligned} \partial _t (\partial _x u) + \partial _x(\hat{\lambda }(t,x) \partial _x u) = 0 \ \text { in a distributional sense}, \end{aligned}$$

(6)

where

$$\begin{aligned} \hat{\lambda }(\bar{t}, \bar{x}) := {\left\{ \begin{array}{ll} F'(u(\bar{t},\bar{x})) &{} \text { if } x \mapsto u(\bar{t},x) \text { is continuous at } \bar{x}, \\ \frac{F(u(\bar{t}, \bar{x}+)) - F(u(\bar{t}, \bar{x}-))}{u(\bar{t}, \bar{x}+) - u(\bar{t}, \bar{x} -)} &{} \text { if } x \mapsto u(\bar{t},x) \text { has a jump at } \bar{x}. \end{array}\right. } \end{aligned}$$

(7)

Mimicking (1)–(2)–(3), we give the following definition.

Definition 1.

A Lagrangian representation for the entropic solution u to (5) is a triple $(W, \mathtt X, \rho )$, where

• 1. $W \subseteq \mathbb {R}$ is a bounded interval; its elements are denoted by w and are called waves;

• 2. $\mathtt X: [0, \infty ) \times W \rightarrow \mathbb {R}$ is $\Vert F'\Vert _{L^\infty }$-Lipschitz in t for fixed w and increasing in w for fixed t, and it is called flow or position function;

• 3. $\rho : W \rightarrow [-1, 1]$ is called density function,

such that for a.e. time $t \in [0, \infty )$

$$\begin{aligned} \partial _x u(t, \cdot ) = \mathtt X(t, \cdot )_\sharp \big ( \rho \mathscr {L}^1|_{W}) \text { in the sense of measures} \end{aligned}$$

(8)

and

$$\begin{aligned} \frac{\partial \mathtt X}{\partial t}(t,w) = \hat{\lambda }(t, \mathtt X(t,w)) \text { for }|\rho | \mathscr {L}^1-\text {a.e.} w \in W. \end{aligned}$$

(9)

Remark 1.

Notice that (8) is the analog of (2) and (9) is the analog of (3); only two differences must be observed:

• In (8), the term which is transported is an absolute continuous measure w.r.t. $\mathscr {L}^1$, even if the initial datum $\partial _x \bar{u}$ has a jump part or a Cantor part;

• In (9), in general $\mathtt X(0, w) \ne w$; i.e., w is just the label of a particle with no relationship with its starting point.

Definition 1 provides a (hopefully) good notion of Lagrangian representation. How can we now explicitly construct the objects $W, \mathtt X, \rho$ satisfying the properties above?

As usual in the theory of conservation laws, the idea is to consider a sequence of approximate solutions $(u^q)_{q \in \mathbb {N}}$ solving the approximate Cauchy problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u^q + \partial _x F^q(u^q) = 0, \\ u^q(0, x) = \bar{u}^q(x), \end{array}\right. } \end{aligned}$$

(10)

where $F^q$ is the piecewise affine interpolation of F with grid size $2^{-q}$ and $\bar{u}^q$ is a piecewise constant function taking values in $2^{-q}\mathbb {Z}$ such that $\Vert \bar{u}^q - \bar{u}\Vert _{L^1} \rightarrow 0$ as $q \rightarrow \infty$. The solution $u^q$ to (10) can be constructed by means of the wavefront tracking algorithm (see [11, Chap. 4]), and it is a piecewise constant function with values in $2^{-q}\mathbb {Z}$ for any time t which converges strongly in $L^1$ to the entropic solution of (5), as $q \rightarrow \infty$.

Since $u^q(t, \cdot )$ is piecewise constant, it is not difficult to construct by hand a Lagrangian representation of it.⁵ Now, the family $\{\mathtt X^q\}_q$ is pre-compact in $L^1([0, \infty ) \times \mathbb {R})$, since, by Definition 1, each $\mathtt X^q$ is $\Vert F'\Vert$-Lipschitz in t for fixed w and increasing in w for fixed t; the family $\{\rho ^q\}$ is weakly$^*$ pre-compact in $L^\infty (W)$. Therefore, up to subsequences, $\mathtt X^q \rightarrow \mathtt X$ strongly in $L^1$ and $\rho ^q \rightarrow \rho$ weakly$^*$ in $L^\infty$.

Equation (8) is then easily obtained passing to the limit in the corresponding equation for approximations

$$\begin{aligned} \partial _x u^q(t) = \mathtt X^q(t)_\sharp \big ( \rho ^q \mathscr {L}^1|_{W}) \end{aligned}$$

(11)

and using that $u^q \rightarrow u$ in $L^1$.

On the contrary, Eq. (9) cannot be deduced directly from the corresponding equation for the approximations

$$\begin{aligned} \partial _t \mathtt X^q(t,w) = \hat{\lambda }^q(t, \mathtt X^q(t,w)), \end{aligned}$$

(12)

since, in general, for fixed t, $\hat{\lambda }^q(t) \circ \mathtt X^q(t) \nrightarrow \hat{\lambda }(t) \circ \mathtt X(t)$, as the following example shows.

Example 1.

Assume that u is a solution of the scalar conservation law $\partial _t u + \partial _x F(u) = 0$, taking values in the finite set $\{u^L, u^M, u^R\}$, with $u^L, u^M, u^R \in \mathbb {R}$ and $u^L< u^M < u^R$, as described in Fig. 1.

Fig. 1

The solution u(t, x) to the scalar conservation law $\partial _t u + \partial _x F(u) = 0$ and the function F(u)

Assume that the sequence of approximations $(u^q)_q$ is given by $u^q(t,x) := u(t - 1/q, x)$. Notice now that at time $\bar{t}$, $u^q(\bar{t}, \cdot )$ is made by two consecutive jumps, while $u(\bar{t}, \cdot )$ is made by a single jump, which is given, roughly speaking, by the juxtaposition of the two jumps in the approximations.

By (9), all the waves w located in $x_1^q$ have speed

$$\begin{aligned} \sigma _1 := \partial _t \mathtt X^q(\bar{t},w) = \hat{\lambda }^q (\bar{t}, x_1^q) = \frac{F(u^M) - F(u^L)}{u^M - u^L}, \end{aligned}$$

all the waves w located in $x_2^q$ have speed

$$\begin{aligned} \sigma _2 := \partial _t \mathtt X^q(\bar{t},w) = \hat{\lambda }^q (\bar{t}, x_2^q) = \frac{F(u^R) - F(u^M)}{u^R - u^M}, \end{aligned}$$

while in the exact solution all the waves w should have speed

$$\begin{aligned} \sigma := \partial _t \mathtt X(\bar{t},w) = \hat{\lambda }(\bar{t}, \bar{x}) = \frac{F(u^R) - F(u^L)}{u^R - u^L}. \end{aligned}$$

Unfortunately, in general $\sigma _1, \sigma _2 \ne \sigma$ and thus $\hat{\lambda }^q(\bar{t}) \circ \mathtt X^q(\bar{t}) \nrightarrow \hat{\lambda }(\bar{t}) \circ \mathtt X(\bar{t})$ as $q \rightarrow \infty$.

To overcome this problem and recover (9), we can proceed as follows (the argument is taken from [4]). From (6) and (8), we get for every $\varphi \in C^\infty _c((0, \infty ) \times \mathbb {R})$

$$\begin{aligned} \begin{aligned} \iint u \partial _t \partial _x \varphi dxdt&= \iint \partial _x \varphi (t,x) \hat{\lambda }(t,x) \partial _x u(t, dx) dt = \iint \partial _x \varphi (t, \mathtt X(t,w)) \hat{\lambda }(t, \mathtt X(t,w)) \rho (w) dwdt. \end{aligned} \end{aligned}$$

On the other side, testing (8) against $\partial _t \varphi$, we get

$$\begin{aligned} \begin{aligned} \iint u \partial _t \partial _x \varphi dxdt&= - \iint \partial _t \varphi (t, \mathtt X(t, w)) \rho (w) dwdt = \iint \partial _x \varphi (t, \mathtt X(t,w)) \partial _t \mathtt X(t,w) \rho (w) dwdt. \end{aligned} \end{aligned}$$

Therefore,

$$\begin{aligned} \partial _x \bigg [ \mathtt X(t)_\sharp \Big ( \rho (\hat{\lambda }(t, \mathtt X(t, \cdot )) - \partial _t \mathtt X(t, \cdot ))\mathscr {L}^1|_W \Big ) \bigg ] = 0. \end{aligned}$$

Equation (9) follows, just observing that $\mathtt X(t, \cdot )$ takes values in a compact set and that $\partial _t \mathtt X(t,w)$ is constant on waves having the same position (since $w \mapsto \mathtt X(t,w)$ is increasing).

3 Analysis of Linear Systems of Conservation Laws

We wish now to extend the scalar analysis done in the previous section to the system case. As a first step in this direction, let us study the linear system of conservation laws

$$\begin{aligned} \partial _t u + A \partial _x u = 0, \quad \text {where } A \text { is a } n \times n \text { strictly hyperbolic matrix,} \end{aligned}$$

(13)

together with an initial datum $u(0, \cdot ) = \bar{u} \in BV(\mathbb {R})$.

Let $\lambda _1, \dots , \lambda _n$ be the n distinct real eigenvalues of A, $r_1, \dots , r_n$ be the right eigenvectors (i.e., $A r_k = \lambda _k r_k$) normalized such that $|r_k| = 1$, $l_1, \dots , l_n$ be the left eigenvectors (i.e., $l_k A = \lambda _k l_k$), normalized such that $l_k \cdot r_h = \delta _{kh}$.

Our aim is to find a good definition of Lagrangian representation for the solution to the linear system (13) and to explicitly construct such Lagrangian representation. This is easily done, observing that the scalar product of (13) with $l_k$ gives the n scalar equations $\partial _t (l_k \cdot u) + \lambda _k \partial _x (l_k \cdot u) = 0$, with constant field $\lambda _k$.

Therefore, by the analysis in Sect. 2, for each k we can find a set $W_k$ (called the set of k-waves), a flow $\mathtt X_k: [0, \infty ) \times W_k \rightarrow \mathbb {R}$ and a density $\rho _k: W_k \rightarrow [-1,1]$, as in Definition 1, such that

$$\begin{aligned} \partial _x ( l_k \cdot u ) = \mathtt X_k(t)_\sharp \big ( \rho _k \mathscr {L}^1|_{W_k} \big ) \end{aligned}$$

and

$$\begin{aligned} \partial _t \mathtt X_k(t, w) \equiv \lambda _k. \end{aligned}$$

(14)

Definition 2.

A Lagrangian representation of the solution to the linear system (13) is thus defined as a family of n triples $(W_k, \mathtt X_k, \rho _k)$, $k=1, \dots , n$, (with the same regularity properties as the ones described in Definition 1) such that

$$\begin{aligned} \partial _x u(t) = \sum _{k=1}^n \partial _x (l_k \cdot u) r_k = \sum _{k=1}^n\mathtt X_k(t)_\sharp \big ( \rho _k \mathscr {L}^1|_{W_k})r_k \end{aligned}$$

(15)

and the ODE (14) holds for every $k=1, \dots , n$.

The existence of such a Lagrangian representation for the solution to the linear system (13) is then an immediate consequence of the scalar analysis done in Sect. 2.

4 The Riemann Problem

Before moving to the analysis of the nonlinear system (4), we need to recall some basic facts about the entropic solution to the Riemann problem, i.e., the Cauchy problem (4) with a piecewise constant initial datum

$$\begin{aligned} u(0, x) = \bar{u}(x) = {\left\{ \begin{array}{ll} u^L &{} \text {if } x < 0, \\ u^R &{} \text {if } x \ge 0, \end{array}\right. } \quad \text {with } u^L, u^R \in \mathbb {R}^n \text { close enough to } 0. \end{aligned}$$

(16)

It is shown in [3] that for any $k=1, \dots , n$ it is possible to define a neighborhood $D_k \subseteq \mathbb {R}^{n+2}$ of the point $(0, 0, \lambda _k(0)) \in \mathbb {R}^n \times \mathbb {R}\times \mathbb {R}$ and two functions $\tilde{r}_k : D_k \rightarrow \mathbb {R}^n$, $\tilde{\lambda }_k : D_k \rightarrow \mathbb {R}$; $r_k(u_k, v_k, \sigma _k)$ (resp. $\lambda _k(u_k, v_k, \sigma _k)$) is called the kth generalized eigenvector (resp. the kth generalized eigenvalue) at $(u_k, v_k, \sigma _k) \in \mathbb {R}^n \times \mathbb {R}\times \mathbb {R}$.

It is also shown in [3] that, given $u^L, u^R \in \mathbb {R}^n$ close enough to 0, one can find n curves $\gamma _k : I_k \rightarrow D_k \subseteq \mathbb {R}^{n+2}$, $k=1, \dots , n$, defined on the intervals

$$\begin{aligned} I_k := {\left\{ \begin{array}{ll} \,[0, s_k] &{} \text {if } s_k \ge 0, \\ \,[s_k, 0] &{} \text {if } s_k \le 0, \end{array}\right. } \end{aligned}$$

satisfying the fixed point problem (if $s_k>0$⁶)

$$\begin{aligned} {\left\{ \begin{array}{ll} u_k(\tau ) = u_k^L + \int _0^\tau \tilde{r}_k (u_k(\varsigma ), v_k(\varsigma ), \sigma _k(\varsigma )) d \varsigma , \quad u_k^L = {\left\{ \begin{array}{ll} u^L &{} \text {if } k=1, \\ u_{k-1}(s_{k-1}) &{} \text {if } k>1, \end{array}\right. } \\ v_k(\tau ) = f_k(\tau ) - {{\mathrm{conv}}}_{[0, s_k]} f_k(\tau ), \\ \sigma _k(\tau ) = \frac{d}{d\tau } {{\mathrm{conv}}}_{[0, \sigma _k]} f_k(\tau ), \end{array}\right. } \end{aligned}$$

(17)

with $f_k$ defined by $f_k(\tau ) := \int _0^\tau \tilde{\lambda }_k(u_k(\varsigma ), v_k(\varsigma ), \sigma _k(\varsigma )) d\varsigma$ and the ${{\mathrm{conv}}}_{[a.b]}g$ denotes the convex envelope of a function g on the interval [a, b], i.e., the biggest convex function which stays below g.

The right-continuous solution to the Riemann problem (4), (16) is now given by the BV function

$$\begin{aligned} u(t, x) = {\left\{ \begin{array}{ll} u^L &{} \text {if } x/t \le \sigma _1(0), \\ u_k(\tau ) &{} \text {if } x/t = \sigma _k(\tau ), \\ u^R &{} \text {if } x/t \ge \sigma _n(\tau ). \end{array}\right. } \end{aligned}$$

5 Definition of Lagrangian Representation for Systems

We can now finally move to the analysis of the nonlinear system (4). As in the linear case, let $\lambda _1(u), \dots , \lambda _n(u)$ be the n distinct real eigenvalues of $A(u) := DF(u)$, $r_1(u), \dots , r_n(u)$ (resp. $l_1(u), \dots , l_n(u)$) be the right (resp. left) eigenvectors.

Trying to extend Definition 2 (and, in particular, Eqs. (14), (15)) from the linear to the nonlinear case, the first problem we have to face is that $\lambda _k$ and $r_k$ are not constant anymore, but they depend on u. As in the scalar case (see (7)), we have thus to find a good definition of kth eigenvalue $\hat{\lambda }_k(\bar{t}, \bar{x})$ and kth eigenvector $\hat{r}_k(\bar{t}, \bar{x})$ at a given point $(\bar{t}, \bar{x})$.

If $x \mapsto u(\bar{t}, x)$ is continuous at $\bar{x}$, the natural choice is to set $\hat{r}_k(\bar{t}, \bar{x}):= r_k(u(\bar{t}, \bar{x}))$ and $\hat{\lambda }_k(\bar{t}, \bar{x}):= \lambda _k(u(\bar{t}, \bar{x}))$.

If $x \mapsto u(\bar{t}, x)$ has a jump at $\bar{x}$ between $u^L := u(\bar{t}, x-)$ and $u^R:= u(\bar{t}, x+)$, we solve the Riemann problem $(u^L, u^R)$, defining the curves $(u_k (\cdot ), v_k(\cdot ), \sigma _k(\cdot ))$ as in (17), and we set

Notice that, in the case of a scalar equation, the definition of $\hat{\lambda }(\bar{t}, \bar{x})$ given above coincides with (7).

After this preparation, we can now propose the following definition of Lagrangian representation for the solution to the nonlinear system (4). Compare it with Definitions 1 and 2.

Definition 3.

A Lagrangian representation for the entropic solution u to (4) is a family of n triples $(W_k, \mathtt X_k, \rho _k)$, $k=1, \dots , n$, where

• 1. $W_k \subseteq \mathbb {R}$ is a bounded interval, whose elements are called waves of the kth family; we also assume for simplicity that $W_k \cap W_h = \emptyset$ for $k \ne h$;

• 2. $\mathtt X_k: [0, \infty ) \times W_k \rightarrow \mathbb {R}$ is $\Vert DF\Vert _{L^\infty }$-Lipschitz in t for fixed w and increasing in w for fixed t, and it is called kth flow or kth position function;

• 3. $\rho _k: [0, \infty ) \times W_k \rightarrow [-1, 1]$ is uniformly BV in time for a.e. w, and it is called kth density function;

such that for a.e. $t \in [0, \infty )$

$$\begin{aligned} \partial _x u(t) = \sum _{k=1}^n \mathtt X_k(t)_\sharp \big ( \rho _k(t) \mathscr {L}^1|_{W_k}) \hat{r}_k(t) \text { in the sense of measures} \end{aligned}$$

(18)

and

$$\begin{aligned} \frac{\partial \mathtt X_k}{\partial t}(t,w) = \hat{\lambda }_k(t, \mathtt X_k(t,w)) \text { for }|\rho _k(t)| \mathscr {L}^1 \text {-a.e.} w \in W_k. \end{aligned}$$

(19)

Remark 2.

The main difference between Definition 3 and Definitions 1 and 2 is that the density function $\rho = \rho (t,w)$ is now allowed to be a function of time. This seems strange in comparison with Formula (2) for the continuity equation. However, this dependence on time cannot be avoided: It comes from the well-known fact that nonlinear interactions between wavefronts, taking place at times $t>0$, can create new wavefronts.

Nevertheless, the total amount of created waves can be bounded a priori (see [2]): This implies that the length of the set of waves $W_k$ can be bounded by $C(F) \text { Tot.Var.}(\bar{u})$ and that $\rho$ can be chosen uniformly BV in time for a.e. wave. Here, C(F) is a constant which depends only on F.

6 Construction of a Lagrangian Representation

In Sect. 5, we proposed a possible definition of Lagrangian representation for the entropic solution u to the system (4). In this section, we state the main theorem of these notes, i.e., the existence of such a Lagrangian representation, and we present a sketch of its proof.

Theorem 1.

There exists a Lagrangian representation for the entropic solution to the system (4), in the sense of Definition 3.

Sketch of the proof. The proof follows a path similar to the one we used in the scalar case. We start by taking a sequence of piecewise constant approximate solutions $(u^q)_q$ (constructed through the wavefront tracking algorithm or the Glimm scheme) which converges in $L^1$ to the exact entropic solution u to (4).

For each $u^q$, it is not difficult to construct by hand a Lagrangian representation (as we did for the scalar conservation law in Sect. 2), i.e., for each $k=1, \dots , n$, a set of k-waves $W_k$ (which we assume to be independent of q, without restriction), a flow $\mathtt X_k^q: [0, \infty ) \times W_k \rightarrow \mathbb {R}$ and a density $\rho _k^q: [0, \infty ) \times W_k \rightarrow [-1,1]$ such that:

• for a.e. time t $\partial _x u^q(t) \!=\! \sum _{k=1}^n \mathtt X_k^q(t)_\sharp \big ( \rho _k^q(t) \mathscr {L}^1|_{W_k} \big ) \hat{r}_k^q(t, \cdot )$ i.e. for any $\varphi \in C^\infty _c(\mathbb {R})$,

  $$\begin{aligned} - \int \varphi '(x) u^q(t,x) dx = \sum _{k=1}^n \int _{W_k} \varphi (\mathtt X_k^q(t,w)) \rho _k^q(t,w) \hat{r}_k^q(t, \mathtt X_k^q(t,w)) dw; \end{aligned}$$

  (20)

• for a.e. time t and for $|\rho _k^q|\mathscr {L}^1$ almost every $w \in W_k$

  $$\begin{aligned} \partial _t \mathtt X_k^q(t,w) = \lambda _k^q(t, \mathtt X_k^q(t,w)). \end{aligned}$$

  (21)

Exactly as in the scalar case, the regularity properties of $\mathtt X_k^q$, $\rho _k^q$ imply that there exist $\mathtt X_k: [0, \infty ) \times W_k \rightarrow \mathbb {R}$, $\rho _k : [0, \infty ) \times W_k \rightarrow [-1,1]$ such that, up to subsequences, $\mathtt X_k^q(t) \rightarrow \mathtt X_k(t)$ strongly in $L^1(W_k)$ and $\rho _k^q(t) \rightarrow \rho _k(t)$ weakly$^*$ in $L^\infty (W_k)$, for a.e. time t. To complete the proof of Theorem 1, we have thus to pass to the limit in Formulae (20), (21) to get (18), (19), respectively.

In the scalar case, first we passed to the limit in (11) (corresponding here to (20)) to obtain (8) (corresponding here to (18)); then, we used (8) to prove (9) (corresponding here to Eq. (19)). Example 1 showed that it is not possible to obtain (9) directly passing to the limit in its approximate version (12), because in general $\hat{\lambda }^q(t) \circ \mathtt X^q(t) \nrightarrow \hat{\lambda }(t) \circ \mathtt X(t)$.

In the system case, we cannot repeat the same argument (i.e., first passing to the limit in (20) to get (18) and then use (18) to prove (19)), because in (20) there is already a term $\hat{r}_k^q(t) \circ \mathtt X_k^q(t)$ which most likely does not converge in general to $\hat{r}_k(t) \circ \mathtt X_k(t)$, exactly as $\hat{\lambda }^q(t) \circ \mathtt X^q(t)$ did not converge in general to $\hat{\lambda }(t) \circ \mathtt X(t)$ in the scalar case. We thus need some new ideas to pass to the limit in (20), (21).

Example 1 shows that the are times (as time $\bar{t}$ in that example) for which there is no hope for $\hat{r}_k^q(t) \circ \mathtt X_k^q(t)$ (resp. $\hat{\lambda }_k^q(t) \circ \mathtt X_k^q(t))$ to converge to $\hat{r}_k(t) \circ \mathtt X_k(t)$ (resp. $\hat{\lambda }_k(t) \circ \mathtt X_k(t)$). However, the same example suggests that these times are strong interaction times, i.e., roughly speaking, times when many waves undergo a major change of their speed. For instance, in Example 1, $\hat{\lambda }^q(t) \circ \mathtt X^q(t) \rightarrow \hat{\lambda }(t) \circ \mathtt X(t)$ for every time, except the time $\bar{t}$ where a strong interaction between wavefronts takes place.

The strategy is thus to find a way to identify a priori those times of strong interaction in the solution u, to show that the set of such times has zero Lebesgue measure (or even that it is countable), and to prove that, up to those times, we can pass to the limit in (20), (21).

To identify such bad times, we introduce, for each approximate solution $u^q$, the Radon measure $\mu ^q := \sum _{k=1}^n |\partial _t (\rho ^q \partial _t \mathtt X_k^q)|$, which measure the change of the speed of the waves. Being $u^q$ a piecewise constant function with a finite number of discontinuity lines, $\mu ^q$ is just a finite sum of Dirac’s deltas. For instance, for the configuration described in Example 1, $\mu ^q$ is just a single Dirac’s delta, located in the point $(\bar{t}, \bar{x})$, with size $|\sigma _1 - \sigma | |u^M - u^L| + |\sigma _2 - \sigma ||u^R - u^M|$.

Notice that, by construction of the Lagrangian representation in the approximations, for each $u^q$ the times where waves can change their speed, i.e., times of strong interaction, are exactly those times t for which $\mu ^q(\{t\} \times \mathbb {R}) > 0$.

Next, we prove that there is a Radon measure $\mu$ such that $\mu ^q \rightarrow \mu$ weakly$^*$ in the sense of measures (see Remark 3 below for a comment about the existence of $\mu$).

To conclude the proof, it is now enough to prove that if t is not a time of strong interaction; i.e., by definition, if t is a time such that

$$\begin{aligned} \mu \big ( \{t\} \times \mathbb {R}\big ) = 0 \end{aligned}$$

(22)

(and this happens for all but a countable number of times), then we can pass to the limit in (20), (21) to get (18), (19), respectively. This would conclude the proof of Theorem 1.

Proving this last fact (i.e., passing to the limit in (20), (21)) is a major part of the proof of Theorem 1, which, however, requires the introduction of several ad hoc notations and contains rather technical steps. Therefore, in these notes, it is omitted. We just spend some words about the general strategy.

For each approximate solution $u^q$ at each time t, through a fixed point procedure similar to the one described in Sect. 4 for solving the Riemann problem, we associate to each wave $w \in W_k$, a point

$$\begin{aligned} (\hat{u}^q_k(t,w), \hat{v}^q_k(t,w), \hat{\sigma }^q_k(t,w)) \in \mathbb {R}^n \times \mathbb {R}\times \mathbb {R}, \end{aligned}$$

such that for each time t and each point $x \in \mathbb {R}$ for which $\mathtt X_k^q(t)^{-1}(x) \ne \emptyset$,

and, similarly, for the exact solution u at each time t, we associate to each $w \in W_k$ a point

$$\begin{aligned} (\hat{u}_k(t,w), \hat{v}_k(t,w), \hat{\sigma }_k(t,w)) \in \mathbb {R}^n \times \mathbb {R}\times \mathbb {R}, \end{aligned}$$

such that for each time t and each point $x \in \mathbb {R}$ for which $\mathtt X_k(t)^{-1}(x) \ne \emptyset$,

Similar expressions hold for $\lambda _k^q(t,x)$, $\lambda _k(t,x)$. We then prove that if t is not a time of strong interaction, i.e. if (22) holds, then $\hat{u}_k^q \rightarrow \hat{u}_k$, $\hat{v}_k^q \rightarrow \hat{v}_k$, $\hat{\sigma }_k^q \rightarrow \hat{\sigma }_k$ in some appropriate topologies. Using this fact, we finally show that $\hat{r}_k^q(t) \circ \mathtt X^q(t) \rightarrow \hat{r}_k(t) \circ \mathtt X(t)$ and $\hat{\lambda }_k^q(t) \circ \mathtt X^q(t) \rightarrow \hat{\lambda }_k(t) \circ \mathtt X(t)$, thus concluding the proof of Theorem 1. $\square$

Remark 3.

Proving that the sequence $(\mu ^q)_q$ is weakly$^*$ pre-compact in the sense of measure, i.e., proving that $|\mu ^q| \le C(f, \bar{u})$, where C is a constant which depends on f and the initial datum $\bar{u}$, but not on q, is not trivial at all. It amounts to prove that the total amount of change of speed of the waves present in an approximate solution $u^q$

$$\begin{aligned} \mu ^q \big ([0, \infty ) \times \mathbb {R}\big ) = \sum _{k=1}^n \int _{W_k} \text { Tot.Var.}\big (\rho ^q(\cdot , w) \partial _t \mathtt X_k^q(\cdot , w); [0, \infty ) \big ) dw \end{aligned}$$

is uniformly bounded by $C(f, \bar{u})$. Such estimate is proved in [6,7,8], using a quadratic interaction potential.