Construction of Approximate Entropy Measure-Valued Solutions for Hyperbolic Systems of Conservation Laws

“There is no theory for the initial value problem for compressible flows in two space dimensions once shocks show up, much less in three space dimensions. This is a scientific scandal and a challenge.”

P. D. Lax, 2007 Gibbs Lecture [48].

Abstract

Entropy solutions have been widely accepted as the suitable solution framework for systems of conservation laws in several space dimensions. However, recent results in De Lellis and Székelyhidi Jr (Ann Math 170(3):1417–1436, 2009) and Chiodaroli et al. (2013) have demonstrated that entropy solutions may not be unique. In this paper, we present numerical evidence that state-of-the-art numerical schemes need not converge to an entropy solution of systems of conservation laws as the mesh is refined. Combining these two facts, we argue that entropy solutions may not be suitable as a solution framework for systems of conservation laws, particularly in several space dimensions. We advocate entropy measure-valued solutions, first proposed by DiPerna, as the appropriate solution paradigm for systems of conservation laws. To this end, we present a detailed numerical procedure which constructs stable approximations to entropy measure-valued solutions, and provide sufficient conditions that guarantee that these approximations converge to an entropy measure-valued solution as the mesh is refined, thus providing a viable numerical framework for systems of conservation laws in several space dimensions. A large number of numerical experiments that illustrate the proposed paradigm are presented and are utilized to examine several interesting properties of the computed entropy measure-valued solutions.

Appendices

Appendix 1: Young Measures

We provide here a very short introduction to Young measures. The reader may wish to consult [7, 30] on the theory of Radon measures and probability measures and [2, 3] on the theory of Young measures.

1.1 Probability Measures

A.1.1 We denote by $\mathcal {M}(\mathbb {R}^N)$ the set of finite Radon measures on $\mathbb {R}^N$, which are inner regular Borel measures $\mu$ with finite total variation $|\mu |(\mathbb {R}^N)$. Let $C_0(\mathbb {R}^N)$ be the space of continuous real-valued functions on $\mathbb {R}^N$ which vanish at infinity, equipped with the supremum norm. Then it can be shown (see, e.g., [30, Section 7.3]) that $\mathcal {M}(\mathbb {R}^N)$ can be identified with the dual space of $C_0(\mathbb {R}^N)$ through the pairing $\left\langle \mu , g\right\rangle = \int _{\mathbb {R}^N} g(\xi )\ \hbox {d}\mu (\xi )$. We do not distinguish between these two equivalent definitions of $\mathcal {M}$. By a slight abuse of notation, we shall sometimes write $\left\langle \mu , g(\xi )\right\rangle = \int _{\mathbb {R}^N}g(\xi )\ \hbox {d}\mu (\xi ).$ We will be particularly interested in the pairing $\left\langle \mu , {{\mathrm{id}}}\right\rangle = \int _{\mathbb {R}^N}\xi \ \hbox {d}\mu (\xi )$ between $\mu$ and the identity function ${{\mathrm{id}}}(\xi ) = \xi$.

A.1.2 The duality between $C_0(\mathbb {R}^N)$ and $\mathcal {M}(\mathbb {R}^N)$ induces a weak* topology on $\mathcal {M}(\mathbb {R}^N)$, that of weak* convergence. A sequence $\mu ^n\in \mathcal {M}(\mathbb {R}^N)$ converges weak* to $\mu \in \mathcal {M}(\mathbb {R}^N)$ provided $\left\langle \mu ^n, g\right\rangle \rightarrow \left\langle \mu , g\right\rangle$ for all $g\in C_0(\mathbb {R}^N)$. (This is also called weak or vague convergence, see [7, 30].)

A.1.3 The set of probability measures on $\mathbb {R}^N$ is the subset

$$\begin{aligned} \mathcal {P}(\mathbb {R}^N) := \left\{ \mu \in \mathcal {M}(\mathbb {R}^N)\ :\ \mu \geqslant 0,\ \mu (\mathbb {R}^N) = 1\right\} . \end{aligned}$$

Let $\mathcal {P}^p(\mathbb {R}^N) \subset \mathcal {P}(\mathbb {R}^N)$ for $p\in [1,\infty )$ denote the set of probability measures $\mu$ such that $\left\langle \mu , |\xi |^p\right\rangle < \infty$. For $\mu ,\rho \in \mathcal {P}^p(\mathbb {R}^N)$ the Wasserstein metric $W_p$ is defined as

$$\begin{aligned} W_p(\mu ,\rho ) := \inf \left\{ \int _{\mathbb {R}^N\times \mathbb {R}^N}|\xi -\zeta |^p\ \hbox {d}\pi (\xi ,\zeta )\ :\ \pi \in \varPi (\mu ,\rho )\right\} ^{1/p}, \end{aligned}$$

where $\varPi (\mu ,\rho )$ is the set of probability measures on $\mathbb {R}^N\times \mathbb {R}^N$ with marginals $\mu$ and $\rho$:

It can be shown that $W_p$ for any p metrizes the topology of weak convergence on $\mathcal {P}^p(\mathbb {R}^N)$ (see [1, Proposition 7.1.5] or [64, Chapter 7]).

A.1.4 Let $\mu , \rho \in \mathcal {P}(\mathbb {R})$, and let $F, G : \mathbb {R}\rightarrow [0,1]$ be their distribution functions,

$$\begin{aligned} F(x) := \mu ((-\infty ,x]), \quad G(y) := \rho ((-\infty ,y]). \end{aligned}$$

Then it can be shown that

$$\begin{aligned} W_p(\mu , \rho ) = \left( \int _0^1 \left| F^{-1}(s) - G^{-1}(s)\right| ^p\ \hbox {d}s\right) ^{1/p}, \end{aligned}$$

see [64, p. 75]. This gives rise to an efficient algorithm for computing the Wasserstein distance between discrete probability distributions. Let $x_1, \ldots , x_n$ and $y_1, \ldots , y_n$ be random numbers drawn from the probability distributions $\mu$ and $\rho$, respectively, and define the discrete distributions $\mu _n := (\delta _{x_1} + \cdots + \delta _{x_n})/n$ and $\rho _n := (\delta _{y_1} + \cdots + \delta _{y_n})/n$. By the law of large numbers, we have $\mu _n \rightarrow \mu$ and $\rho _n \rightarrow \rho$ weak* as $n\rightarrow \infty$, almost surely. Moreover, their distribution functions are

$$\begin{aligned} F_n(x) = \frac{\#\{x_j\ :\ x_j \leqslant x\}}{n}, \quad G_n(y) = \frac{\#\{y_j\ :\ y_j \leqslant y\}}{n}. \end{aligned}$$

Hence, if the sequences $x_j$ and $y_j$ are sorted in increasing order, then

$$\begin{aligned} W_p(\mu _n, \rho _n)^p = \int _0^1 \left| F_n^{-1}(s) - G_n^{-1}(s)\right| ^p\ \hbox {d}s = \frac{1}{n}\sum _{j=1}^n |x_j-y_j|^p. \end{aligned}$$

The latter expression is very easy to implement on a computer.

The analogous problem when $\mu ,\rho \in \mathcal {P}(\mathbb {R}^N)$ is more complex, but can be solved in $O(n^3)$ time using the so-called Hungarian algorithm, see [56].

1.2 Young Measures

A.2.1 A Young measure from $D\subset \mathbb {R}^k$ to $\mathbb {R}^N$ is a function which maps $z\in D$ to a probability measure on $\mathbb {R}^N$. More precisely, a Young measure is a weak* measurable map $\nu : D \rightarrow \mathcal {P}(\mathbb {R}^N)$, that is, the map** $z\mapsto \left\langle \nu (z), g\right\rangle$ is Borel measurable for every $g\in C_0(\mathbb {R}^N)$. We denote the image of $z\in D$ under $\nu$ by $\nu _z:= \nu (z) \in \mathcal {P}(\mathbb {R}^N)$. The set of all Young measures from D into $\mathbb {R}^N$ is denoted by $\mathbf {Y}(D,\mathbb {R}^N)$. When $N=1$ we write $\mathbf {Y}(D) := \mathbf {Y}(D,\mathbb {R})$.

A.2.2 A Young measure $\nu \in \mathbf {Y}(D,\mathbb {R}^N)$ is uniformly bounded if there is a compact set $K\subset \mathbb {R}^N$ such that ${{\mathrm{supp}}}\nu _z\subset K$ for all $z\in D$. Note that if $\nu$ is atomic, $\nu = \delta _u$, then $\nu$ is uniformly bounded if and only if $\Vert u\Vert _{L^\infty (D)} < \infty$.

A.2.3 If $u:\mathbb {R}^k\rightarrow \mathbb {R}^N$ is any measurable function, then $\nu _{z} := \delta _{u(z)}$ defines a Young measure, and we have $u(z) = \left\langle \nu _z, {{\mathrm{id}}}\right\rangle$ for every $z$. Conversely, we will say that a given Young measure $\nu$ is atomic if it can be written as $\nu = \delta _{u}$ for a measurable function u.

A.2.4 Two topologies on $\mathbf {Y}(D,\mathbb {R}^N)$ arise naturally in the study of Young measures: those of weak* and strong convergence. A sequence $\nu ^n \in \mathbf {Y}(D,\mathbb {R}^N)$ converges weak* to $\nu \in \mathbf {Y}(D,\mathbb {R}^N)$ if $\left\langle \nu ^n, g\right\rangle \overset{*}{\rightharpoonup }\left\langle \nu , g\right\rangle$ in $L^\infty (D)$ for all $g\in C_0(\mathbb {R}^N)$, that is,

$$\begin{aligned} \int _D \varphi (z) \left\langle \nu ^n_z, g\right\rangle \ \hbox {d}z \rightarrow \int _D \varphi (z) \left\langle \nu _z, g\right\rangle \ \hbox {d}z \quad \forall \ \varphi \in L^1(D). \end{aligned}$$

We say that $\nu ^n\in \mathbf {Y}(D,\mathbb {R}^N)$ converges strongly to $\nu \in \mathbf {Y}(D,\mathbb {R}^N)$ if

$$\begin{aligned} \bigl \Vert W_p(\nu ^n,\nu )\bigr \Vert _{L^p(D)} \rightarrow 0 \end{aligned}$$

for some $p\in [1,\infty )$. If $\nu$ is atomic, $\nu = \delta _u$ for some $u:D\rightarrow \mathbb {R}^N$, then $\nu ^n \rightarrow \nu$ strongly if and only if

$$\begin{aligned} \int _D \int _{\mathbb {R}^N} |\xi - u(z)|^p\ \hbox {d}\nu ^n_z(\xi ) \hbox {d}z \rightarrow 0. \end{aligned}$$

A.2.5 The fundamental theorem of Young measures was first introduced by Tartar for $L^\infty$-bounded sequences [63] and then generalized by Schonbek [59] and Ball [3] for sequences of measurable functions. We provide a further generalization: Every sequence $\nu ^n\in \mathbf {Y}(D,\mathbb {R}^N)$ which does not “leak mass at infinity” (condition (40)) has a weak* convergent subsequence:

Theorem 13

Let $\nu ^n \in \mathbf {Y}(D,\mathbb {R}^N)$ for $n\in \mathbb {N}$ be a sequence of Young measures. Then there exists a subsequence $\nu ^m$ which converges weak* to a nonnegative measure-valued function $\nu :D\rightarrow \mathcal {M}_+(\mathbb {R}^N)$ in the sense that

• (i) $\left\langle \nu ^m_z, g\right\rangle \overset{*}{\rightharpoonup }\left\langle \nu , g\right\rangle$ in $L^\infty (D)$ for all $g\in C_0(\mathbb {R}^N)$,

and moreover satisfies

• (ii) $\Vert \nu _z\Vert _{\mathcal {M}(\mathbb {R}^N)} \leqslant 1$ for a.e. $z\in D$;

• (iii) If $K\subset \mathbb {R}^N$ is closed and ${{\mathrm{supp}}}\nu ^n_z\subset K$ for a.e. $z\in D$ and n large, then ${{\mathrm{supp}}}\nu _z\subset K$ for a.e. $z\in D$.

Suppose further that for every bounded, measurable $E \subset D$, there is a nonnegative $\kappa \in C(\mathbb {R}^N)$ with $\lim _{|\xi |\rightarrow \infty }\kappa (\xi )=\infty$ such that

$$\begin{aligned} \sup _n \int _E \left\langle \nu ^n_z, \kappa \right\rangle \ \hbox {d}z < \infty . \end{aligned}$$

(40)

Then

• (iv) $\Vert \nu _z\Vert _{\mathcal {M}(\mathbb {R}^N)}=1$ for a.e. $z\in D$,

whence $\nu \in \mathbf {Y}(D,\mathbb {R}^N)$.

Proof

The proof is a generalization of Ball [3].

Denote by $L_{w}^\infty (D;\mathcal {M}(\mathbb {R}^N))$ the set of weak* measurable functions $\mu :D\rightarrow \mathcal {M}(\mathbb {R}^N)$, equipped with the norm

$$\begin{aligned} \Vert \mu \Vert _{\infty ,\mathcal {M}}:= \mathop {\hbox {ess sup}}\limits _{z\in D}\Vert \mu _z\Vert _\mathcal {M}. \end{aligned}$$

From the fact that $C_0(\mathbb {R}^N)$ is separable, it can be shown (see [24, Theorem 8.18.2]) that $L_{w}^\infty (D;\mathcal {M}(\mathbb {R}^N))$ is isometrically isomorphic to the dual of $L^1(D;C_0(\mathbb {R}^N))$. The sequence $\mu ^n$ is bounded in $L_{w}^\infty (D;\mathcal {M}(\mathbb {R}^N))$ since $\Vert \mu ^n\Vert _{\infty ,\mathcal {M}}\equiv 1$, and hence there is a $\mu \in L_{w}^\infty (D;\mathcal {M}(\mathbb {R}^N))$ and a weak* convergent subsequence $\mu ^m$ of $\mu ^n$ such that $\left\langle \mu ^m, \varPsi \right\rangle _{\infty ,\mathcal {M}}\rightarrow \left\langle \mu , \varPsi \right\rangle _{\infty ,\mathcal {M}}$, or equivalently,

$$\begin{aligned} \int _{D} \left\langle \mu ^m_z, \varPsi (z,\cdot )\right\rangle \hbox {d}z\rightarrow \int _{D} \left\langle \mu _z, \varPsi (z,\cdot )\right\rangle \hbox {d}z\quad \text {as }m\rightarrow \infty \end{aligned}$$

for all $\varPsi \in L^1(D;C_0(\mathbb {R}^N))$. In particular, letting $\varPsi (z,\xi )=\varphi (z)g(\xi )$ for $\varphi \in L^1(D)$ and $g\in C_0(\mathbb {R}^N)$, we obtain (i). We claim that $\mu _z\geqslant 0$ for a.e. $z\in D$. If not, then there would be a nonnegative $\varPsi \in L^1(D;C_0(\mathbb {R}^N))$ such that $\int _D\left\langle \mu _z, \varPsi (z,\cdot )\right\rangle \ \hbox {d}z< 0$. But then

$$\begin{aligned} 0 > \int _{D} \left\langle \mu _z, \varPsi (z,\cdot )\right\rangle \hbox {d}z= \lim _{m\rightarrow \infty } \int _D \left\langle \mu ^m_z, \varPsi (z,\cdot )\right\rangle \hbox {d}z\geqslant 0 \end{aligned}$$

(since $\mu ^m_z\geqslant 0$ for all $z$), a contradiction.

(ii) follows from the weak* lower semi-continuity of the norm $\Vert \cdot \Vert _{\infty ,\mathcal {M}}$. To see that (iii) holds, let $g\in C_0(\mathbb {R}^N)$ be such that $g\bigr |_{K} = 0$. Since $\mu ^m \rightarrow K$ in measure, it follows that $\left\langle \mu ^m, g\right\rangle \rightarrow 0$ in measure (that is, $|\{z\in D:|\left\langle \mu ^m_z, g\right\rangle |>\delta \}| \rightarrow 0$ for all $\delta >0$). Hence,

$$\begin{aligned} \int _D \varphi (z)\left\langle \mu _z, g\right\rangle \hbox {d}z= \lim _m \int _D \varphi (z)\left\langle \mu ^m_z, g\right\rangle \hbox {d}z= 0 \end{aligned}$$

and therefore $\left\langle \mu _z, g\right\rangle = 0$ for a.e.  $z\in D$. This is precisely (ii).

Assume now that (40) holds. Fix a set $E\subset D$ of finite, nonzero Lebesgue measure |E|, and denote the average integral over E as $-\!\!\!\!\!\!\int _E = \frac{1}{|E|}\int _E$. For every $R>0$, we define

$$\begin{aligned} \theta _R(\xi ) = {\left\{ \begin{array}{ll} 1 &{} \kappa (\xi ) \leqslant R \\ 1+R-\kappa (\xi ) &{} R<\kappa (\xi )\leqslant R+1 \\ 0 &{} R+1 < \kappa (\xi ). \end{array}\right. } \end{aligned}$$

Then $\theta _R \in C_0(\mathbb {R}^N)$, so

$$\begin{aligned} \lim _m-\!\!\!\!\!\!\int _E \left\langle \mu ^m_z, \theta _R\right\rangle \hbox {d}z= -\!\!\!\!\!\!\int _E \left\langle \mu _z, \theta _R\right\rangle \hbox {d}z\leqslant -\!\!\!\!\!\!\int _E \Vert \mu _z\Vert _\mathbb {R}\hbox {d}z\leqslant 1, \end{aligned}$$

the last inequality following from the fact that $\Vert \mu _z\Vert _\mathbb {R}\leqslant 1$ for all $z$. Conversely,

$$\begin{aligned} 0 \leqslant -\!\!\!\!\!\!\int _E \left( 1-\left\langle \mu ^m_z, \theta _R\right\rangle \right) \ \hbox {d}z= -\!\!\!\!\!\!\int _E \left\langle \mu ^m_z, 1-\theta _R\right\rangle \ \hbox {d}z\leqslant \frac{1}{R}-\!\!\!\!\!\!\int _E \left\langle \mu ^m_z, \kappa \right\rangle \ \hbox {d}z, \end{aligned}$$

so (40) gives

$$\begin{aligned} 1&\leqslant \lim _{R\rightarrow \infty }\lim _m-\!\!\!\!\!\!\int _E \left\langle \mu ^m_z, \theta _R\right\rangle \hbox {d}z+ \lim _{R\rightarrow \infty } \sup _m \frac{1}{R}-\!\!\!\!\!\!\int _E \left\langle \mu ^m_z, \kappa \right\rangle \ \hbox {d}z\\&= \lim _{R\rightarrow \infty }-\!\!\!\!\!\!\int _E \left\langle \mu _z, \theta _R\right\rangle \hbox {d}z\\&\leqslant -\!\!\!\!\!\!\int _E \Vert \mu _z\Vert _{\mathcal {M}(\mathbb {R}^N)} \hbox {d}z\leqslant 1, \end{aligned}$$

whence $-\!\!\!\!\!\!\int _E \Vert \mu _z\Vert _{\mathcal {M}(\mathbb {R}^N)}\ \hbox {d}z= 1$. Since $E\subset D$ is arbitrary, (iv) follows. $\square$

A.2.6 An important special case of (40) is when $\kappa (\xi ) = |\xi |^p$ for $1\leqslant p < \infty$, which translates to the $L^p$ bound

$$\begin{aligned} \sup _n\int _D \left\langle \mu ^n, |\xi |^p\right\rangle \ \hbox {d}z < \infty . \end{aligned}$$

The case $p=\infty$ translates to the support of $\nu ^n_z$ lying in a compact set $K\subset \mathbb {R}^N$ for a.e. z and all n. Part (iii) of Theorem 13 then holds for all $g\in C(\mathbb {R}^N)$, and condition (40) is automatically satisfied for any such $\kappa$. The latter is the original form of the theorem given by Tartar [63].

1.3 Random Fields and Young Measures

A.3.1 If $(\varOmega ,\mathcal {F},P)$ is a probability space, $D\subset \mathbb {R}^k$ is a Borel set and $u : \varOmega \times D \rightarrow \mathbb {R}^N$ is a random field (i.e., a jointly measurable function), then we can define its law by

$$\begin{aligned} \nu _{z}(F) := P\left( u(z) \in F\right) = P\left( \left\{ \omega \ :\ u(\omega ,z) \in F\right\} \right) \end{aligned}$$

(41a)

for Borel subsets $F\subset \mathbb {R}^N$ of phase space, or equivalently,

$$\begin{aligned} \left\langle \nu _z, g\right\rangle := \int _\varOmega g(u(\omega ,z))\ \hbox {d}P(\omega ) \end{aligned}$$

(41b)

for $g\in C_0(\mathbb {R}^N)$. This defines a Young measure:

Proposition 1

If $u:\varOmega \times D \rightarrow \mathbb {R}^N$ is jointly measurable, then (41) defines a Young measure from D to $\mathbb {R}^N$.

Proof

First of all, for fixed $z\in D$ the set $\bigl \{\omega : u(\omega ,z) \in U\bigr \}$ is P-measurable for Borel sets U. Indeed, if $w(\omega ) := u(\omega ,z)$ denotes the $z$-section of the measurable function $(\omega ,y) \mapsto u(\omega ,y)$, then $\bigl \{\omega : u(\omega ,z) \in U\bigr \} = w^{-1}(U)$ is measurable.

We need to show that the definition of $\nu$ is independent of the choice of map** in the equivalence classes of map**s from $\varOmega \times D \rightarrow \mathbb {R}^N$. Let $\hat{u}, \tilde{u}: \varOmega \times D \rightarrow \mathbb {R}^N$ be two map**s such that $\hat{u}(\omega ,z) = \tilde{u}(\omega ,z)$ for $P\times \lambda$-a.e. $(\omega ,z)$. We apply Tonelli’s theorem to find that

$$\begin{aligned} 0 = \int _{\varOmega \times D} \mathbbm {1}_{\{\hat{u}\ne \tilde{u}\}}(\omega ,z)\ d(P\times \lambda )(\omega ,z) = \int _D P(\{\hat{u}(z) \ne \tilde{u}(z) \})\ \hbox {d}z. \end{aligned}$$

Hence, $P(\hat{u}(z) \ne \tilde{u}(z)) = 0$ for a.e. $z\in D$, so for every Borel set $U\subset \mathbb {R}^N$,

$$\begin{aligned} P\left( \hat{u}(z) \in U\right) = P\left( \tilde{u}(z) \in U\right) \end{aligned}$$

for a.e. $z\in D$.

Finally, $\nu$ is weak* measurable since

$$\begin{aligned} \left\langle \nu _z, g\right\rangle = \int _{\mathbb {R}^N}g(\xi )\ \hbox {d}\nu _z(\xi ) = \int _{\varOmega }g(u(\omega ,z))\ \hbox {d}P(\omega ), \end{aligned}$$

which is measurable in $z$ for any $g\in C_0(\mathbb {R}^N)$. $\square$

A.3.2 It is well known that every measure on $\mathbb {R}^N$ can be realized as the law of a random variable. Here we show that for every Young measure $\nu$, there is always a random field with law $\nu$.

Proposition 2

For every Young measure $\nu \in \mathbf {Y}(D,\mathbb {R}^N)$, there exist a probability space $(\varOmega , \mathcal {F}, P)$ and a Borel measurable function $u : \varOmega \times D \rightarrow \mathbb {R}^N$ such that u has law $\nu$, i.e., for all Borel sets E,

$$\begin{aligned} \nu _z(E) = P(u(\omega ,z) \in E). \end{aligned}$$

In particular, we can choose $(\varOmega , \mathcal {F}, P)$ to be the Borel $\sigma$-algebra on $\varOmega =[0,1)$ with Lebesgue measure.

Proof

The method of proof is standard, see, e.g., [6, Theorem 5.3].

We assume that $N=1$. The generalization to $N>1$ is straightforward but tedious. For $n\in \mathbb {N}$ and $j\in \mathbb {Z}$, we set

$$\begin{aligned} F_n^j := {\left\{ \begin{array}{ll} (-\infty , -2^{n}) &{} \text {if }j = -2^{2n} \\ {\big [2^{-n}(j-1), 2^{-n}j\big )} &{} \text {if } j = -2^{2n}+1, \ldots , 2^{2n} \\ {[2^n, \infty )} &{} \text {if } j = 2^{2n}+1\\ \emptyset &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

Let $p_n^j(z) := \sum _{l\leqslant j} \nu _z(F_n^l)$. Note that $p_n^j : \mathbb {R}\rightarrow [0,1]$ is measurable for all n, j, and that $0\leqslant p_n^{-j}\leqslant \dots \leqslant p_n^j=1$ for j large enough. Choose any $\xi _n^j \in F_n^j$, and for $\omega \in \varOmega :=[0,1)$, define

$$\begin{aligned} u_n(\omega ,z) := \xi _n^j \quad \text {for { j} such that } p_n^{j-1}(z) \leqslant \omega < p_n^{j}. \end{aligned}$$

We claim that $u_n$ is measurable on the product $\sigma$-algebra between $\mathcal {F}$ and the Borel $\sigma$-algebra on D. Each function $u_n$ takes only finitely many values $\xi _n^j$, so it suffices to show that $u_n^{-1}(\{\xi _n^j\})$ is measurable for every $\xi _n^j$. Indeed,

$$\begin{aligned} u_n^{-1}(\{\xi _n^j\})&= \Bigl \{(\omega ,z)\in \varOmega \times D\ :\ p_n^j(z) \leqslant \omega < p_n^{j+1}(z) \Bigr \} \\&= \Bigl (\varOmega \times D\Bigr ) \cap \Bigl \{(\omega ,z)\in \mathbb {R}\times D\ :\ p_n^j(z) \leqslant \omega \Bigr \} \\&\quad \cap \Bigl \{(\omega ,z)\in \mathbb {R}\times D\ :\ \omega < p_n^{j+1}(z) \Bigr \}, \end{aligned}$$

the intersection between the epigraph of $p_n^j$ and the hypograph of $p_n^{j+1}$, which are measurable by the measurability of the functions $p_n^j$ and $p_n^{j+1}$.

Because the partition $\{F_m^j\}_{j\in \mathbb {Z}}$ is a refinement of $\{F_n^j\}_{j\in \mathbb {Z}}$ whenever $m>n$, it follows that $|u_n(\omega ,z) - u_m(\omega ,z)| < \mathrm{diam}(F_n^j) = 2^{-n}$ for any $(\omega ,z)$ whenever m, n are large enough. Hence, $u_n$ converges pointwise to some function $u : \varOmega \times D \rightarrow \mathbb {R}$, which is measurable by the measurability of each $u_n$.

Finally, for every $g\in C_0(\mathbb {R})$ and almost every $z\in D$, we have by Lebesgue’s dominated convergence theorem

$$\begin{aligned} \int _\varOmega g(u(\omega ,z))\ \hbox {d}P(\omega )&= \lim _n \int _\varOmega g(u_n(\omega ,z))\ \hbox {d}P(\omega ) \\&= \lim _n \sum _j \nu _z(F_n^j)g(\xi _n^j) = \int _\mathbb {R}g(\xi )\ \hbox {d}\nu _z(\xi ). \end{aligned}$$

Hence, $u(\cdot ,z)$ has law $\nu _z$. $\square$

Appendix 2: Proof of Theorem 11

Proof

For any random field $\zeta : \varOmega \rightarrow L^1(\mathbb {R}^d \times \mathbb {R}_+) \cap L^{\infty }(\mathbb {R}^d \times \mathbb {R}_+)$ on $(\varOmega ,\mathcal {F},P)$, we denote the expectation with respect to the probability measure P as

$$\begin{aligned} \mathbb {E}(\zeta ) := \int \limits _{\varOmega } \zeta (\omega ) \hbox {d}P(\omega ). \end{aligned}$$

For $1 \leqslant k \leqslant M$, denote

$$\begin{aligned} \begin{aligned} G(\omega )&= \int _{\mathbb {R}_+}\int _{\mathbb {R}^d} \psi (x,t) g\left( u^{{\Delta x}}(\omega ;x,t)\right) \hbox {d}x \hbox {d}t, \\ G_k(\omega )&= \int _{\mathbb {R}_+}\int _{\mathbb {R}^d} \psi (x,t) g\left( u^{{\Delta x},k}(\omega ;x,t)\right) \hbox {d}x \hbox {d}t. \end{aligned} \end{aligned}$$

(42)

Henceforth, we suppress the $\omega$-dependence of G and $G_k$ for notational convenience. The $L^2(P)$ error in the approximation can be written as

$$\begin{aligned}&\mathbb {E}\left( \left( \mathbb {E}(G) - \frac{1}{M}\sum \limits _{k=1}^M G_k \right) ^2\right) = \mathbb {E}\left( \frac{1}{M^2}\left( \sum \limits _{k=1}^M (\mathbb {E}(G) - G_k) \right) ^2 \right) , \\&\quad = \mathbb {E}\left( \frac{1}{M^2} \left( \sum \limits _{k=1}^M \bigl (\mathbb {E}(G) - G_k\bigr )^2 + 2\sum _{k=1}^{M}\sum _{l \ne k} \bigl (\mathbb {E}(G) - G_k\bigr )\bigl (\mathbb {E}(G) - G_l\bigr )\right) \right) \\&\quad = \underbrace{\frac{1}{M^2}\sum \limits _{k=1}^M \mathbb {E}\left( \bigl (\mathbb {E}(G) - G_k\bigr )^2\right) }_{=:\ T_1} + \frac{2}{M^2} \sum _{k=1}^{M}\sum _{l \ne k}\underbrace{\mathbb {E}\Bigl ( \bigl (\mathbb {E}(G) - G_k\bigr ) \bigl (\mathbb {E}(G) - G_l\bigr )\Bigr )}_{=:\ T^{kl}_2}. \end{aligned}$$

As $u^{{\Delta x},1}, \ldots , u^{{\Delta x},M}$ are independent and identically distributed, it follows from the definition of $G_k$ that $G_1,\ldots ,G_M$ are independent and identically distributed random variables. Hence, $\mathbb {E}(G_k) = \mathbb {E}(G)$ and $\mathbb {E}(G_kG_l) = \mathbb {E}(G_k)\mathbb {E}(G_l)$ for all k, l. Consequently, a simple calculation shows that $T^{kl}_2=0$ for all $1 \leqslant k,l \leqslant M$ and $k \ne l$.

The fact that $G_1,\ldots ,G_M$ are independent and identically distributed yields

$$\begin{aligned} T_1 = \frac{1}{M} \left( \mathbb {E}(G^2) - \mathbb {E}(G)^2\right) . \end{aligned}$$

Hence,

In conclusion, the sample mean

$$\begin{aligned} \frac{1}{M} \sum _{k=1}^{M} \int _{\mathbb {R}_+}\int _{\mathbb {R}^d} \psi (x,t) g\left( u^{{\Delta x},k}(x,t)\right) \ \hbox {d}x\hbox {d}t \end{aligned}$$

converges to the corresponding ensemble average, $\int _{\mathbb {R}_+}\int _{\mathbb {R}^d} \psi (x,t) \left\langle \nu ^{{\Delta x}}_{x,t}, g\right\rangle \ \hbox {d}x\hbox {d}t$ in $L^2(\varOmega ;P)$, with a convergence rate of $\frac{1}{\sqrt{M}}$. Taking a subsequence $M'\rightarrow \infty$, the convergence also holds P-almost surely. $\square$

Appendix 3: Time Continuity of Approximations

From the time integration procedure (17b), we can show that the approximate MV solutions are time continuous. Consequently, the initial data is attained in a certain sense, and moreover, it is meaningful to evaluate the MV solution at a specific time t.

We state the theorem without proof, since the results are straightforward generalizations of “deterministic” counterparts.

Theorem 14

Let $\psi \in C_c^1(\mathbb {R})$ and assume that (19a) and (19b) are satisfied. Let $\nu ^{\Delta x}$ be generated by Algorithm 5. Then the functions

$$\begin{aligned} \varPsi ^{\Delta x}(t) := \int _\mathbb {R}\psi (x)\left\langle \nu ^{\Delta x}_{(x,t)}, {{\mathrm{id}}}\right\rangle \ \hbox {d}x \end{aligned}$$

and

$$\begin{aligned} \varPsi (t) := \int _\mathbb {R}\psi (x) \left\langle \nu _{(x,t)}, {{\mathrm{id}}}\right\rangle \ \hbox {d}x \end{aligned}$$

are Hölder continuous with exponent $\gamma := \frac{r-1}{r}$ and with constant independent of ${\Delta x}$, and $\varPsi ^{\Delta x}(t) \rightarrow \varPsi (t)$ as ${\Delta x}\rightarrow 0$ for a.e. $t\in [0,T]$. Moreover,

$$\begin{aligned} \varPsi (0) = \lim _{t\rightarrow 0} \varPsi (t) = \int _\mathbb {R}\psi (x) \left\langle \sigma _{x}, {{\mathrm{id}}}\right\rangle \ \hbox {d}x. \end{aligned}$$