Small dispersion approximation of shock wave dynamics

Abstract

We introduce a dispersion approximation model for weak, entropy solutions of multidimensional scalar conservation laws using variational kinetic representation, where equilibrium densities satisfy Gibb’s entropy minimization principle for a piecewise linear, convex entropy. For such solutions, we show that small scale discontinuities, measured by the entropy increments, propagate with characteristic velocities, while the large-scale, shock-type discontinuities propagate with speeds close to the speeds of classical shock waves. In the zero-limit of the scale parameter, approximate solutions converge to a unique, entropy solution of a scalar conservation law.

1 Introduction

Consider the Cauchy problem for a quasilinear system

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t U{}+{}\sum _{i=1}^d\partial _{x_i} F_i(U){}={}0,\quad (x,t)\in \mathbb {R}^{d+1}_+,\\ U(x,0)=U_0(x),\quad x\in \mathbb {R}^d, \end{array}\right. } \end{aligned}$$

(1)

where $U:\mathbb {R}^{d+1}_+\rightarrow \mathbb {R}^m,$ $F_i:\mathbb {R}^m\rightarrow \mathbb {R}^{m}.$ The main difficulty in constructing weak solutions for quasilinear systems (1) is the lack of apriori estimates on solutions in norms that control oscillations. This limits the application of such methods as viscosity or relaxation approximations of (1) for which pointwise convergence of approximate solutions is hard to establish.

The difficulty is well illustrated on an example of a shock wave. For systems with a convex entropy, weak solutions are typically restricted to verify entropy dissipation balance:

$$\begin{aligned} \partial _t\eta (U)+\mathrm{div}_x\, q(U){}={}r,\quad r{}\le 0, \end{aligned}$$

which provides apriori estimate on the total entropy at time t and total dissipated entropy up to time t in terms of the entropy of the initial data. This type of control is, however, too weak. For example, for a shock wave contained inside an interval [a, b],  the total dissipated entropy $\int \limits _0^t\int \limits _a^b r\,\mathrm{d}x$ is cubic in the strength of the shock, see theorem 8.5.1 of Dafermos [4]. Thus, in a regime of increasing number of small shock waves, the entropy does not control the oscillations as measured by the sum of all shock wave strengths.

In this paper, we explore the possibility of constructing approximate solutions of (1) for which entropy inequality implies strong compactness, at the price of distorting certain small scale details of the original solutions. More specifically, we will seek approximate, weak solutions of (1) such that

• 1. large shocks propagate with speeds close to the speeds computed from the original system (1);

• 2. the discontinuities, for which the change in the entropy is smaller than a certain threshold value $\varepsilon ,$ are transported with characteristic velocities.

Thus, the approximation of this type involves small-scale dispersion effects. The present work shows how this type of approximation can be implemented for scalar conservation laws in multi-dimensions. We consider equation

$$\begin{aligned} \partial _t \rho {}+{}\mathrm{div}_x A(\rho ){}={}0,&\quad&(x,t)\in \mathbb {R}^{d+1}_+, \end{aligned}$$

(2)

where $A:\mathbb {R}\rightarrow \mathbb {R}^d$ is a smooth vector function of fluxes. The theory of unique entropy solutions for scalar conservation laws was developed by Kruzhkov [8] using viscosity approximation. Later, different approaches have been used to build such solutions, see [1,2,3, 6, 7, 10]. Our approach is based on the kinetic representation of entropy weak solutions of (2) developed by Brenier [1, 2], Brenier and Corrias [3], Giga and Miyakawa [6], and Lions et al. [10]. According to the theory, an admissible $\rho (x,t)$ is represented as a moment of an “equilibrium” kinetic density $f_{\mathrm{eq}}:$

$$\begin{aligned} \rho (x,t){}={}\int f_{\mathrm{eq}}(x,t,v)\,\mathrm{d}v,\quad f_{\mathrm{eq}}(x,t,v){}={} \mathbb {I}_{[0,\rho (x,t)]}(v), \end{aligned}$$

(3)

with $f_{\mathrm{eq}}$ solving a kinetic equation

$$\begin{aligned} \partial _t f{}+{}A'(v)\cdot \nabla {}_x f{}={}\partial _v m, \end{aligned}$$

(4)

where m is non-negative Radon measure on $\mathbb {R}^{d+2}_+.$ Here, for the simplicity of the presentation we assume that $\rho$ is non-negative. Conversely, any solution of (4) constrained by condition (3) for some $\rho (x,t)$ defines an admissible weak solution of conservation law in (2), see [10]. Moreover, for any strictly convex function $\eta ,$ and a.e. (x, t),  $f_{\mathrm{eq}}(x,t,v)$ is the unique minimizer of the problem

$$\begin{aligned} \min \left\{ \int \eta '(v)\tilde{f}(v)\,\mathrm{d}v\,:\, \tilde{f}(v)\in [0,1],\,\int \tilde{f}\,\mathrm{d}v{}={}\rho (x,t)\right\} . \end{aligned}$$

(5)

Solutions of (4) can be obtained as limits of solutions of a relaxation problem

$$\begin{aligned} \partial _t f{}+{}A'(v)\cdot \nabla {}_x f{}={}h^{-1}(M_f-f), \end{aligned}$$

(6)

where $M_f$ is the minimizer of (5) with $\rho =\int f\,\mathrm{d}v.$ Strong compactness of a family of solutions with $h\rightarrow 0$ can be obtained through the uniform $L^1$ continuity ( [1, 2, 6]), or the compensated compactness method ([10]).

To obtain the approximate solutions with properties 1 and 2, described above, we will use the variational kinetic formulation (5) and (6), in which we introduce a scale parameter $\varepsilon .$ For that purpose we replace a strictly convex function $\eta$ by a continuous, piecewise linear approximate entropy $\eta _\varepsilon .$ With the new entropy function, the minimization problem admits multiple solutions, with indeterminacy on small $\varepsilon$–scales. A particular minimizer $M_f$ will be selected so that $L^1$ norm of $f-M_f$ can be estimated by the entropy increment $\int \eta _\varepsilon '(v)(f-M_f)\,\mathrm{d}v.$

Our main result, Theorem 1 describes the properties kinetic functions obtained from this kinetic relaxation approach. Such kinetic functions verify Eq. (4) where, in addition, the right-hand side is a signed Radon measure, with the total variation controlled by the entropy:

$$\begin{aligned} ||\partial _v m||{}\le {}\frac{2}{\varepsilon }\int \eta _{\varepsilon }(\rho _0)\,\mathrm{d}x. \end{aligned}$$

Furthermore, we show that moments $\rho =\int f\,\mathrm{d}v,$ and $\phi =\int A'(v)f\,\mathrm{d}v,$ solve the balance equation

$$\begin{aligned} \partial _t\rho +\mathrm{div}_x\phi {}={}0, \end{aligned}$$

and $\phi (x,t){}={}A(\rho (x,t))+O(\varepsilon ^2).$ In particular, if there is a co-dimension one discontinuity of $\rho$ with values $\rho ^+,\rho ^-,$ (such discontinuities do develop in the solutions), such that $|\rho ^+-\rho ^-|>\varepsilon ,$ then it propagates with the velocity

$$\begin{aligned} \sigma {}={}\frac{A(\rho ^+)-A(\rho ^-)}{\rho ^+-\rho ^-} {}+{}O(\varepsilon ). \end{aligned}$$

The kinetic function f,  as well as its moments, depends on the scale parameter $\varepsilon .$ In Theorem 2 we show that in the limit of $\varepsilon \rightarrow 0,$ $\rho =\rho ^\varepsilon (x,t)$ converges to an admissible solutions of (2).

In summary, we describe a new type of approximation of scalar conservation laws with properties distinct from the well-known viscosity approximation of Kruzhkov [8], kinetic relaxation approximation of Brenier [1] and Giga and Miyakawa [6], or semi-linear relaxation of Katsoulakis and Tzavaras [7]. In this approximation, large shocks propagate as sharp profiles (not smoothed) with velocities approximately verifying Rankine–Hugoniot conditions, while the entropy balance controls small-scale oscillations.

2 Main result

Let $A\in C^2(\mathbb {R})^d.$ Without loss of generality, we will assume that $\rho _0$ is non-negative and bounded, so that all kinetic functions are defined for the range of the kinetic variable $v\in [0,L],$ for some $L>0.$ Unless it is specified otherwise, in the inegrals below, the integration in v is over [0, L],  in x is over $\mathbb {R}^d,$ and in t is over $\mathbb {R}_+.$ Let $\varepsilon >0.$ Define a piecewise constant function $\eta _\varepsilon$ as

$$\begin{aligned} \eta _\varepsilon (v){}={}k,\quad v\in [k\varepsilon ,(k+1)\varepsilon ),\,k=0..\lceil L/\varepsilon \rceil . \end{aligned}$$

$\eta _\varepsilon$ approximates the derivative of the quadratic entropy function. Here, for notational convenience, we use $\eta _\varepsilon$ to denote the derivative of the entropy function described in the introduction.

Theorem 1

Let $f_0\in L^1(\mathbb {R}^d\times [0,L])$ with values $\{0,1\}.$ For any $\varepsilon >0$ there is $f\in L^1(\mathbb {R}^{d+1}_+\times [0,L])$ with values in [0, 1] and m – a non-negative Radon measure on $\mathbb {R}^{d+1}_+\times \mathbb {R}_+$ such that $\partial _v m$ is a signed Radon measure on $\mathbb {R}^{d+1}_+\times [0,L]$ with the following properties:

• i. (Kinetic equation) f and m verify (in distributional sense) equation

  $$\begin{aligned} \partial _t f{}+{}A'(v)\cdot \nabla {}_x f{}={}\partial _v m. \end{aligned}$$

  (7)

  Moreover,

  $$\begin{aligned} ||\partial _v m||_{\mathbb {R}^{d+1}_+\times [0,L]}{}\le {}\frac{2}{\varepsilon }\iint \eta _{\varepsilon }f_0\,\mathrm{d}x\mathrm{d}v; \end{aligned}$$

  (8)

• ii. (Optimality) for a.e. (x, t),  f is a minimizer of

  $$\begin{aligned} \min \left\{ \int \eta _\varepsilon (v)f(v)\,\mathrm{d}v\,:\, f(v)\in [0,1],\,\int f\,\mathrm{d}v{}={}\rho (x,t)\right\} ; \end{aligned}$$

  (9)

• iii. (Equi-continuity) for a.e. $t>0,$ and any $\xi \in \mathbb {R}^d,$

  $$\begin{aligned} \iint |f(x+\xi ,t,v)-f(x,t,v)|\,\mathrm{d}x\mathrm{d}v {}\le {}\iint |f_0(x+\xi ,v)-f_0(x,v)|\,\mathrm{d}x\mathrm{d}v. \end{aligned}$$

  (10)

Remark 1

Estimate (10) was derived in [1, 6]. We use it to show strong compactness of moments of a time discerete approximation, in the proof of Theorem 1, and to verify (9). This estimate seems to be restricted to scalar conservation laws, and does not apply to systems. However, there is an alternative way to obtain strong compactness of moments by using only entopy estimate (8), through an kinetic averaging lemma of Gérard, [5].

Kinetic functions from Theorem 1 give rise to the approximate solutions of the conservation law (2), as described in the next theorem.

Theorem 2

For function f from the previous theorem, moments

$$\begin{aligned} \rho (x,t){}={}\int f(x,t,v)\,\mathrm{d}v,\quad \phi (x,t){}={}\int A'(v)f(x,t,v)\,\mathrm{d}v \end{aligned}$$

(11)

have the following properties.

• i. $\rho ,\phi \in L^\infty (\mathbb {R}^{d+1}_+)$ and verify (in distributional sense) conservation law

  $$\begin{aligned} \partial _t \rho {}+{}\mathrm{div}_x \phi {}={}0. \end{aligned}$$

  (12)

  For any $\psi \in C^\infty _0(\mathbb {R}^d),$ $\int \rho (x,t)\psi (x)\,\mathrm{d}x$ is continuous in t and

  $$\begin{aligned} \lim _{t\rightarrow 0+}\int \rho (x,t)\psi (x)\,\mathrm{d}x = \iint f_0(x,v)\psi (x)\,\mathrm{d}v\mathrm{d}x; \end{aligned}$$

• ii. for any two pairs of values $(\rho (x,t),\phi _i(x,t))$ and $(\rho (y,\tau ),\phi _i(y,\tau )),$ such that $|\rho (x,t)-\rho (y,t)|\ge c_0\varepsilon ,$ it holds:

  $$\begin{aligned} \frac{\phi _i(x,t)-\phi _i(y,\tau )}{\rho (x,t)-\rho (y,\tau )}{}={} \frac{A_i(\rho (x,t))-A_i(\rho (y,\tau ))}{\rho (x,t)-\rho (y,\tau )}{}+{}O(\varepsilon ),\quad i=1..d; \end{aligned}$$

  (13)

• iii. (Limit to Kruzhkov’s solution) Considered as a function of $\varepsilon ,$ $\rho =\rho _\varepsilon ,$ there is a sequence $\varepsilon \rightarrow 0$ on which $\rho _\varepsilon$ converges to the unique, entropy solution of the conservation law (2) in $L^p_{loc}(\mathbb {R}^{d+1}_+),$ for any $p\in [0,+\infty ).$

2.1 Proof of Theorem 1

For a non-negative constant $\rho \in [0,L]$ consider a minimization problem

$$\begin{aligned} \min \left\{ \int \eta _\varepsilon (v)f(v)\,\mathrm{d}v\,:\, f(v)\in [0,1],\,\int f\,\mathrm{d}v{}={}\rho \right\} . \end{aligned}$$

(14)

In the next lemma $\mathbb {I}_A(v)$ stands for a characteristic function of set A.

Lemma 1

Let $n{}={}\lfloor \rho /\varepsilon \rfloor .$ The minimum in problem (14) equals

$$\begin{aligned} \left\{ \begin{array}{ll} \varepsilon \sum _{k=0}^{n-1}k+ \varepsilon n(\rho -n\varepsilon ), &{} n\ge 1,\\ 0, &{} n=0. \end{array} \right. \end{aligned}$$

It is achieved on minimizers

$$\begin{aligned} f_{min}(v){}={}\mathbb {I}_{[0,n\varepsilon ]}(v){}+{}\tilde{f}(v), \end{aligned}$$

where $\tilde{f}$ is an arbitrary function verifying conditions:

$$\begin{aligned}&\tilde{f}(v)\in [0,1],\quad \forall v\in [0,L];\quad \text{ supp }\; \tilde{f}\subset [n\varepsilon ,(n+1)\varepsilon ]; \end{aligned}$$

(15)

$$\begin{aligned}&\int \tilde{f}\,\mathrm{d}v{}={}\rho - n\varepsilon . \end{aligned}$$

(16)

Proof

$\eta _\varepsilon (v)$ is a non-decreasing function. To minimize functional $\displaystyle {\int \eta _\varepsilon f\,\mathrm{d}v}$ one needs to pick f that has all its mass as close to $v=0$ as possible, and is less than or equal 1. This shows the first statement. On interval $[n\varepsilon ,(n+1)\varepsilon ],$ a minimizer $f_{min}$ can be arbitrarily re-arranged without changing the value of its $\eta _\varepsilon$ moment. This leads to the second part of the lemma. $\square$

Given a kinetic density f we select a particular minimizer of (14) with $\rho =\int f\,\mathrm{d}v$ in the following way. If $\int \limits _{(n+1)\varepsilon }^Lf\,\mathrm{d}v > n\varepsilon - \int \limits _0^{n\varepsilon }f\,\mathrm{d}v,$ we set

$$\begin{aligned} M_f(v) = \mathbb {I}_{[0,n\varepsilon +v_0]}(v) + f(v)\mathbb {I}_{(n\varepsilon +v_0,(n+1)\varepsilon )}(v), \end{aligned}$$

(17)

where $v_0\in (0,\varepsilon )$ is determined by the relation $\int M_f\,\mathrm{d}v{}={}\int f\,\mathrm{d}v.$ It is the smallest number such that

$$\begin{aligned} \int \limits _0^{n\varepsilon +v_0}1-f\,\mathrm{d}v{}={}\int \limits _{(n+1)\varepsilon }^L f\,\mathrm{d}v. \end{aligned}$$

If $\int \limits _{(n+1)\varepsilon }^Lf\,\mathrm{d}v \ge n\varepsilon - \int \limits _0^{n\varepsilon }f\,\mathrm{d}v,$ we set

$$\begin{aligned} M_f(v) = \mathbb {I}_{[0,n\varepsilon ]}(v) + f(v)\mathbb {I}_{(n\varepsilon ,n\varepsilon +v_0)}(v), \end{aligned}$$

(18)

where $v_0\in (0,\varepsilon )$ is uniquely determined as the smallest number such that

$$\begin{aligned} \int \limits _0^{n\varepsilon }1-f\,\mathrm{d}v{}={}\int \limits _{n\varepsilon +v_0}^L f\,\mathrm{d}v. \end{aligned}$$

This minimizer can be thought of as a rearrangement of mass f obtained by shifting its pieces to the locations with smaller values of $\eta _\varepsilon (v).$

The key properties of the minimizer $f_{min}$ are listed in the next lemma.

Lemma 2

Let f be any function with values in [0, 1] and supported on [0, L]. For $M_f$, defined above

$$\begin{aligned} \int |f-M_f|\,\mathrm{d}v \le \frac{2}{\varepsilon }\int \eta _\varepsilon (v)(f-M_f)\,\mathrm{d}v. \end{aligned}$$

(19)

For any non-decreasing function $\eta ,$

$$\begin{aligned} \int \eta (v) (f(v)-M_f(v))\,\mathrm{d}v\ge 0. \end{aligned}$$

(20)

For any two functions $f_1,$ $f_2$ with values in $\{0,1\}$ and supported on [0, L], 

$$\begin{aligned} \int |M_{f_1} - M_{f_2}|\,\mathrm{d}v{}\le {}\int |f_1 - f_2|\,\mathrm{d}v, \end{aligned}$$

(21)

where $M_{f_1},M_{f_2}$ are the corresponding minimizers.

Proof

Let n be as in the previous lemma. Consider case (17).

$$\begin{aligned} \int |f-M_f|\,\mathrm{d}v {}={}\int \limits _0^{n\varepsilon +v_0} 1-f\,\mathrm{d}v{}+{}\int \limits _{(n+1)\varepsilon }^L f\,\mathrm{d}v{}={}2\int \limits _{(n+1)\varepsilon }^L f\,\mathrm{d}v {}\le {}\frac{2}{\varepsilon }\int \eta _\varepsilon (v)(f-M_f)\,\mathrm{d}v, \end{aligned}$$

(22)

where the last inequality holds since all mass of f on interval $[(n+1)\varepsilon ,L]$ has been removed from that interval. Similarly, in case (18)

$$\begin{aligned} \int |f-M_f|\,\mathrm{d}v {}={}\int \limits _0^{n\varepsilon } 1-f\,\mathrm{d}v{}+{}\int \limits _{n\varepsilon +v_0}^L f\,\mathrm{d}v{}={}2\int \limits _{n\varepsilon +v_0}^L f\,\mathrm{d}v {}\le {}\frac{2}{\varepsilon }\int \eta _\varepsilon (v)(f-M_f)\,\mathrm{d}v. \end{aligned}$$

(23)

For a non-decreasing function $\eta$, (20) follows from the definition of $M_f.$

To prove (21) it suffices to show that

$$\begin{aligned} \int f_1 f_2\,\mathrm{d}v{}\le {}\int M_{f_1}M_{f_2}\,\mathrm{d}v, \end{aligned}$$

(24)

since functions take only values 0 or 1. Let $n_1,v_{1,0}$ and $n_2,v_{2,0}$ be the corresponding values of n and $v_0$ from (17), (18) for functions $f_1$ and $f_2.$

Consider the case $n_1> n_2$ first. Here

$$\begin{aligned} \int M_{f_1}M_{f_2}\,\mathrm{d}v {}={}\int \limits _0^{(n_2+1)\varepsilon }M_{f_2}\,\mathrm{d}v{}={}\int f_2\,\mathrm{d}v\ge \int f_1 f_2\,\mathrm{d}v. \end{aligned}$$

Next, consider the case $n_1=n_2$ ($=n$). Suppose that representation (17) applies to both functions $f_1,f_2,$ and assume $v_{1,0}\ge v_{2,0}.$ Then,

$$\begin{aligned} \int M_{f_1}M_{f_2}\,\mathrm{d}v {}\ge {}\int \limits _{n\varepsilon +v_{1,0}}^{(n+1)\varepsilon }f_1f_2\,\mathrm{d}v +\int \limits _0^{n\varepsilon +v_{1,0}}f_2\,\mathrm{d}v {}+{}\int \limits _{(n+1)\varepsilon }^L f_2\,\mathrm{d}v{}\ge \int f_1 f_2\,\mathrm{d}v. \end{aligned}$$

Suppose that representation (18) applies to both functions $f_1,f_2,$ and assume $v_{1,0}\ge v_{2,0}.$ Then,

$$\begin{aligned} \int M_{f_1}M_{f_2}\,\mathrm{d}v {}\ge {} \int \limits _0^{n\varepsilon } f_2\,\mathrm{d}v{}+{} \int \limits _{n\varepsilon +v_{2,0}}^Lf_2\,\mathrm{d}v{}+{}\int \limits _{n\varepsilon }^{n\varepsilon +v_{2,0}}f_1f_2\,\mathrm{d}v{}\ge {}\int f_1f_2\,\mathrm{d}v. \end{aligned}$$

Suppose that (18) applies to function $f_1$ and (17) to $f_2.$ If $v_{1,0}\ge v_{2,0}$ then

$$\begin{aligned} \int M_{f_1}M_{f_2}\,\mathrm{d}v {}\ge {} \int \limits _0^{n\varepsilon +v_{2,0}} f_1\,\mathrm{d}v{}+{} \int \limits _{n\varepsilon +v_{1,0}}^Lf_1\,\mathrm{d}v{}+{}\int \limits _{n\varepsilon +v_{2,0}}^{n\varepsilon +v_{1,0}}f_1f_2\,\mathrm{d}v{}\ge {}\int f_1f_2\,\mathrm{d}v. \end{aligned}$$

If $v_{1,0}<v_{2,0}$ then

$$\begin{aligned} \int M_{f_1}M_{f_2}\,\mathrm{d}v {}\ge {} \int \limits _0^L f_1\,\mathrm{d}v{}\ge {}\int f_1f_2\,\mathrm{d}v. \end{aligned}$$

The contraction property (21) is proved now. $\square$

Now we consider a discrete-time approximation, with time step $h>0$ and $t_n=nh,$ $n=0,1,2...$ Given $f_{n-1}(x,v)$ the next period kinetic function

$$\begin{aligned} f_{n}(x,v) = M_{\hat{f_{n}}},\quad \hat{f_n}(x,v) = f_{n-1}(x-A'(v)h), \end{aligned}$$

with $f_0$ being the initial data. A continuous time approximate is defined as

$$\begin{aligned} f^h(x,v,t){}={}\left\{ \begin{array}{ll} f_{n-1}(x-A'(v)(t-nh)), &{} t\in [(n-1)h,nh),\\ f_n(x,v), &{} t=nh. \end{array} \right. \end{aligned}$$

(25)

Remark 2

It can be easily seen that in dimension one, if initial data $f_0$ is such that $f_0(x,v)=1,$ for $0\le v\le k\varepsilon$ and $f_0(x,v)=0$ for $v>((k+1)\varepsilon$, then $f_n$ is evolved by simple translation with kinetic velocities v,  leading to dispersion effect. On the other hand if initial data, for example, has a form

$$\begin{aligned} f_0(x,v) = \left\{ \begin{array}{ll} \mathbb {I}_{[0,v_1]}(v),&{} x<0,\\ \mathbb {I}_{[0,v_2]}(v), &{} x>0, \end{array} \right. \end{aligned}$$

with $v_1-v_2>\varepsilon$ and $A(v)=v$ (corresponding to Burger’s equation) then $f_n$ evolves as a classical shock wave in a discrete-time approximation.

The following properties of $f^h$ follow from its definition and properties established in Lemma 2.

Lemma 3

It holds:

• i. for any (x, v, t),  $f^h\in \{0,1\};$

• ii. for any (x, t),  $\mathrm{supp }f^h\subset [0,L];$

• iii. for any $t>0,$

  $$\begin{aligned} \iint f^h(x,v,t)\,\mathrm{d}v\mathrm{d}x{}\le {}f_0(x,v)\,\mathrm{d}v\mathrm{d}x; \end{aligned}$$

  (26)
  $$\begin{aligned} \iint \eta _\varepsilon (v)f^h(x,v,t)\,\mathrm{d}v\mathrm{d}x{}\le {}\eta _\varepsilon (v)f_0(x,v)\,\mathrm{d}v\mathrm{d}x; \end{aligned}$$

  (27)

• iv. $f^h$ is a weak solution of the equation

  $$\begin{aligned} \partial _tf^h{}+{}A'(v)\cdot \nabla {}_xf^h{}={}R^h, \end{aligned}$$

  (28)

  where

  $$\begin{aligned} R^h{}={}\sum _{n=1}^\infty \delta (t-nh)(f_n(x,v) - f_{n-1}(x-A'(v)h)); \end{aligned}$$

  (29)

• v. for any $t>0$ and any $\xi \in \mathbb {R}^d,$

  $$\begin{aligned} \iint |f^h(x+\xi ,t,v)-f^h(x,t)|\,\mathrm{d}x\mathrm{d}v {}\le {}\iint |f_0(x+\xi ,v)-f_0(x,v)|\,\mathrm{d}x\mathrm{d}v. \end{aligned}$$

Next, we estimate the interaction term $R^h$ in Eq. (28)

Lemma 4

For any $t>0,$

$$\begin{aligned} \iint R^h\,\mathrm{d}v\mathrm{d}x{}\le {}\sum _{n=1}^\infty \delta (t-nh)\iint |(f_n(x,v) - f_{n-1}(x-A'(v)h))|\,\mathrm{d}x\mathrm{d}v; \end{aligned}$$

and

$$\begin{aligned} \int \limits _0^\infty \iint |R^h|\,\mathrm{d}v\mathrm{d}x\mathrm{d}t{}\le \frac{2}{\varepsilon }\iint \eta _\varepsilon (v)f_0(x,v)\,\mathrm{d}v\mathrm{d}x. \end{aligned}$$

Proof

The first inequality is obvious. Using Eq. (28) we find that

$$\begin{aligned} \sum _{n=1}^\infty \iint \eta _\varepsilon (v)(f_n(x,v)-f_{n-1}(x-A'(v)h,v))\,\mathrm{d}x\mathrm{d}v{}\le {} \iint \eta _\varepsilon (v)f_0(x,v)\,\mathrm{d}x\mathrm{d}v. \end{aligned}$$

Since $f_n{}={}M_{f_{n-1}(x-A'(v)h,v)},$ using inequality (19) we get

$$\begin{aligned} \sum _{n=1}^\infty \iint |f_n(x,v)-f_{n-1}(x-A'(v)h,v))|,\mathrm{d}x\mathrm{d}v{}\le {} \frac{2}{\varepsilon }\iint \eta _\varepsilon (v)f_0(x,v)\,\mathrm{d}x\mathrm{d}v, \end{aligned}$$

from which the second inequality of the lemma follows. $\square$

With the information from the last two lemma, we consider compactness properties of $f^h$ as $h\rightarrow 0.$ There is f with a.e. values in [0, 1] and a signed Radon measure $\tilde{m}$ such that on a suitable subsequence $h_k\rightarrow 0,$

$$\begin{aligned} f^{h_k}\rightarrow f\quad \text{*-weakly } \text{ in } L^\infty (\mathbb {R}^{d+1}_+\times [0,L]),\\ R^{h_k}\rightarrow \tilde{m}\quad \text{*-weakly } \text{ in } \mathcal {M}_{loc}(\mathbb {R}^{d+1}_+\times [0,L]), \end{aligned}$$

for a.e. $t>0,$

$$\begin{aligned} \iint f(x,v,t)\,\mathrm{d}v\mathrm{d}x{}\le {}\iint f_0(x,v)\,\mathrm{d}v\mathrm{d}x,\\ \iint \eta _\varepsilon (v)f(x,v,t)\,\mathrm{d}v\mathrm{d}x{}\le {}\iint \eta _\varepsilon (v)f_0(x,v)\,\mathrm{d}v\mathrm{d}x, \end{aligned}$$

and inequalities (8) and (10) hold.

Inequality (20) implies that $\langle \tilde{m},\eta (v)\psi (x,t)\rangle {}\le {} 0$ for any continuously non-decreasing function $\eta ,$ and any non-negative $\psi \in C^\infty _0(\mathbb {R}^{d+1}_+).$ Thus, $\tilde{m}{}={}\partial _v m$ for a non-negative Radon measure.

Now we show that v–moments of $f^h$ are compact in $L^p$ norms.

Lemma 5

Let $\omega (v)$ be a measurable, bounded function on [0, L]. Then, the set of moments

$$\begin{aligned} \left\{ \int \omega (v)f^h(x,v,t)\,\mathrm{d}v\right\} \quad \text{ is } \text{ pre-compact } \text{ in } L^p_{loc}(\mathbb {R}^{d+1}_+),\,p\in [0,+\infty ). \end{aligned}$$

Proof

Denote by $\rho ^h_{\omega }{}={}\int \omega (v)f^h(x,v,t)\,\mathrm{d}v.$ $\rho ^h_{\omega }$ is bounded in $L^\infty (\mathbb {R}^{d+1}_+).$ It follows from part v. of Lemma 3 that for any $\xi \in \mathbb {R}^d,$ and any $T>0,$ and $p\in [1,+\infty ),$

$$\begin{aligned} \Vert \rho ^h_\omega (x+\xi ,t)-\rho ^h_{\omega }(x,t)\Vert _{L^\infty ((0,T); L^p(\mathbb {R}^d))}\rightarrow 0,\quad |\xi |\rightarrow 0, \end{aligned}$$

uniformly in h. It follows from Eq. (28) that for any $T>0$ and $p\in [0,+\infty ),$

$$\begin{aligned} \left\{ \partial _t \rho ^h_\omega \right\} \quad \text{ is } \text{ bounded } \text{ in } \mathcal {M}((0,T);L^p(\mathbb {R}^d)) {}+{} L^\infty ((0,T); W^{-1,p}_{loc}(\mathbb {R}^d)). \end{aligned}$$

Under these conditions, compactness lemma 5.1 of Lions [9] ensures that on a suitable sequence of values of $h\rightarrow 0,$ $(\rho ^h_\omega )^2\rightarrow (\rho _\omega )^2$ in distributional sense, where $\rho _\omega$ is a limiting point of $\rho ^h_\omega$ in *-weak topology of $L^\infty (\mathbb {R}^{d+1}_+).$ This implies the statement of the lemma. $\square$

A little bit more can be said about moments $\rho ^h{}={}\int f^h(x,v,t)\,\mathrm{d}v.$ Indeed,

$$\begin{aligned} \{\partial _t\rho ^h \} \quad \text{ is } \text{ bounded } \text{ in } L^\infty ((0,T); W^{-1,p}_{loc}(\mathbb {R}^d)),\,p\in [1,\infty ). \end{aligned}$$

Thus, $\rho ^h$ converges for a limiting point $\rho ,$ in $C([0,T]; W^{-1,p}_{loc}(\mathbb {R}^d)).$ This shows, in particular, that $\rho (x,0){}={}\int f_0(x,v)\,\mathrm{d}v.$

We consider the moments of $f^h$ from the set $\omega \in \{1,\eta _\varepsilon (v),A_1(v),..,A_d(v)\}$ and select a sequence $h=h_k\rightarrow 0$ on which $f^h$ and $\rho ^h_\omega$ converge in the topologies described above to their limiting values.

To finish the proof of Theorem 1 it remains to establish (9). Let $\hat{\rho }^h=\int \hat{f}^h\,\mathrm{d}v.$ For each (x, t),  $\hat{f}^h(x,t,v)$ is a minimizer of the problem (9) with $\rho =\hat{\rho }^h(x,t).$ Since this problem depends continuously on the value of the constraint $\hat{\rho }^h$ and the latter converges a.e. (x, t) to $\rho (x,t),$ then the limit of the minimizers $\tilde{f}^h$ is a minimizer corresponding to $\rho .$

2.2 Proof of Theorem 2

Part i. of the Theorem 2 was established in proving Theorem 1. Part ii. follows from from (9) and Lemma 1. Indeed, let $\rho ,$ and $\phi$ be given by (11), and (x, t) is such that $f(x,t,\cdot )$ is the minimizer of (9). Let n and $\tilde{f}$ be as in Lemma 1. We can write for any $i=1..d,$

$$\begin{aligned} \phi _i(x,t){}={}\int A_i'(v)f(x,t,v)\,\mathrm{d}v{}= & {} {} A_i(\rho (x,t)){}+{}\int \limits _{n\varepsilon }^{(n+1)\varepsilon } A_i'(v)\left( \tilde{f}-\mathbb {I}_{[0,\rho ]}(v)\right) \,\mathrm{d}v\\ {}= & {} {}A_i(\rho (x,t)){}+{}\int \limits _{n\varepsilon }^{(n+1)\varepsilon } \left( A_i'(v)-A_i'(n\varepsilon )\right) \left( \tilde{f}-\mathbb {I}_{[0,\rho ]}(v)\right) \,\mathrm{d}v\\ {}= & {} {}A_i(\rho (x,t)){}+{}O(\varepsilon ^2), \end{aligned}$$

which establishes (13).

To show part iii of the theorem, we consider the sequence of kinetic functions $f^\varepsilon$ and their moments $\rho ^\varepsilon =\int f^\varepsilon \,\mathrm{d}v,$ $\phi ^\varepsilon _i{}={}\int A_i'(v)f^\varepsilon \,\mathrm{d}v$ from Theorem 1 in the limit $\varepsilon \rightarrow 0.$

Given the uniform bounds on the sequence $f^\varepsilon$, continuity estimate (10) and Eq. (7), one can repeat the arguments of the proof of Theorem 1 to establish that v–moments of $f^\varepsilon$ are pre-compact in $L^p_{loc}(\mathbb {R}^{d+1}_+).$ In particular, $(1, A_1(v),..,A_d(v))$ moments of $f^\varepsilon$ converge (on a subsequence) to a pair $(\rho ,\phi )$ – a solution of (12). $f^\varepsilon$ itself converges weakly to a function that f that verifies the kinetic equation (7), Moreover, a.e. (x, t),  f is a minimizer of problem (9) with function $\eta (v)=v,$ in place of $\eta _\varepsilon .$ This means that f has a structure of an equilibrium density $f(x,\cdot ,t){}={}\mathbb {I}_{\rho (x,t)}(\cdot )$ and, thus, $\phi (x,t){}={}A(\rho (x,t)$ a.e. (x, t).

This new problem

$$\begin{aligned} \min \left\{ \int \eta (v)f(v)\,\mathrm{d}v\,:\, f(v)\in [0,1],\,\int f\,\mathrm{d}v{}={}\rho (x,t)\right\} \end{aligned}$$

has a unique minimizer in the form $f(x,v,t){}={}\mathbb {I}_{[0,\rho (x,t)]}(v).$ Thus, $\phi (x,t) {}={}A(\rho (x,t))$ a.e. (x, t) and $\rho$ is a unique entropy solution of the conservation law (2). The uniqueness implies that the sequence $\rho ^\varepsilon$ converges to $\rho$ in the limit of $\varepsilon \rightarrow 0.$