Kruzhkov's Doubling of Variables and the Entropy Solution

Contents

Statement
The inputs this treatise does not build
From the admissible entropies to Kruzhkov's family
Doubling the variables
The limit and the cone estimate
Existence by vanishing viscosity

This appendix proves Theorem 14.103 of Partial Differential Equations: a scalar conservation law Equation (14.81) with bounded measurable datum has at most one entropy solution in the sense of Definition 14.101, two entropy solutions contract in the mean, and at least one exists. The uniqueness half is Kruzhkov's argument by doubling of variables, and it is the whole content of the section; the existence half is the vanishing-viscosity limit already announced in Remark 14.100, whose architecture is set out in Existence by vanishing viscosity with its two compactness inputs named.

Three features of the argument are worth flagging before it begins, because each is what makes it work. The first is the choice of entropy: the family $\eta(u) = \abs{u-k}$, indexed by a real constant $k$, is not $C^{2}$ and so is not admitted by Definition 14.101 directly; it is reached by approximation in From the admissible entropies to Kruzhkov's family, and it is the family for which the entropy flux of two solutions can be compared. The second is that the comparison is carried out in four variables — $u$ is evaluated at $(t,x)$ and $v$ at $(s,y)$ — with each solution's own entropy inequality tested against the same function and the constant $k$ taken to be the other solution. The third is the cancellation that makes the whole thing finite: with a test function of the form $\chi\bigl(\tfrac{t+s}{2},\tfrac{x+y}{2}\bigr)$ times a mollifier in the differences, the two entropy inequalities add so that the mollifier is differentiated in neither of them.

Throughout, $F \in C^{1}(\R)$, $u$ and $v$ are entropy solutions of Equation (14.81) with bounded measurable data $u_{0}, v_{0}$,

\begin{equation}\tag{A17.1} \abs{u}, \abs{v} \le M_{0} \quad\text{almost everywhere}\ec\qquad M = \max_{\abs{w}\le M_{0}}\abs{F'(w)}\ec \end{equation}

and $\varphi$ denotes a non-negative test function of the kind appearing in Equation (14.92). Convexity of $F$, assumed in Theorem 14.103, is nowhere used below: Kruzhkov's theorem needs only that $F$ be $C^{1}$, which is a strictly stronger result and is recorded in Remark A17.16.

Statement

Definition A17.1 (Kruzhkov entropy pair).

For $k \in \R$ put

\begin{equation}\tag{A17.2} \eta_{k}(w) = \abs{w-k}\ec\qquad q_{k}(w) = \sgn(w-k)\bigl(F(w)-F(k)\bigr)\ep \end{equation}

Rests on Definitions 14.93 and 14.101.

Theorem A17.2 ($L^{1}$ contraction on a cone).

Let $u,v$ be entropy solutions in the sense of Definition 14.101 obeying Equation (A17.1). Then for almost every $0 < \tau_{1} < \tau_{2}$ and every $R>0$

\begin{equation}\tag{A17.3} \int_{\abs{x}\le R}\abs{u(\tau_{2},x)-v(\tau_{2},x)}\,\dd x \le \int_{\abs{x}\le R+M(\tau_{2}-\tau_{1})} \abs{u(\tau_{1},x)-v(\tau_{1},x)}\,\dd x\ec \end{equation}

and, using Lemma A17.5, the same inequality holds with $\tau_{1}=0$ and $u,v$ replaced there by $u_{0},v_{0}$. If in addition $u_{0}-v_{0}$ is integrable on $\R$, then for almost every $t>0$

\begin{equation}\tag{A17.4} \int_{\R}\abs{u(t,x)-v(t,x)}\,\dd x \le \int_{\R}\abs{u_{0}(x)-v_{0}(x)}\,\dd x\ep \end{equation}

Rests on Definitions 14.95, 14.101 and A17.1.

Corollary A17.3 (Uniqueness).

An entropy solution with a given bounded measurable datum is unique up to a null set, and the solution map $u_{0}\longmapsto u(t,\cdot)$ is a contraction for the $L^{1}$ distance — which is condition (iii) of Definition 14.24 with the mean as the norm. Rests on Theorem A17.2 and Definition 14.24.

Proof.

Derives Corollary A17.3. Let $u_{0} = v_{0}$ almost everywhere. Then the right-hand side of Equation (A17.3) with $\tau_{1}=0$ vanishes for every $R$, so $u(t,\cdot) = v(t,\cdot)$ almost everywhere on every bounded interval and hence on $\R$, for almost every $t$. Continuous dependence is Equation (A17.4) itself, the map being $1$-Lipschitz from $L^{1}$ to $L^{1}$.

∎

The inputs this treatise does not build

Remark A17.4 (What is quoted here).

Unlike the rest of Partial Differential Equations, the subject of this section is not formulated in terms of the Riemann integral and cannot be. A weak solution in the sense of Definition 14.95 is a bounded measurable function, its defining identity is an integral over a region of the plane against an arbitrary test function, and the conclusions are statements valid almost everywhere; every one of these words belongs to Lebesgue's theory, which Real Analysis does not develop — it builds the Riemann integral only, as Probability and Statistics also records where it needs dominated convergence and declines to use it. The following facts are therefore assumed, and they are assumed for the definitions as much as for the proofs:

The Lebesgue integral of a bounded measurable function over a measurable subset of $\R^{d}$, its linearity and monotonicity, and the fact that a function vanishing almost everywhere has zero integral.
Fubini's theorem for a bounded measurable function on a product, used in Proposition A17.10 to integrate one solution's entropy inequality over the other solution's variables, and again to change variables from $(t,x,s,y)$ to sum and difference coordinates.
Dominated convergence, used at each passage to a limit.
Continuity of translation in $L^{1}$: for $w$ bounded and measurable and $K$ compact, $\int_{K}\abs{w(\cdot+h)-w} \longrightarrow 0$ as $h \to 0$. Equivalently, almost every point is a Lebesgue point. This is the one substantial analytic fact of Proposition A17.13, and it is exactly what replaces the continuity that a classical solution would have had.
Lebesgue's differentiation theorem in one variable, used in The limit and the cone estimate to pass from an average over a short time interval to a value at almost every instant.
Kruzhkov's initial-layer lemma (Lemma A17.5): an entropy solution attains its datum in the local mean. This is the only theorem of the subject that is quoted rather than proved, and it is isolated as such below.
For the existence half only (Existence by vanishing viscosity): classical solvability of the viscous problem, and a compactness theorem for families of uniformly bounded functions of uniformly bounded variation.

Everything else — the passage from the $C^{2}$ entropies of Definition 14.101 to the Kruzhkov family, the doubling, the cancellation, the limit, the cone estimate, and the derivation of the entropy inequality from the viscous equation — is carried out in full. The primary source is [Kruzhkov:1970], where the entropy class, the doubling of variables and the $L^{1}$ contraction first appear together.

Lemma A17.5 (Quoted: the initial layer).

Let $u$ be an entropy solution in the sense of Definition 14.101 with datum $u_{0}$. Then for every compact $K \subset \R$

\begin{equation}\tag{A17.5} \lim_{t\to0^{+}}\int_{K}\abs{u(t,x)-u_{0}(x)}\,\dd x = 0\ec \end{equation}

the limit being taken through the instants at which $u(t,\cdot)$ is defined.

Remark A17.6.

Equation (A17.5) is stronger than what the definition plainly gives, and the gap is worth naming. Testing Equation (14.85) with $\varphi(x,t)=\rho(x)\alpha(t)$ shows that $t \longmapsto \int u(t,x)\rho(x)\,\dd x$ agrees almost everywhere with an absolutely continuous function taking the value $\int u_{0}\rho$ at $t = 0$, so the datum is attained weakly; that much is a two-line computation from the definition. Weak attainment is not enough for Theorem A17.2, because the functional $w \longmapsto \int\abs{w}$ is only lower semicontinuous under weak convergence and the inequality needed runs the other way. Kruzhkov obtains Equation (A17.5) from the entropy inequality itself applied with constant $k$ and a test function concentrating at $t=0$, which yields an upper bound on $\limsup_{t\to0}\int\abs{u(t,x)-k}\rho$ by $\int\abs{u_{0}-k}\rho$; a covering argument over a countable dense set of values $k$, together with approximation of $u_{0}$ by step functions, upgrades that to the strong statement. The argument is measure-theoretic throughout and is not reproduced here.

From the admissible entropies to Kruzhkov's family

Proposition A17.7 (The Kruzhkov inequality).

Let $u$ be an entropy solution in the sense of Definition 14.101. Then for every $k \in \R$ and every non-negative test function $\varphi$,

\begin{equation}\tag{A17.6} \int_{0}^{\infty}\!\!\int_{\R} \left[\abs{u-k}\,\pp_{t}\varphi + q_{k}(u)\,\pp_{x}\varphi\right]\dd x\,\dd t + \int_{\R}\abs{u_{0}(x)-k}\,\varphi(x,0)\,\dd x \ge 0\ec \end{equation}

with $q_{k}$ of Equation (A17.2). Rests on Definition 14.101, Definition A17.1 and Equation (14.92).

Proof.

Derives Proposition A17.7. For $\delta>0$ set

\begin{equation}\tag{A17.7} \eta_{\delta}(w) = \sqrt{(w-k)^{2}+\delta^{2}} - \delta\ec\qquad q_{\delta}(w) = \int_{k}^{w}\eta_{\delta}'(\lambda)\,F'(\lambda)\, \dd\lambda\ep \end{equation}

Then $\eta_{\delta}$ is $C^{\infty}$, with

\begin{equation}\tag{A17.8} \eta_{\delta}'(w) = \frac{w-k}{\sqrt{(w-k)^{2}+\delta^{2}}}\ec\qquad \eta_{\delta}''(w) = \frac{\delta^{2}}{\bigl[(w-k)^{2}+\delta^{2}\bigr]^{3/2}} > 0\ec \end{equation}

so it is convex; $q_{\delta}$ is $C^{1}$ with $q_{\delta}' = \eta_{\delta}'F'$ by the fundamental theorem of calculus. Thus $(\eta_{\delta},q_{\delta})$ is an admissible entropy pair in the sense of Definition 14.101, and Equation (14.92) holds for it.

Uniform convergence of the entropy. From $\abs{w-k} \le \sqrt{(w-k)^{2}+\delta^{2}} \le \abs{w-k}+\delta$,

\begin{equation}\tag{A17.9} \abs{\eta_{\delta}(w) - \abs{w-k}} \le \delta \qquad\text{for every } w\ep \end{equation}

Uniform convergence of the flux on the range. Let $\abs{w} \le M_{0}$ and abbreviate $r_{\delta}(\lambda) = \sgn(\lambda-k) - \eta_{\delta}'(\lambda)$, so that $\abs{r_{\delta}} \le 2$ and, for $\abs{\lambda-k}\ge\varepsilon$,

\begin{equation}\tag{A17.10} \abs{r_{\delta}(\lambda)} = 1 - \frac{\abs{\lambda-k}}{\sqrt{(\lambda-k)^{2}+\delta^{2}}} \le 1 - \frac{\varepsilon}{\sqrt{\varepsilon^{2}+\delta^{2}}}\ep \end{equation}

Since $\sgn(\lambda-k)$ is constant and equal to $\sgn(w-k)$ for $\lambda$ strictly between $k$ and $w$, $\int_{k}^{w}\sgn(\lambda-k)F'(\lambda)\,\dd\lambda = \sgn(w-k)\bigl(F(w)-F(k)\bigr) = q_{k}(w)$, whence

\begin{equation}\tag{A17.11} \abs{q_{k}(w)-q_{\delta}(w)} \le \left|\int_{k}^{w}\abs{r_{\delta}(\lambda)}\, \abs{F'(\lambda)}\,\dd\lambda\right| \le M\left[2\varepsilon + 2M_{0}\left(1-\frac{\varepsilon} {\sqrt{\varepsilon^{2}+\delta^{2}}}\right)\right]\ec \end{equation}

the interval of integration having been split at $\abs{\lambda-k}=\varepsilon$ and its length bounded by $2M_{0}$ — here $\abs{k} \le M_{0}$ may be assumed, since for $\abs{k}>M_{0}$ the sign of $u-k$ is constant and Equation (A17.6) reduces to Equation (14.85) with a sign. Choosing first $\varepsilon$ and then $\delta$, the right-hand side of Equation (A17.11) is made arbitrarily small uniformly in $w$: $q_{\delta} \longrightarrow q_{k}$ uniformly on $[-M_{0},M_{0}]$.

Passage to the limit. Apply Equation (14.92) to $(\eta_{\delta},q_{\delta})$ and subtract the corresponding expression built from $(\eta_{k},q_{k})$. Each of the three differences is bounded in modulus by the uniform bounds Equation (A17.9) and Equation (A17.11) times the integral of $\abs{\pp_{t}\varphi}$, $\abs{\pp_{x}\varphi}$ or $\abs{\varphi(\cdot,0)}$ over the compact support of $\varphi$, a fixed finite number. Letting $\delta\to0$ therefore gives Equation (A17.6). No convergence theorem is needed at this step: the convergence is uniform on a set of finite measure.

∎

Lemma A17.8 (Two elementary identities).

For all $a,b \in [-M_{0},M_{0}]$, with $\Phi(a,b) = \sgn(a-b)\bigl(F(a)-F(b)\bigr)$,

\begin{equation}\tag{A17.12} \Phi(a,b) = F\bigl(\max(a,b)\bigr) - F\bigl(\min(a,b)\bigr)\ec \end{equation}

so that $\Phi$ is symmetric, continuous, and Lipschitz:

\begin{equation}\tag{A17.13} \abs{\Phi(a,b)} \le M\,\abs{a-b}\ec\qquad \abs{\Phi(a,b)-\Phi(a',b')} \le M\bigl(\abs{a-a'}+\abs{b-b'}\bigr)\ep \end{equation}

Rests on Definition A17.1 and Equation (A17.1).

Proof.

Derives Lemma A17.8. If $a>b$ then $\sgn(a-b)=1$, $\max=a$, $\min=b$ and both sides of Equation (A17.12) equal $F(a)-F(b)$; if $a<b$ then $\sgn(a-b)=-1$, $\max=b$, $\min=a$ and both sides equal $F(b)-F(a)$; if $a=b$ both vanish. Symmetry and continuity are then manifest, $\max$ and $\min$ being continuous. For the bounds, the mean value theorem gives $\abs{F(\max)-F(\min)} \le M\abs{\max-\min} = M\abs{a-b}$, which is the first; and since $a \longmapsto \max(a,b)$ and $a\longmapsto\min(a,b)$ are $1$-Lipschitz in each argument, so is $F\circ\max$ and $F\circ\min$ up to the factor $M$, giving the second.

∎

Remark A17.9 (Why $\abs{u-k}$ and not a general convex entropy).

Equation (A17.12) is the reason Kruzhkov's family is the right one. What the doubling will produce is the pair $\bigl(\abs{u-v},\,\Phi(u,v)\bigr)$ built from two solutions, and for this to be usable it must be a genuine entropy pair in each variable separately with the other frozen — which is exactly what Definition A17.1 provides, the frozen solution playing the role of the constant $k$. No single convex $\eta$ of Definition 14.101 has that property, because a general $\eta$ knows nothing about the second solution. The price is that $\abs{u-k}$ is not $C^{2}$, and Proposition A17.7 is what pays it.

Doubling the variables

Proposition A17.10 (The doubled inequality).

Let $\psi = \psi(t,x,s,y) \ge 0$ be smooth with compact support in $\set{t>0}\times\R\times\set{s>0}\times\R$. Then

\begin{equation}\tag{A17.14} \int\!\!\int\!\!\int\!\!\int \Bigl[\abs{u(t,x)-v(s,y)}\, \bigl(\pp_{t}+\pp_{s}\bigr)\psi + \Phi\bigl(u(t,x),v(s,y)\bigr)\, \bigl(\pp_{x}+\pp_{y}\bigr)\psi\Bigr] \ge 0\ec \end{equation}

the integral being over $(t,x,s,y)$ with $t,s>0$. Rests on Proposition A17.7 and Lemma A17.8.

Proof.

Derives Proposition A17.10. Fix $(s,y)$ with $s>0$ and apply Equation (A17.6) to the solution $u$, with the constant $k = v(s,y)$ — legitimate for almost every $(s,y)$, since $\abs{v(s,y)}\le M_{0}$ is then a genuine real number — and with the non-negative test function $(t,x)\longmapsto\psi(t,x,s,y)$, which is smooth with compact support in $\set{t>0}\times\R$. The initial term is absent because $\psi$ vanishes for $t$ near $0$. Recalling $q_{v(s,y)}\bigl(u(t,x)\bigr) = \Phi\bigl(u(t,x),v(s,y)\bigr)$ from Equation (A17.2),

\begin{equation}\tag{A17.15} \int_{0}^{\infty}\!\!\int_{\R} \Bigl[\abs{u-v(s,y)}\,\pp_{t}\psi + \Phi\bigl(u,v(s,y)\bigr)\,\pp_{x}\psi\Bigr] \dd x\,\dd t \ge 0\ep \end{equation}

The left-hand side of Equation (A17.15) is a bounded measurable function of $(s,y)$ — bounded because the integrand is bounded by $2M_{0}\abs{\pp_{t}\psi}+2MM_{0}\abs{\pp_{x}\psi}$ using Equation (A17.13) — so it may be integrated over $\set{s>0}\times\R$, and Fubini's theorem rearranges the result into a single integral over the four variables. This gives Equation (A17.14) with only the $\pp_{t}$ and $\pp_{x}$ terms.

Now exchange the roles. Fix $(t,x)$ with $t>0$ and apply Equation (A17.6) to $v$, with $k = u(t,x)$ and the test function $(s,y)\longmapsto\psi(t,x,s,y)$; by the symmetry of $\Phi$ recorded in Lemma A17.8, $q_{u(t,x)}\bigl(v(s,y)\bigr) = \Phi\bigl(v(s,y),u(t,x)\bigr) = \Phi\bigl(u(t,x),v(s,y)\bigr)$, and $\abs{v-u} = \abs{u-v}$. Integrating over $(t,x)$ gives Equation (A17.14) with the $\pp_{s}$ and $\pp_{y}$ terms. Adding the two inequalities gives Equation (A17.14).

∎

Proposition A17.11 (The cancellation).

Let $\chi \ge 0$ be smooth with compact support in $(0,\infty)\times\R$, let $\theta \in C^{\infty}(\R)$ be even and non-negative with support in $(-1,1)$ and $\int\theta = 1$, and put $\theta_{\delta}(\sigma) = \delta^{-1}\theta(\sigma/\delta)$ and

\begin{equation}\tag{A17.16} \psi_{\delta}(t,x,s,y) = \chi\!\left(\frac{t+s}{2},\frac{x+y}{2}\right)\, \theta_{\delta}\!\left(\frac{t-s}{2}\right)\, \theta_{\delta}\!\left(\frac{x-y}{2}\right)\ep \end{equation}

For $\delta$ small enough $\psi_{\delta}$ is admissible in Proposition A17.10, and in the sum-and-difference coordinates

\begin{equation}\tag{A17.17} \tau = \frac{t+s}{2}\ec\quad \zeta = \frac{x+y}{2}\ec\quad \sigma = \frac{t-s}{2}\ec\quad \omega = \frac{x-y}{2}\ec \end{equation}

one has

\begin{equation}\tag{A17.18} \bigl(\pp_{t}+\pp_{s}\bigr)\psi_{\delta} = \bigl(\pp_{\tau}\chi\bigr)\, \theta_{\delta}(\sigma)\theta_{\delta}(\omega)\ec \qquad \bigl(\pp_{x}+\pp_{y}\bigr)\psi_{\delta} = \bigl(\pp_{\zeta}\chi\bigr)\, \theta_{\delta}(\sigma)\theta_{\delta}(\omega)\ep \end{equation}

Rests on Proposition A17.10 and Equation (A17.16).

Proof.

Derives Proposition A17.11. By the chain rule applied to Equation (A17.17), $\pp_{t} = \tfrac12\pp_{\tau}+\tfrac12\pp_{\sigma}$ and $\pp_{s} = \tfrac12\pp_{\tau}-\tfrac12\pp_{\sigma}$, so $\pp_{t}+\pp_{s} = \pp_{\tau}$; likewise $\pp_{x}+\pp_{y} = \pp_{\zeta}$. In Equation (A17.16) the factors $\theta_{\delta}$ depend only on $\sigma$ and $\omega$ and are therefore annihilated by $\pp_{\tau}$ and $\pp_{\zeta}$, while $\chi$ depends only on $(\tau,\zeta)$; that is Equation (A17.18). Admissibility: the support of $\psi_{\delta}$ lies within distance $2\delta$ of the diagonal $t=s$, $x=y$ over the support of $\chi$, so if $2\delta$ is smaller than the distance from the support of $\chi$ to $\set{\tau = 0}$ then $t,s>0$ throughout.

∎

Remark A17.12.

This is the step the method is named for and the only one that could fail. Had the mollifier been differentiated, each derivative would have produced a factor $\delta^{-1}$, and the limit $\delta\to0$ would have been meaningless. It survives because the two entropy inequalities are added, not subtracted, and because the sum of the two time derivatives — one for each copy of the equation — is a derivative along the diagonal, in whose direction the mollifier is constant. In effect the argument tests the difference of two solutions against a function that is smooth along the diagonal and singular across it, and only the diagonal direction is ever differentiated.

The limit and the cone estimate

Proposition A17.13 (The limit $\delta\to0$).

For every non-negative $\chi$ smooth with compact support in $(0,\infty)\times\R$,

\begin{equation}\tag{A17.19} \int_{0}^{\infty}\!\!\int_{\R} \Bigl[\abs{u-v}\,\pp_{t}\chi + \Phi(u,v)\,\pp_{x}\chi\Bigr]\dd x\,\dd t \ge 0\ec \end{equation}

that is, $\pp_{t}\abs{u-v} + \pp_{x}\Phi(u,v) \le 0$ in the sense of distributions on $(0,\infty)\times\R$. Rests on Proposition A17.11 and Lemma A17.8.

Proof.

Derives Proposition A17.13. Insert Equation (A17.18) into Equation (A17.14) and change variables to Equation (A17.17). The map $(t,x,s,y)\longmapsto(\tau,\zeta,\sigma,\omega)$ is linear with $\dd t\,\dd s = 2\,\dd\tau\,\dd\sigma$ and $\dd x\,\dd y = 2\,\dd\zeta\,\dd\omega$, so

\begin{equation}\tag{A17.20} \int\!\!\int \left[\int\!\!\int \Bigl(G_{\sigma\omega}\,\pp_{\tau}\chi + H_{\sigma\omega}\,\pp_{\zeta}\chi\Bigr) \theta_{\delta}(\sigma)\theta_{\delta}(\omega)\, \dd\sigma\,\dd\omega\right]\dd\zeta\,\dd\tau \ge 0\ec \end{equation}

after dividing by the positive constant $4$, where

\begin{equation}\tag{A17.21} G_{\sigma\omega}(\tau,\zeta) = \abs{u(\tau+\sigma,\zeta+\omega)-v(\tau-\sigma,\zeta-\omega)}\ec \quad H_{\sigma\omega}(\tau,\zeta) = \Phi\bigl(u(\tau+\sigma,\zeta+\omega), v(\tau-\sigma,\zeta-\omega)\bigr)\ep \end{equation}

Let $K$ be a compact neighbourhood of the support of $\chi$ in $(0,\infty)\times\R$. By continuity of translation in $L^{1}$ (item 4 of Remark A17.4) both $(\tau,\zeta)\longmapsto u(\tau+\sigma,\zeta+\omega)$ and $(\tau,\zeta)\longmapsto v(\tau-\sigma,\zeta-\omega)$ converge to $u$ and $v$ in $L^{1}(K)$ as $(\sigma,\omega)\to(0,0)$; hence, the maps $(a,b)\longmapsto\abs{a-b}$ and $(a,b)\longmapsto\Phi(a,b)$ being Lipschitz by Equation (A17.13),

\begin{equation}\tag{A17.22} \int_{K}\abs{G_{\sigma\omega}-\abs{u-v}} \longrightarrow 0\ec\qquad \int_{K}\abs{H_{\sigma\omega}-\Phi(u,v)} \longrightarrow 0\ep \end{equation}

Denote by $\Xi(\sigma,\omega)$ the inner bracket of Equation (A17.20) integrated against $\chi$'s derivatives, i.e. the quantity $\iint\bigl(G_{\sigma\omega}\pp_{\tau}\chi + H_{\sigma\omega}\pp_{\zeta}\chi\bigr)\dd\zeta\,\dd\tau$, and by $\Xi_{0}$ the left-hand side of Equation (A17.19). Then $\abs{\Xi(\sigma,\omega)-\Xi_{0}} \le \norm{\nabla\chi}_{\infty}\int_{K} \bigl(\abs{G_{\sigma\omega}-\abs{u-v}} + \abs{H_{\sigma\omega}-\Phi(u,v)}\bigr)$, which by Equation (A17.22) tends to $0$ as $(\sigma,\omega)\to0$. But Equation (A17.20) says exactly $\iint\Xi(\sigma,\omega)\theta_{\delta}(\sigma)\theta_{\delta}(\omega) \ge 0$, an average of $\Xi$ over the square $\abs{\sigma},\abs{\omega}<\delta$ with a non-negative weight of total mass $1$; as $\delta\to0$ that average converges to $\Xi_{0}$. Hence $\Xi_{0}\ge0$, which is Equation (A17.19).

∎

Proof of Theorem A17.2. Derives Theorem A17.2. Write $w = \abs{u-v}$ and $\Psi = \Phi(u,v)$, so that Equation (A17.19) holds and $\abs{\Psi} \le M\,w$ by Equation (A17.13).

The cone cutoff. Fix $R>0$ and $T>0$. For $\epsilon>0$ let $\Lambda_{\epsilon} \in C^{\infty}(\R)$ be non-decreasing with $\Lambda_{\epsilon} = 0$ on $(-\infty,0]$, $\Lambda_{\epsilon}=1$ on $[\epsilon,\infty)$ and $0 \le \Lambda_{\epsilon}' \le 2/\epsilon$; and let $n_{\epsilon} \in C^{\infty}(\R)$ be even with $\abs{n_{\epsilon}'} \le 1$ and $\abs{n_{\epsilon}(x)-\abs{x}} \le \epsilon$ — for instance $n_{\epsilon} = \sqrt{x^{2}+\epsilon^{2}}$. Put

\begin{equation}\tag{A17.23} \beta(t,x) = \Lambda_{\epsilon}\bigl(R + M(T-t) - n_{\epsilon}(x)\bigr)\ec \end{equation}

a smooth function with values in $[0,1]$, equal to $1$ well inside the backward cone $\set{n_{\epsilon}(x) \le R+M(T-t)}$ and to $0$ outside it. Let $\alpha \ge 0$ be smooth with compact support in $(0,T]$ and take $\chi = \alpha\beta$, which is admissible in Equation (A17.19) after multiplication by a fixed spatial cutoff equal to $1$ on the (bounded) support of $\beta$ — the cone is bounded for $t \ge 0$ — so that $\chi$ has compact support.

The lateral term has a sign. Since $\pp_{t}\chi = \alpha'\beta + \alpha\,\pp_{t}\beta$ and $\pp_{x}\chi = \alpha\,\pp_{x}\beta$, with

\begin{equation}\tag{A17.24} \pp_{t}\beta = -M\,\Lambda_{\epsilon}'\ec\qquad \pp_{x}\beta = -\Lambda_{\epsilon}'\,n_{\epsilon}'(x)\ec \end{equation}

the contribution of the second group to Equation (A17.19) is

\begin{equation}\tag{A17.25} \int\!\!\int \alpha\left[w\,\pp_{t}\beta + \Psi\,\pp_{x}\beta\right] = -\int\!\!\int \alpha\,\Lambda_{\epsilon}' \left[M\,w + \Psi\,n_{\epsilon}'\right] \le 0\ec \end{equation}

because $\alpha \ge 0$, $\Lambda_{\epsilon}'\ge0$, and $M w + \Psi n_{\epsilon}' \ge Mw - \abs{\Psi} \ge 0$ by $\abs{n_{\epsilon}'}\le1$ and $\abs{\Psi}\le Mw$. This is the point of choosing the slope of the cone to be $M$: the flux can never carry mass out through the lateral boundary faster than the boundary retreats. Subtracting Equation (A17.25) from Equation (A17.19) leaves

\begin{equation}\tag{A17.26} \int_{0}^{\infty}\alpha'(t)\, \left[\int_{\R}\beta(t,x)\,w(t,x)\,\dd x\right]\dd t \ge 0\ep \end{equation}

From an average to a value. Let $0<\tau_{1}<\tau_{2}\le T$ and let $\alpha = \alpha_{h}$ rise smoothly from $0$ to $1$ on $[\tau_{1},\tau_{1}+h]$ and fall back to $0$ on $[\tau_{2},\tau_{2}+h]$, with $\alpha_{h}'$ bounded by $2/h$ and of one sign on each interval. Then Equation (A17.26) reads, in the limit of the standard smooth ramps,

\begin{equation}\tag{A17.27} \frac{1}{h}\int_{\tau_{1}}^{\tau_{1}+h}\Theta(t)\,\dd t \ \ge\ \frac{1}{h}\int_{\tau_{2}}^{\tau_{2}+h}\Theta(t)\,\dd t\ec \qquad \Theta(t) = \int_{\R}\beta(t,x)\,w(t,x)\,\dd x\ec \end{equation}

up to an error tending to $0$ with the width of the ramps. $\Theta$ is bounded and measurable, so by Lebesgue's differentiation theorem (item 5 of Remark A17.4) the two averages converge, as $h\to0$, to $\Theta(\tau_{1})$ and $\Theta(\tau_{2})$ for almost every $\tau_{1},\tau_{2}$. Hence $\Theta(\tau_{2}) \le \Theta(\tau_{1})$ for almost all $\tau_{1}<\tau_{2}$.

Removing the smoothing. As $\epsilon\to0$, $\beta(t,x)$ tends to $1$ at every $x$ with $\abs{x} < R+M(T-t)$ and to $0$ at every $x$ with $\abs{x} > R+M(T-t)$, and $0\le\beta\le1$ throughout; by dominated convergence, applied with the integrable majorant $2M_{0}\,\indic{\abs{x}\le R+MT}$,

\begin{equation}\tag{A17.28} \int_{\abs{x}\le R+M(T-\tau_{2})}w(\tau_{2},x)\,\dd x \ \le\ \int_{\abs{x}\le R+M(T-\tau_{1})}w(\tau_{1},x)\,\dd x\ep \end{equation}

Choosing $T = \tau_{2}$ turns Equation (A17.28) into Equation (A17.3).

Down to $t=0$. By Lemma A17.5 applied to $u$ and to $v$ on the compact set $\abs{x}\le R+M\tau_{2}$, $w(\tau_{1},\cdot)\longrightarrow\abs{u_{0}-v_{0}}$ in the mean over that set as $\tau_{1}\to0^{+}$ — the triangle inequality gives $\bigl|\,\abs{u(\tau_{1})-v(\tau_{1})} - \abs{u_{0}-v_{0}}\,\bigr| \le \abs{u(\tau_{1})-u_{0}} + \abs{v(\tau_{1})-v_{0}}$ — so the right-hand side of Equation (A17.3) converges to $\int_{\abs{x}\le R+M\tau_{2}}\abs{u_{0}-v_{0}}$ along a sequence of admissible $\tau_{1}$, which proves the version with $\tau_{1}=0$.

The global statement. If $u_{0}-v_{0}$ is integrable, let $R\to\infty$ in that version: the left-hand side increases to $\int_{\R}w(t,x)\,\dd x$ by monotone passage on an increasing family of sets, and the right-hand side increases to $\int_{\R}\abs{u_{0}-v_{0}}$, giving Equation (A17.4).

∎

Existence by vanishing viscosity

The uniqueness half is now complete. Existence is the limit $\varepsilon \to 0^{+}$ of the viscous problem, and the one step that is genuinely about entropy — and the only one that would be invisible in a purely functional-analytic account — is carried out here in full.

Proposition A17.14 (The viscous problem produces the entropy inequality).

Let $u^{\varepsilon}$ be a bounded classical solution of

\begin{equation}\tag{A17.29} \pp_{t}u^{\varepsilon} + \pp_{x}F(u^{\varepsilon}) = \varepsilon\,\pp_{x}^{2}u^{\varepsilon}\ec\qquad u^{\varepsilon}(0,\cdot) = u_{0}^{\varepsilon}\ec \end{equation}

with $\abs{u^{\varepsilon}} \le M_{0}$, and let $(\eta,q)$ be an entropy pair in the sense of Definition 14.101. Then for every non-negative test function $\varphi$

\begin{equation}\tag{A17.30} \int_{0}^{\infty}\!\!\int_{\R} \left[\eta(u^{\varepsilon})\pp_{t}\varphi + q(u^{\varepsilon})\pp_{x}\varphi\right]\dd x\,\dd t + \int_{\R}\eta\bigl(u_{0}^{\varepsilon}\bigr)\varphi(x,0)\,\dd x \ \ge\ -\varepsilon\,C_{\varphi}\ec \end{equation}

with $C_{\varphi} = \max_{\abs{w}\le M_{0}}\abs{\eta(w)}\cdot \iint\abs{\pp_{x}^{2}\varphi}$, a constant independent of $\varepsilon$. Rests on Definition 14.101, Equation (14.81) and Theorem 14.80.

Proof.

Derives Proposition A17.14. Multiply Equation (A17.29) by $\eta'(u^{\varepsilon})$. On the left, the chain rule and $q' = \eta'F'$ give $\eta'(u^{\varepsilon})\pp_{t}u^{\varepsilon} = \pp_{t}\eta(u^{\varepsilon})$ and $\eta'(u^{\varepsilon})\pp_{x}F(u^{\varepsilon}) = \eta'(u^{\varepsilon})F'(u^{\varepsilon})\pp_{x}u^{\varepsilon} = \pp_{x}q(u^{\varepsilon})$. On the right,

\begin{equation}\tag{A17.31} \varepsilon\,\eta'(u^{\varepsilon})\,\pp_{x}^{2}u^{\varepsilon} = \varepsilon\,\pp_{x}^{2}\eta(u^{\varepsilon}) - \varepsilon\,\eta''(u^{\varepsilon}) \bigl(\pp_{x}u^{\varepsilon}\bigr)^{2} \ \le\ \varepsilon\,\pp_{x}^{2}\eta(u^{\varepsilon})\ec \end{equation}

the identity being the chain rule applied twice and the inequality being $\eta'' \ge 0$, which is the convexity demanded by Definition 14.101 and the only place it is used. Hence

\begin{equation}\tag{A17.32} \pp_{t}\eta(u^{\varepsilon}) + \pp_{x}q(u^{\varepsilon}) \le \varepsilon\,\pp_{x}^{2}\eta(u^{\varepsilon}) \end{equation}

pointwise. Multiply Equation (A17.32) by $\varphi \ge 0$, integrate over $\set{t>0}\times\R$ and integrate by parts, all boundary terms at spatial infinity vanishing with $\varphi$ and the one at $t=0$ producing $-\int\eta(u_{0}^{\varepsilon})\varphi(x,0)\,\dd x$:

\begin{equation*} -\int\!\!\int\left[\eta\,\pp_{t}\varphi + q\,\pp_{x}\varphi\right] - \int\eta\bigl(u_{0}^{\varepsilon}\bigr)\varphi(x,0)\,\dd x \ \le\ \varepsilon\int\!\!\int \eta(u^{\varepsilon})\,\pp_{x}^{2}\varphi\ec \end{equation*}

and the right-hand side is bounded in modulus by $\varepsilon C_{\varphi}$, which is Equation (A17.30). The two integrations by parts on the viscous term are the reason the estimate is $O(\varepsilon)$ and not $O(1)$: all the differentiation has been moved onto the test function, and no derivative of $u^{\varepsilon}$ survives.

∎

Remark A17.15 (The remaining inputs for existence).

Proposition A17.14 is the whole of the entropy argument; what it does not supply is a limit to take it to. Three further steps are needed, of which the first is elementary in this treatise and the other two are the quoted items (7) of Remark A17.4.

The supremum bound. $\abs{u^{\varepsilon}} \le \norm{u_{0}^{\varepsilon}}_{\infty} \le \norm{u_{0}}_{\infty}$, by the parabolic maximum principle Theorem 14.80 applied to Equation (A17.29) on large space-time rectangles: the equation is of the form treated there with the extra first-order term $F'(u^{\varepsilon})\pp_{x}u^{\varepsilon}$, which vanishes at an interior extremum and so does not disturb the argument. This is why $M_{0}$ and hence $M$ in Equation (A17.1) can be fixed independently of $\varepsilon$.
Classical solvability of Equation (A17.29) for smooth bounded data $u_{0}^{\varepsilon} = u_{0}$ mollified, on $\set{t>0}\times\R$. This is parabolic theory and is quoted.
Compactness. The spatial derivative $p = \pp_{x}u^{\varepsilon}$ satisfies the linear parabolic equation $\pp_{t}p + \pp_{x}\bigl(F'(u^{\varepsilon})p\bigr) = \varepsilon\pp_{x}^{2}p$, from which the total variation $\int\abs{p(t,x)}\,\dd x$ is non-increasing in $t$; together with (i) this gives a family uniformly bounded and of uniformly bounded variation, and a compactness theorem for such families produces a sequence $\varepsilon_{j}\to0$ with $u^{\varepsilon_{j}}\to u$ in $L^{1}$ on compact sets and almost everywhere. The compactness theorem is quoted.

Granting these, pass to the limit in Equation (A17.30): $\eta$ and $q$ are continuous and $u^{\varepsilon_{j}}\to u$ almost everywhere with values in a fixed bounded interval, so dominated convergence carries both integrals, the data converge because $u_{0}^{\varepsilon}\to u_{0}$ in $L^{1}$ on compact sets, and the right-hand side tends to $0$. The limit is exactly Equation (14.92). Taking $\eta(w)=w$ and $q = F$ — a legitimate, if degenerate, member of Definition 14.101, for which Equation (A17.31) is an equality — gives the same statement with equality, which is Equation (14.85): the limit is a weak solution.

Proof of Theorem 14.103. Derives Theorem 14.103. Existence is Remark A17.15: the vanishing-viscosity limit is a weak solution in the sense of Definition 14.95 satisfying Equation (14.92) for every entropy pair, hence an entropy solution in the sense of Definition 14.101. Uniqueness and continuous dependence in the mean are Corollary A17.3, which rests on Theorem A17.2. The last clause of the chapter's statement — that where the solution is piecewise $C^{1}$ across a $C^{1}$ curve its jumps obey Equation (14.91) — is Proposition 14.102, proved in the chapter, and is not re-derived here.

∎

Remark A17.16 (Convexity of the flux is not needed).

Theorem 14.103 assumes $F$ convex, and the chapter has good reason to: convexity is what makes Proposition 14.102 equate the entropy inequality with the geometric condition Equation (14.91), and it is what makes the shock picture of Section 14.6.3 the whole story. But the proof given above never used it. Only $F \in C^{1}$ entered, through the bound $M$ of Equation (A17.1) and the Lipschitz estimate Equation (A17.13). What is lost without convexity is not uniqueness but the description of the admissible discontinuities: for a non-convex flux the entropy inequality still selects one solution, but the admissible jumps are described by a chord condition — the chord joining the two states must lie on one side of the graph of $F$ between them — rather than by the simple inequality $u_{-} \ge u_{+}$. The function $G$ of Equation (14.94) is exactly that chord difference, and the chapter's proof of Proposition 14.102 is where convexity is spent: it is used to pass from $G(u_{-}) = G(u_{+}) = 0$ to $G \le 0$ between the two states, and that implication is what fails when $F$ is not convex.

Remark A17.17 (Why the entropy inequality and not the jump condition).

The chapter already observes, in the paragraph following Definition 14.101, that Equation (14.91) presupposes one-sided limits that a bounded measurable weak solution need not have. The proof above shows what is gained by not presupposing them: at no point did the argument speak of a discontinuity, a curve, or a one-sided limit. It compared two solutions through their entropy inequalities alone, and the only regularity used — continuity of translation in the mean — is a property every bounded measurable function has. Example 14.98 is settled by the same stroke without any examination of the two candidate solutions: whichever of them satisfies Equation (14.92) is the only one that does.

Remark A17.18.

Kruzhkov's Doubling of Variables and the Entropy Solution discharges the proof obligation of Theorem 14.103 in Partial Differential Equations, and with it closes the pattern that Remark 14.104 describes: a nonlinear equation destroys its own classical solution in finite time (Proposition 14.94), weak solutions restore existence at the cost of uniqueness (Example 14.98), and the entropy inequality Equation (14.92) restores uniqueness, giving back all three of Hadamard's conditions (Definition 14.24) with the mean as the norm. The physical reading is Remark 14.100, and the gas-dynamical form of the admissible jump is Fluid Dynamics. Six inputs are assumed and are listed in Remark A17.4; five of them are the Lebesgue theory that this treatise deliberately does not build, and the sixth is Lemma A17.5.