Weak and entropy solutions of the scalar conservation law $u_t+(u^2\mathrm{/}2{)}_x=0$

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the actual Lebesgue measure foundation used by the bounded Borel weak/entropy class.

Write $F(u):=\frac{1}{2}{u}^2$ and let ${\lambda }_2$ be two-dimensional Lebesgue measure on ${\mathbb{R}}^2$ (Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$).

Every space-time expression written as ${\int }_0^{\mathrm{\infty }}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\int }_{\mathbb{R}}(\cdots )dxdt$ below means the single signed Lebesgue integral over $[0,\mathrm{\infty })\times \mathbb{R}$ with respect to ${\lambda }_2$; it is not an iterated-integral convention. The initial-data integral is with respect to one-dimensional Lebesgue measure ${\lambda }_1$. In (W) and (E), each integrand is Borel measurable, bounded, and supported in the compact support of the test function (or its initial trace). A compact support is contained in a bounded rectangle of finite Lebesgue measure, by A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included; monotonicity and homogeneity of the nonnegative integral give $\int \mathrm{\mid }q\mathrm{\mid }\le \mathrm{\parallel }q{\mathrm{\parallel }}_{\mathrm{\infty }}{\lambda }_d(K)<\mathrm{\infty }$ for such an integrand $q$ supported in $K$ (Monotonicity and nonnegative homogeneity of the nonnegative integral). Thus all displayed signed integrals are defined as Lebesgue integrals of integrable functions (Integrable real and complex functions, and their integrals).

A test function is a function $\phi :[0,\mathrm{\infty })\times \mathbb{R}\to \mathbb{R}$ of class $C^1$ up to the boundary: it is the restriction of a $C^1$ function on an open neighborhood of this closed half-space, and its support within the half-space is compact. No vanishing or continuation by zero for $t<0$ is required. Thus the trace $\phi (0,x)$ may be nonzero. Derivatives at $t=0$ are the continuous one-sided derivatives. Write $C_c^1([0,\mathrm{\infty })\times \mathbb{R})$ for this class; admissible means belonging to this class and nonnegative means $\phi \ge 0$ on the half-space.

(a) Weak solutions. Let $f:\mathbb{R}\to \mathbb{R}$ be bounded and Borel measurable (Extended-real-valued measurable functions) and let $u:(0,\mathrm{\infty })\times \mathbb{R}\to \mathbb{R}$ be bounded and Borel measurable. Then $u$ is a weak solution of

$$\begin{array}{cccc}& u_t+(F(u){)}_x=0\text{with }u(0,\cdot )=f& & \text{(CL)}\end{array}$$

when

$$\begin{array}{cccc}& {\int }_0^{\mathrm{\infty }}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\int }_{\mathbb{R}}[u(t,x){\phi }_t(t,x)+F(u(t,x)){\phi }_x(t,x)]dxdt+{\int }_{\mathbb{R}}f(x)\phi (0,x)dx=0& & \text{(W)}\end{array}$$

for every test function $\phi$ (Integral over a measurable subset, Integrable real and complex functions, and their integrals, The Lebesgue integral is linear on $L^1(\mu )$). The integrals exist because $u,f$ are bounded and the test functions have compact support; the identity depends on $u$ and $f$ only through their almost-everywhere classes.

(b) Entropy pairs. An entropy pair is a pair $(\eta ,q)$ of $C^1$ functions $\eta ,q:\mathbb{R}\to \mathbb{R}$ (The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set) such that

$q^{\mathrm{\prime }}(u)={\eta }^{\mathrm{\prime }}(u)F^{\mathrm{\prime }}(u)=u{\eta }^{\mathrm{\prime }}(u)\text{for every }u\in \mathbb{R};$

the function $\eta$ is then called an entropy and $q$ its entropy flux. The entropy is convex when $\eta ((1-a)r+as)\le (1-a)\eta (r)+a\eta (s)$ for all real $r,s$ and $a\in [0,1]$.

(c) Entropy solutions. A weak solution $u$ of (CL) is an entropy solution when, for every convex entropy pair $(\eta ,q)$ with $q^{\mathrm{\prime }}=u{\eta }^{\mathrm{\prime }}$, the distributional inequality

$(\eta (u){)}_t+(q(u){)}_x\le 0$

holds, by definition meaning that

$$\begin{array}{cccc}& {\int }_0^{\mathrm{\infty }}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\int }_{\mathbb{R}}\left[\eta \right(u){\phi }_t+q(u\left){\phi }_x\right]dxdt+{\int }_{\mathbb{R}}\eta (f(x))\phi (0,x)dx\ge 0& & \text{(E)}\end{array}$$

for every nonnegative test function $\phi$.

(d) Riemann data. For reals $u_L,u_R$ the Riemann initial data are

$f(x):=\{\begin{array}{ll}{u}_L,& x<0,\\ u_R,& x\ge 0.\end{array}$

• The sign convention is fixed by (E): with $\phi \ge 0$ the identity says that the distribution $\eta (u{)}_t+q(u{)}_x$ is a nonpositive measure. Under the opposite convention the inequality and the displayed integral would both flip sign; the two conventions are not mixed anywhere in this collection.

• The distinguished family of entropies used in the examples is ${\eta }_k(u):=\mathrm{\mid }u-k\mathrm{\mid }$, which is convex and only Lipschitz at $u=k$, together with $q_k(u)=\mathrm{sgn}(u-k)(u^2-k^2)\mathrm{/}2$ off $u=k$ and $q_k(k)=0$. Here $\mathrm{sgn}(r):=1$ for $r>0$, $0$ for $r=0$, and $-1$ for $r<0$. The identity $q_k^{\mathrm{\prime }}=u{\eta }_k^{\mathrm{\prime }}$ holds off $u=k$, and the pair is applied to piecewise $C^1$ solutions for which the distributional computation is carried out piecewise. The definition of a general entropy solution in (c) is stated for $C^1$ entropies; the piecewise application to ${\eta }_k$ is part of the corresponding theorem and is checked there.

• The definition asserts neither existence nor uniqueness of weak or entropy solutions, and no selection principle. Integrals are Lebesgue integrals over the stated domains and the test-function class is exactly as declared; evaluating a weak solution at $t=0$ pointwise is not part of the definition, only the initial term in (W).