Kruzhkov entropy solutions

Let $n\ge 1$, $T>0$, $f\in C^1(\mathbb{R};{\mathbb{R}}^n)$, and ${\mathrm{\Pi }}_T={\mathbb{R}}^n\times (0,T)$. The unknown and initial datum are equivalence classes $[u]\in L^{\mathrm{\infty }}({\mathrm{\Pi }}_T)$ and $[u_0]\in L^{\mathrm{\infty }}({\mathbb{R}}^n)\cap L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^n)$, where equality is Lebesgue-a.e. (The space $L^p(\mu )$ as the quotient by null functions, A locally integrable function on ${\mathbb{R}}^n$).

For $k\in \mathbb{R}$ define the Kruzhkov entropy pair ${\eta }_k(s)=\mathrm{\mid }s-k\mathrm{\mid },q_k(s)=\mathrm{sgn}(s-k)(f(s)-f(k)),$ with $\mathrm{sgn}(0)=0$ (Absolute value in an ordered field). Then ${\eta }_k$ is convex and $q_k^{\mathrm{\prime }}={\eta }_k^{\mathrm{\prime }}{f}^{\mathrm{\prime }}$ almost everywhere, so $({\eta }_k,q_k)$ is a locally Lipschitz convex entropy--entropy flux pair (Convex entropy--entropy flux pairs).

The class $[u]$ is a Kruzhkov entropy solution of $u_t+{\mathrm{div}}_{x}f(u)=0$ with initial trace $[u_0]$ if:

(i) for every $k\in \mathbb{R}$ and every nonnegative $\phi \in C_c^{\mathrm{\infty }}({\mathrm{\Pi }}_T)$, ${\int }_{{\mathrm{\Pi }}_T}({\eta }_k(u){\phi }_t+q_k(u)\cdot {\mathrm{\nabla }}_x\phi )dxdt\ge 0,$ that is, ${\mathrm{\partial }}_t{\eta }_k(u)+{\mathrm{div}}_x{q}_k(u)\le 0$ in ${\mathcal{D}}^{\mathrm{\prime }}({\mathrm{\Pi }}_T)$; and

(ii) for every compact $K\subseteq {\mathbb{R}}^n$, ${\lim }_{\delta \downarrow 0}{\mathrm{ess\hspace{0.17em}sup}}_{0<t<\min \{\delta ,T\}}{\int }_K\mathrm{\mid }u(t,x)-u_0(x)\mathrm{\mid }dx=0.$

The integral in (i) is independent of the representative because its integrand is unchanged almost everywhere. For (ii), Fubini's theorem gives locally integrable spatial sections for almost every $t$ (Fubini's theorem for L^1 functions on a sigma-finite product); the displayed slice integral is defined for those times and its essential supremum ignores the exceptional null set. If $u$ or $u_0$ is changed on a null set in its respective space, Fubini's theorem shows that the slice-integral function changes only for a null set of times, so the trace condition is well defined on the equivalence classes: this is the strong local $L^1$ initial trace.

Taking $k$ above and below the essential range of $u$ makes ${\eta }_k$ equal $k-u$ and $u-k$, whose $t$- and $x$-derivatives cancel the constant terms against compactly supported test functions, so the entropy inequalities imply the weak conservation law of Distributional weak solutions of the Cauchy problem tested against nonnegative test functions, hence by linearity against all test functions.

To recover the Cauchy boundary term, apply the interior weak identity to $\phi (t,x)\zeta (t\mathrm{/}\delta )$, where $\zeta$ is smooth and nondecreasing, $\zeta =0$ on $(-\mathrm{\infty },1\mathrm{/}2]$ and $\zeta =1$ on $[1,\mathrm{\infty })$. For $\delta >0$ this product is supported away from $t=0$, so it is an admissible interior test. The term containing ${\zeta }^{\mathrm{\prime }}\mathrm{/}\delta$ converges to $\int u_0(x)\phi (0,x)dx$ by (ii), while the other terms converge on the compact support. This proves the full weak formulation with initial datum $u_0$, rather than only its interior equation.