Distributional weak solutions of the Cauchy problem

Let $n\ge 1$, $T>0$, ${\mathrm{\Pi }}_T={\mathbb{R}}^n\times (0,T)$, let $f\text{\hspace{0.17em}⁣}:\mathbb{R}\to {\mathbb{R}}^n$ be continuous and let $u_0\in L^{\mathrm{\infty }}({\mathbb{R}}^n)\cap L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^n)$. A bounded measurable $u\text{\hspace{0.17em}⁣}:{\mathrm{\Pi }}_T\to \mathbb{R}$ is a distributional weak solution of the Cauchy problem $u_t+{\mathrm{div}}_{x}f(u)=0$, $u(\cdot ,0)=u_0$, if for every $\phi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n\times (-\mathrm{\infty },T))$ --- test functions whose support may meet the initial plane $t=0$ --- one has ${\int }_{{\mathrm{\Pi }}_T}(u{\phi }_t+f(u)\cdot {\mathrm{\nabla }}_x\phi )dxdt+{\int }_{{\mathbb{R}}^n}{u}_0(x)\phi (x,0)dx=0.$ Equivalently $u_t+{\mathrm{div}}_{x}f(u)=0$ in ${\mathcal{D}}^{\mathrm{\prime }}({\mathrm{\Pi }}_T)$, together with the displayed initial boundary term (Distribution, Distributional derivative, Test function space d of an open set). The integrals are absolutely convergent: $u$ is bounded, $f(u)$ is bounded on the bounded range of $u$, and $\phi$ has compact support, so the pairings are $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$ pairings and are representative-independent (A locally integrable function on ${\mathbb{R}}^n$, The space $L^p(\mu )$ as the quotient by null functions; the product-space identities are those of Fubini's theorem for L^1 functions on a sigma-finite product and Tonelli's theorem for nonnegative measurable functions on a sigma-finite product). The condition is an equality in the $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$ classes of $u$ and $f(u)$; it involves no pointwise assignment of $u$ on $\{t=0\}$, and the initial datum enters only through the boundary term (Scalar conservation laws, fluxes and Cauchy data). For the entropy formulation of this page the initial condition is instead imposed as a strong local $L^1$ trace (Kruzhkov entropy solutions).