[proofplan] We restrict to test functions supported away from $\{t=0\}$, split the weak formulation across the shock curve $\Gamma$, and use the classical PDE on each side to convert the integrand into a total divergence. The [Gauss-Green Theorem](/theorems/28) reduces to boundary integrals along $\Gamma$. Explicit parametrisation of $\Gamma$ yields integrals involving the shock speed, and the [Fundamental Lemma of Calculus of Variations](/theorems/45) extracts the pointwise Rankine--Hugoniot condition. [/proofplan] [step:Restrict to test functions supported away from $\{t = 0\}$ and split across $\Gamma$] Let $\varphi \in C_c^\infty(\mathbb{R} \times (0, T))$ (vanishing near $t = 0$ and $t = T$). The initial data term drops out: \begin{align*} \int_0^T \int_{\mathbb{R}} \left(u\,\partial_t\varphi + f(u)\,\partial_x\varphi\right) d\mathcal{L}^1(x) \, d\mathcal{L}^1(t) = 0. \end{align*} Split the domain as $\mathbb{R} \times (0, T) = \Omega_L \cup \Gamma \cup \Omega_R$. On each of $\Omega_L$ and $\Omega_R$, the product rule gives: \begin{align*} u\,\partial_t\varphi + f(u)\,\partial_x\varphi = \partial_t(u\varphi) + \partial_x(f(u)\varphi) - \varphi\bigl(\partial_t u + \partial_x f(u)\bigr). \end{align*} Since $u$ satisfies $\partial_t u + \partial_x f(u) = 0$ classically on $\Omega_L$ and $\Omega_R$, the last term vanishes. [/step] [step:Apply the divergence theorem to reduce to boundary integrals along $\Gamma$] Define the vector field $F := (f(u)\varphi,\, u\varphi)$ in the $(x,t)$-plane. By the [Gauss-Green Theorem](/theorems/28) applied to $\Omega_L$ and $\Omega_R$: \begin{align*} 0 = \int_\Gamma F\big|_L \cdot n_L \, d\mathcal{H}^1 + \int_\Gamma F\big|_R \cdot n_R \, d\mathcal{H}^1, \end{align*} where $n_L$, $n_R$ are the outward normals to $\Omega_L$, $\Omega_R$ on $\Gamma$ (with $n_R = -n_L$). The contributions from spatial infinity and $t = 0, T$ vanish because $\varphi$ has compact support in $\mathbb{R} \times (0,T)$. [/step] [step:Parametrise $\Gamma$ and compute the boundary integrals explicitly] Parametrise $\Gamma$ by $\gamma(\tau) = (s(\tau), \tau)$ for $\tau \in (0,T)$. The tangent vector is $\gamma'(\tau) = (\dot{s}(\tau), 1)$ with speed $|\gamma'(\tau)| = \sqrt{1 + \dot{s}(\tau)^2}$. The outward normal to $\Omega_L = \{x < s(t)\}$ is: \begin{align*} n_L(\gamma(\tau)) = \frac{(1, -\dot{s}(\tau))}{\sqrt{1 + \dot{s}(\tau)^2}}. \end{align*} By the substitution formula $\int_\Gamma h \, d\mathcal{H}^1 = \int_0^T h(\gamma(\tau))\,|\gamma'(\tau)| \, d\mathcal{L}^1(\tau)$, the speed factors cancel with the normalisation of $n_L$: \begin{align*} \int_\Gamma F\big|_L \cdot n_L \, d\mathcal{H}^1 = \int_0^T \varphi(s(\tau),\tau)\bigl(f(u_L(\tau)) - \dot{s}(\tau)\,u_L(\tau)\bigr) \, d\mathcal{L}^1(\tau). \end{align*} For $\Omega_R$, $n_R = -n_L$, giving: \begin{align*} \int_\Gamma F\big|_R \cdot n_R \, d\mathcal{H}^1 = -\int_0^T \varphi(s(\tau),\tau)\bigl(f(u_R(\tau)) - \dot{s}(\tau)\,u_R(\tau)\bigr) \, d\mathcal{L}^1(\tau). \end{align*} [/step] [step:Extract the pointwise jump condition via the fundamental lemma] Adding the two boundary contributions: \begin{align*} \int_0^T \varphi(s(t),t)\bigl[(f(u_L) - f(u_R)) - \dot{s}(t)(u_L - u_R)\bigr] \, d\mathcal{L}^1(t) = 0. \end{align*} The freedom in $\varphi$ allows $\psi(t) := \varphi(s(t),t)$ to range over all of $C_c^\infty((0,T))$: given $\psi_0 \in C_c^\infty((0,T))$, choose $\varphi(x,t) = \psi_0(t)\,\eta(x - s(t))$ with $\eta \in C_c^\infty(\mathbb{R})$ satisfying $\eta(0) = 1$. The integral condition becomes: \begin{align*} \int_0^T \psi(t)\,h(t) \, d\mathcal{L}^1(t) = 0 \quad \text{for all } \psi \in C_c^\infty((0,T)), \end{align*} where $h(t) := (f(u_L) - f(u_R)) - \dot{s}(t)(u_L - u_R)$ is continuous. By the [Fundamental Lemma of Calculus of Variations](/theorems/45), $h(t) = 0$ for all $t \in (0,T)$: \begin{align*} f(u_L(t)) - f(u_R(t)) = \dot{s}(t)\bigl(u_L(t) - u_R(t)\bigr). \end{align*} [/step]