[proofplan] We regularise the conservation law by adding viscosity $\varepsilon\,\partial_x^2 u^\varepsilon$, which gives a smooth parabolic solution. Uniform $L^\infty$ and $BV$ bounds (from the maximum principle) provide compactness via Helly's theorem. We extract a convergent subsequence and verify the Kru\u017ekov entropy condition in the limit by multiplying the viscous PDE by $\eta'(u^\varepsilon)$ for convex $\eta$ and showing the viscous error term vanishes as $\varepsilon \to 0$. [/proofplan] [step:Solve the regularised viscous conservation law] For $\varepsilon > 0$, consider the viscous conservation law: \begin{align*} \partial_t u^\varepsilon + \partial_x f(u^\varepsilon) = \varepsilon\,\partial_x^2 u^\varepsilon \quad \text{in } \mathbb{R} \times (0,T), \qquad u^\varepsilon(\cdot, 0) = u_0. \end{align*} This is a semilinear parabolic equation. Standard theory (fixed-point arguments combined with parabolic regularity) gives a unique smooth solution $u^\varepsilon \in C^\infty(\mathbb{R} \times (0,T))$. [/step] [step:Establish uniform $L^\infty$ and $BV$ estimates] [claim:$\|u^\varepsilon(\cdot,t)\|_{L^\infty(\mathbb{R})} \leq \|u_0\|_{L^\infty(\mathbb{R})}$ for all $t > 0$ and $\varepsilon > 0$] [/claim] [proof] The maximum principle for the parabolic equation $\partial_t u^\varepsilon - \varepsilon\,\partial_x^2 u^\varepsilon = -\partial_x f(u^\varepsilon)$ gives the bound. The right side $-\partial_x f(u^\varepsilon) = -f'(u^\varepsilon)\,\partial_x u^\varepsilon$ is a first-order term, so the comparison principle applies: if $u^\varepsilon$ attains its maximum at an interior point, then $\partial_t u^\varepsilon \leq 0$ and $\partial_x^2 u^\varepsilon \leq 0$ there, forcing $0 \leq -\partial_x f(u^\varepsilon) = 0$. The bound follows. [/proof] The $BV$ estimate $\|u^\varepsilon(\cdot,t)\|_{BV(\mathbb{R})} \leq \|u_0\|_{BV(\mathbb{R})}$ is obtained by differentiating the equation in $x$ and applying the maximum principle to $\partial_x u^\varepsilon$. [/step] [step:Extract a convergent subsequence by compactness] The uniform $L^\infty$ and $BV$ bounds give, by Helly's theorem in $x$ and an Aubin--Lions-type argument in $t$, a subsequence $u^{\varepsilon_j}$ converging in $L^1_{\mathrm{loc}}(\mathbb{R} \times (0,T))$ to a function $u \in L^\infty(\mathbb{R} \times (0,T))$. [/step] [step:Verify the Kru\u017ekov entropy condition in the limit] For smooth $u^\varepsilon$, multiply the PDE by $\eta'(u^\varepsilon)$ for any convex $\eta$ with entropy flux $q' = \eta' f'$: \begin{align*} \partial_t\eta(u^\varepsilon) + \partial_x q(u^\varepsilon) = \varepsilon\,\eta'(u^\varepsilon)\,\partial_x^2 u^\varepsilon = \varepsilon\,\partial_x(\eta'(u^\varepsilon)\,\partial_x u^\varepsilon) - \varepsilon\,\eta''(u^\varepsilon)\,(\partial_x u^\varepsilon)^2. \end{align*} Since $\eta$ is convex ($\eta'' \geq 0$), the last term is non-positive: \begin{align*} \partial_t\eta(u^\varepsilon) + \partial_x q(u^\varepsilon) \leq \varepsilon\,\partial_x(\eta'(u^\varepsilon)\,\partial_x u^\varepsilon). \end{align*} Testing against non-negative $\varphi \in C_c^\infty(\mathbb{R} \times [0,T))$ and integrating by parts: the right side becomes $-\varepsilon\int \eta'(u^\varepsilon)\,\partial_x u^\varepsilon\,\partial_x\varphi$, which tends to $0$ as $\varepsilon \to 0$ (since $\eta'(u^\varepsilon)$ is uniformly bounded and $\varepsilon\,\partial_x u^\varepsilon \to 0$ distributionally). The left side converges by $L^1_{\mathrm{loc}}$ convergence of $u^\varepsilon \to u$. Taking $\eta(u) = |u - k|$ and $q(u) = \operatorname{sgn}(u-k)(f(u)-f(k))$ recovers the Kru\u017ekov entropy condition. [/step] [step:Verify the initial data] The convergence $u^\varepsilon \to u$ in $L^1_{\mathrm{loc}}$ and the uniform $L^\infty$ bound ensure $u(\cdot, t) \to u_0$ in $L^1_{\mathrm{loc}}$ as $t \to 0$, so $u$ attains the initial data. [/step]