**Weak Solution:** A weak solution of

$\displaystyle \vec{u}_t + (\vec{f}(\vec{u}))_x = 0, \quad \vec{u}|_{t=0} = \vec{u}_0(x)$

is a function $\displaystyle \vec{u}(x,t):\mathbb{R}\times\mathbb{R}^+ \rightarrow \mathbb{R}^n$ such that

$\displaystyle \int_0^{\infty}\!\!\!\int_{-\infty}^{\infty}\!\left[\vec{u}(x,t)\cdot \vec{\phi}_t(x,t) + \vec{f}(\vec{u}(x,t))\cdot \vec{\phi}_x(x,t)\right]\dx\,dt + \int_{-\infty}^{\infty}\!\vec{u}_0(x)\vec{\phi}(x,0)\, dx = 0$

for all $\displaystyle \vec{\phi}(x,t) \in C^1_c(\mathbb{R}\times \mathbb{R}^+)$, where

$\displaystyle C^1_c(\mathbb{R}\times \mathbb{R}^+) = \left\{\vec{\phi} \in C^1_c(\mathbb{R}\times \mathbb{R}^+) | \phi \equiv 0 \text{ for } (x,t)\not \in B_r(0,0) \bigcap (\mathbb{R}\times \mathbb{R}^+) \text{ for some } r>0\right\}$