= Solution
A <weak solution> is a field $\mathbf u$ for which the composed densities $\mathbf P(x,t,\mathbf u)$ and fluxes $\mathbf Q(x,t,\mathbf u)$ are locally integrable and
$$
\boxed{\iint\bigl(\mathbf P\cdot\partial_t\boldsymbol\varphi+\mathbf Q\cdot\partial_x\boldsymbol\varphi\bigr)\,dx\,dt=0}
$$
for every smooth compactly supported <vector> <test function> $\boldsymbol\varphi$ in the spacetime domain. This is the distributional form of the <conservation law>, obtained by moving <derivatives> onto the <test function>. If tests are allowed to meet an initial boundary $t=0$, the specified initial <density> contributes the additional term $\int\mathbf P(x,0,\mathbf u_0)\cdot\boldsymbol\varphi(x,0)dx$ in the usual initial-value formulation.
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