Background
Consider the inviscid Burgers' equation. \begin{align*} u_t + f(u)_x&=0 \end{align*} or, using $f(u)=\frac{1}{2}u^2$, \begin{align*} u_t + u u_x &=0 \end{align*} In the case of non-smooth initial data $u_0(x)$, a classical solution does not exist.
A function $u(x,t)$ is called a weak solution if $$ \int_0^\infty\int_{-\infty}^\infty \phi_t u + \phi_x f(u) dxdt = -\int_{-\infty}^\infty\phi(x,0)u(x,0)dx $$ for all test functions $\phi \in C_0^1(\mathbb{R} \times \mathbb{R}^+)$
Question
Is there a way to gather some physical intuition about the meaning of this condition? Looking at the right hand side, the integral is a sort of weighted average of $\phi$ by the concentration $u$. But what would the left hand side represent?
I'm fine with simply using the test functions as a mathematical tool - but it is somewhat disappointing to abandon the wonderfully intuitive "control volume" formulation of a weak solution: $$ \int_{x_1}^{x_2} u(x,t_2)dx = \int_{x_1}^{x_2} u(x,t_1) dx + \int_{t_1}^{t_2} f(u(x_1,t))dt - \int_{t_1}^{t_2}f(u(x_2,t))dt $$ i.e., the concentration of $u$ in the control volume at $t_2$ is simply whatever was in the volume at $t_1$ plus the net flux into the volume.