Consider the conservation law given by $$u_t+f(u)_x=0$$ We know that in general weak solutions are not smooth but are bounded in $L^{\infty}$ norm (they do not belong to Sobolev spaces).
However while deriving the numerical schemes most of the books say, integrating the conservation law over $(a,b) \times (t_1,t_2)$ and applying fundamental theorem of calculus we get $$\int_a^b u(x,t_1)dx - \int_a^b u(x,t_2)dx= -\int_{t_1}^{t_2} f\big(u(b,t)\big)dt+ \int_{t_1}^{t_2} f\big(u(a,t)\big)dt$$ I have the following doubts:
How can we perform integration by parts as the solution does not possess any regularity?
If a function satisfies the above integral formulation, can we say that it is a weak solution? Conversely, if $u$ is a weak solution, will it satisfy the above integral formulation? If so how to prove it? Thank you.
In continuum physics, things usually happen the other way round. First we have the integral form of the balance law $$ \frac{\text d}{\text d t} \int_{a}^{b} u\,\text d x = f(u|_{x=a}) - f(u|_{x=b}) $$ that describes the variation of the total quantity of $u$ over an arbitrary spatial domain $[a, b]$. Then, we rewrite the latter as $\int_{a}^{b} u_t\,\text d x = \int_{b}^{a}\! f(u)_x \text d x$, which yields the local PDE form $$u_t + f(u)_x = 0$$ of the conservation law.
Note that there is no harm to integrate the first equation from $t=t_1$ to $t=t_2$, which is exactly the integral form used in the derivation of finite volume methods.
Both formulations above (integral balance and local PDE) are written for functions. Thus, they are both strong forms. The weak form makes sense for distributions, and involves test functions. Usually, there are some requirements that allow to prove the equivalence between the above strong forms and the weak form.