Conservation of Mass in the Inviscid Burgers' Equation
Theorem
$$ \begin{cases} u_{t} + u u_{x} = 0 & , t>0 \\ u(t,x) = f(x) & , t=0 \end{cases} $$
For the solution $u$ of the inviscid Burgers’ equation above, let us define the line mass $M$ over the interval $[a,b]$ as follows.
$$ M_{a,b}(t) := \int_{a}^{b} u(t,x) dx $$
And if the breaking time is denoted by $t_{\ast}$, then for $t \in ( 0 , t_{\ast})$ the following holds.
$$ {{d} \over {dt}} M_{a,b}(t) = - \left( {{1} \over {2}} u^2 (t,b) - {{1} \over {2}} u^2 (t,a) \right) $$
Explanation
The breaking time refers to the point at which, mathematically, the function ceases to be a function, and physically, multiple states become superimposed simultaneously.
To put the equation in simple terms, it means that the change in mass equals the net sum of what comes in and what goes out.
On the right-hand side, $\displaystyle {{1} \over {2}} u^2 (t,b)$ is called the outflux, and $\displaystyle {{1} \over {2}} u^2 (t,a)$ is called the influx. The fact that the change of $M_{a,b}(t)$ over time is negative means that $u$ is decreasing on $[a,b]$, so the outflux is greater than the influx. Conversely, being positive means that the mass is increasing, so the influx is greater than the outflux, which fits well with the terminology and its meaning.
Collectively, if we let $\displaystyle F(u) := {{1} \over {2}} u^2$ be the flux function, we can express it cleanly as follows.
$$ {{\partial u} \over {\partial t}} + {{\partial u} \over {\partial x}} F(u) =0 $$
In this sense, the Burgers’ equation is also called the conservation of mass in $1$ dimension.
Derivation
Since $u$ is a continuous function, the following holds.
$$ {{d } \over {dt} } \int_{a}^{b} u(t,x) dx = \int_{a}^{b} {{\partial } \over { \partial t} } u(t,x) dx $$
Since $u_{t} = - u u_{x}$, the following holds.
$$ \int_{a}^{b} {{\partial } \over { \partial t} } u(t,x) dx = - \int_{a}^{b} u u_{x} dx $$
Since $\displaystyle u u_{x} = {{\partial} \over {\partial x}} \left( {{1 } \over {2}} u^2 \right)$ and $u_{t} = - u u_{x}$, the following holds.
$$ - \int_{a}^{b} u u_{x} dx = - \int_{a}^{b} {{\partial} \over {\partial x}} \left( {{1} \over {2}} u^2 \right) dx = - \left( {{1} \over {2}} u^2 (t,b) - {{1} \over {2}} u^2 (t,a) \right) $$
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In particular, if $\displaystyle \int_{-\infty}^{\infty} f(x) dx < \infty$, then for $t \in [0,t_{\ast})$ the following holds.
$$ \int_{-\infty}^{\infty} u(t,x) dx = \int_{-\infty}^{\infty} f(x) dx $$