I have this exercise that asks me to find the entropy solution for the inviscid Burgers' equation ($\partial_t u+u~\partial_x u=0$) with initial condition
$u(x,0)=u_0(x) = \begin{cases} 0 &\quad\text{if }\quad x<0, \\ 1 &\quad\text{if}\quad 0\leq x\leq 1, \\ 2 &\quad\text{if }\quad 1<x\leq 2,\\ 0 &\quad\text{if} \quad x>2\\ \end{cases} $
I see I should use Rankine Hugoniot condition and so I did and found that the discontinuity should be for $x=t/2,x=t, x=3/2t$. How can I find the proper solution? Is that an entropy solution? How do I check it? I have seen there are similar questions but my issue is about entropy solution. Moreover I know nothing about shocks and they are often cited and used. Thanks
