Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 1 / 30E / b

Inside the fan $u=x/t$, one has $u_t=-x/t^2$ and $u{u}_x=x/t^2$, while the constant regions also satisfy the equation. Across $x=0$ and $x=t$, the two traces of both $u$ and its flux $u^2/2$ agree. Integrating by parts separately in the three regions therefore cancels all interface terms and givesfor every smooth compactly supported test function. The initial term is correct because $u(\cdot ,t)\to 1_{\{x>0\}}$ in local $L^1$: the difference is supported in $(0,t)$ and its integral is at most $t$. Thus the fan is a weak solution of the Burgers equation with the required initial data.