= Solution
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 gives
$$
\int_0^\infty\int_{\mathbb R}\left(u\phi_t+\frac{u^2}{2}\phi_x\right)dx\,dt
+\int_{\mathbb R}u_0(x)\phi(x,0)\,dx=0
$$
for every smooth compactly supported <test function>. The initial term is correct because $u(\cdot,t)\to\mathbf1_{\{x>0\}}$ in local $L^1$: the difference is supported in $(0,t)$ and its integral is at most $t$. Thus \b[the fan is a <weak solution> of the Burgers equation with the required initial data].
Back to article page
