 Periodic square-wave relaxation for the inviscid Burgers equation

Take period-one data equal to one on $[k-1/2,k]$ and zero on $(k,k+1/2)$. Before $t=1$, the separate rectangular pulse solutions of the inviscid Burgers equation do not overlap in their interiors. At $t=1$ each shock meets the two neighboring fans. Afterward, on the interval between shocks $s_{k-1}$ and $s_k$, the solution remains $(x-k+1/2)/t$. The Rankine-Hugoniot condition gives $s_k^{\prime }=(s_k-k)/t$ and $s_k(1)=k+1/2$, hence $s_k=k+t/2$.

The title formula uses the fractional part and chooses right-hand shock values. Its left and right states are $1/2+1/(2t)$ and $1/2-1/(2t)$, making every shock a causal entropy shock of speed $1/2$. Therefore ${\sup }_x|u(x,t)-1/2|\le 1/(2t)$: the periodic entropy solution converges uniformly to its conserved mean even though it has a jump at every finite time. Periodizing a nonlinear solution is valid here only before the supports interact; after interaction the matching shock calculation is essential.