Past exam of the applied partial differential equations course of the University of Oxford / 2023 / 2 / c / iii

Let $s_k(t)$ be the shock separating the fans from $k-1/2$ and $k+1/2$. Their traces giveThus $\boxed{s_k(t)=k+t/2}$. Between consecutive shocks, $u=(x-k+1/2)/t$. Writing $\{r\}=r-\lfloor r\rfloor$ for the fractional part, the period-one solution can be expressed, with right-hand shock traces, asTo put it in the requested fixed-support sum, takeThe half-open cells partition the line, so exactly one summand contributes at each point. This is a representation of the periodic solution, not a superposition principle for arbitrary solutions of a nonlinear equation.

The left and right shock states are $1/2+1/(2t)$ and $1/2-1/(2t)$; both average to the shock speed $1/2$, and the strict inequalities certify causality. The periodic square-wave relaxation for the inviscid Burgers equation has the uniform estimateThe limit is the conserved mean $1/2$.