Corner compatibility for constant-speed transport
= Corner compatibility for constant-speed transport
{title2=$f^{(j)}(0)=(-a)^j u_0^{(j)}(0)$}
For the <transport equation> $u_t+a u_x=0$ on the quadrant with $a>0$, smooth initial and inflow data glue across $x=at$ to order $m$ precisely when $f^{(j)}(0)=(-a)^j u_0^{(j)}(0)$ for $0\le j\le m$. Matching all orders yields a smooth solution; matching orders zero and one yields a <classical solution> up to the two boundaries.