Inflow transport boundary condition ($u(t,0)=f(t)$)

For a transport equation on a domain, prescribe boundary data only where the velocity points into the domain. On $x>0$ with speed $a>0$, a bounded weak solution with initial data $u_0$ and inflow data $f$ satisfiesThe integrals are over the quadrant and its two boundary rays. Backtracking characteristic curves gives $u_0(x-at)$ when $x\ge at$ and $f(t-x/a)$ otherwise. Corner values need not match for a bounded weak solution.

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.