= Inflow transport boundary condition
{title2=$u(t,0)=f(t)$}
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 $f$ satisfies
$$
\int u(\varphi_t+a\varphi_x)+\int u_0\varphi(0,x)+a\int f(t)\varphi(t,0)=0.
$$
The 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>.
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