Transport equation
= Transport equation
{title2=$u_t+b\mathbin\cdot\nabla u=0$}
{wiki}
= Advection equation
{synonym}
A <transport equation> carries a quantity along the flow of a <velocity field> $b$. For a smooth solution of $u_t+b\mathbin\cdot\nabla u=0$, the value of $u$ is constant along each <characteristic curve>. With constant speed $c$, the <chain rule> gives $\frac{d}{dt}u(x_0+ct,t)=u_t+c u_x=0$, hence $u(x,t)=g(x-ct)$. This is exact <translation>: it preserves the shape rather than smoothing it, as the <periodic heat and transport comparison> shows.