= 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.
For a spatially varying <velocity field>, distinguish an advected value from a conserved <mass density>. The <continuity equation> is $\rho_t+\operatorname{div}(b\rho)=0$, which expands to $\rho_t+b\cdot\nabla\rho+(\operatorname{div}b)\rho=0$. Thus along the <characteristic flow map> $X_t$, it is $\rho(t,X_t(\alpha))\det DX_t(\alpha)$ that remains constant. For $b(x)=x$ on the real line, scalar <transport equation> gives $u(t,x)=g(e^{-t}x)$, whereas the <continuity equation> gives $\rho(t,x)=e^{-t}g(e^{-t}x)$. For integrable $g$, <change of variables> yields $\int u(t,x)\,dx=e^t\int g$ and $\int\rho(t,x)\,dx=\int g$. The expanding flow preserves carried values and spreads conserved <mass> over a growing volume.
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