= Transport of a Dirac point mass
{title2=$u(x,t)=\delta_0(x-ct)$}
= Moving Dirac point mass
{synonym}
For the constant-velocity <linear transport equation> $u_t+c\cdot\nabla_xu=0$, a <Dirac delta distribution> initially at zero moves along $x=ct$. Its spacetime <distribution> pairs by $\langle u,\varphi\rangle=\int\varphi(ct,t)\,dt$, and its initial trace is obtained by <weak convergence of distributions>. The <Fourier representation of the Dirac delta function> writes it as an <oscillatory integral> with <phase function> $(x-ct)\cdot\theta$. Its <singular support> is precisely the trajectory; it has <order-zero distribution> regularity as a measure, without becoming a <smooth function>.
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