= Transported step forcing in an advection-diffusion equation
{title2=$u(t,x)=-\tfrac12(x-t)_+^2$}
In coordinates $s=t$, $y=x-t$, the transport derivative $u_t+u_x$ becomes $U_s$. A <Heaviside step function> moving at that speed becomes stationary, and its twice-integrated profile gives the displayed continuously differentiable solution of $u_t+u_x=u_{xx}+H(x-t)$. Its second spatial derivative has a jump, so the equation holds away from the moving interface and in the distributional sense there.
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