the advection equation
Advection is the technical word for being carried along by a moving medium. When a leaf is swept down a stream, or warm air is pushed across a city by the wind, the leaf and the warmth are advected: they move because the stuff they sit in moves. The advection equation is the PDE that says a quantity is purely carried by a known flow, neither created, destroyed, nor spread out on its own.
It is the same equation as the transport equation, u_t + c u_x = 0, but the name advances a particular reading and a richer generality. The flow speed need not be constant: in u_t + c(x,t) u_x = 0 the carrying velocity can vary in space and time, and in higher dimensions it becomes u_t + v · grad u = 0, where v is a velocity vector field and grad u is the spatial gradient. The dot product v · grad u measures how fast u changes as you ride along the flow, and setting it against u_t to zero says u is constant for an observer drifting with the medium.
Advection sits opposite diffusion, which spreads things out and smooths them (the heat equation). Real models often combine the two into an advection-diffusion equation, u_t + v · grad u = D Laplacian u, the workhorse for pollutants in air and water, heat carried by a current, and salt in the ocean. Recognizing the bare advection term tells you something will be transported with finite speed and sharp features preserved — quite unlike diffusion, which would blur them instantly.
A factory releases a puff of smoke. With a steady 5 m/s wind and no turbulence, pure advection moves the puff 5 metres downwind each second while keeping its shape; turbulence (diffusion) is what actually fattens and fades it.
Advection moves a pattern; diffusion blurs it. Most real flows do both at once.
Advection and transport name the same equation; the choice of word is convention. When velocity v depends on the unknown u itself (v depends on u), the equation becomes quasilinear and can form shocks — that is no longer plain advection.