What are transport, diffusion, dispersion, dissipation, and viscosity terms in a PDE exactly?

Question

Consider the PDE $$\partial_t u + au + b\partial_x u + c\partial^2_{xx} u + d\partial^3_{xxx} u + e\partial^4_{xxxx} u + f\partial^5_{xxxxx} u + \dots= 0.$$ in $(0,\infty) \times \mathbb{R}$, with $a,b,c,d,f, \dots \in \mathbb{R}$.

I've seen multiple times people referring to certain terms in such equations as transport term, diffusion term, dispersion term, dissipation term, viscosity term, and so on.

I can see why one would call $c\partial^2_{xx}u$ a viscosity term (from the vanishing viscosity technique), but I don't know which terms of the equation the other names apply to and why. Can you offer some explanation?

Answer 1

"Transport" is easy: if you think of some distribution of mass $f$ that is moving at some velocity $v$ (which may be constant or point/time-dependent), then the equation for the (varying) mass density at a fixed point $x$ is $\partial_t f+v\cdot \nabla f=0$ (the classical transport equation; there are many variants where the velocity $v$ itself is depending on $f$, in which case more often than not $f$ is no longer a mass density, but the linear term is still often referred to as the transport one).

"Diffusion, dissipation, dispersion, viscosity" can all refer to the $u_{xx}$ term, but not only. You can have a fractional power of the Laplacian instead or higher order terms under that name. Let the PDE experts correct me if I'm wrong, but my impression is that the usage of those words is more context-dependent than equation-dependent. In general there should be something elliptic about them (so that, if you have them alone, there will be some smoothing and decay in the solution of the corresponding evolution equation), and one may prefer the word "diffusion" when thinking of mass propagation, "dissipation" if $u$ has the meaning of energy, and "viscosity" if $u$ has the meaning of speed. I'm not sure what exactly is the most suitable context for "dispersion". It looks like most people make little difference between that and "dissipation" but there may be some unwritten usage rules I'm just not aware of.

In general, don't break your head over the terminology too much. Much of it is just an "expert slang" aimed at speeding up the communication and bringing up certain associations. In each particular case, the meaning should be clear from the context.

Answer 2

I don't know about all the terms you mentioned, but in my particular field, if you take the Navier-Stokes equations for example:

${\partial}_t \mathbf{u} -\nu \Delta \mathbf{u} + (\mathbf{u} \cdot \nabla) \mathbf{u} + \nabla p = \mathbf{f}$

The first term is often referred to as the evolutionary term (Without any system evolution this term is null.. so that makes sense),

The second is the diffusion term or the viscosity term. This comes from the fact that diffusion theory in physics (see Fick's law) has a second derivative (generalizes to the Laplacian $\Delta=(\partial_{x_1x_1}+\partial_{x_2x_2}+...+\partial_{x_Nx_N})$, in higher dimensions) and the viscosity name comes from the fact $\nu$ is the viscosity constant.

The third in the convection term - Arises from convection-diffusion equations.

The fourth is the pressure gradient. Speaks for itself - it the gradient of the pressure.