Hi I come from a mathematical stochastic background. In PDE / ODE / analysis I sometimes hear the terminology 'transport equation' or 'transport part of the equation'.
Can anyone explain $\textbf{In General}$ what someone is referring to when they talk about $\textit{'A transport equation or the transport part of the equation'}$. If anyone can explain this from a mathematical or physical perspective that would be amazing.
I just want enough so that I can have a general idea of what someone is talking about when they mention this 'transport'.
Note I am not talking about Optimal Transport theory which Im confident is not related.
In its most simple form, this is the equation $$ 0=u_t+cu_x $$ which has solutions of the form $u(x,t)=v(x-ct)$, which also identifies $c$ as the velocity of the transport. This means that the form of the solution at time zero is transported unchanged to the right on the $x$ axis with velocity $c$.
Note that you can in reverse also compute traveling solutions in some equation $$ u_t=u_{xx}+f(u) $$ by inserting just that form $u(x,t)=v(x-ct)$ to get a second order ODE $$ 0=v''+cv'+f(v) $$ which resembles the kinetic equation of a particle in a force field and under frictional loss of energy. Usually you get boundary conditions for $v(x)$ like ending in roots of $f$ at $x=\pm\infty$