Linear and almost linear Partial differential equations examples in Sciences

Question

I am interested in learning linear and almost linear PDEs of first order to describe some system or process however I want to learn by real world examples of such a application.Do you know any such system or process which is described by a linear or almost linear PDEs of first order?

Answer 1

This is a little to broad of a topic - but if you want an example of such an equation you can look at Navier-Stokes which in certain limits/assumptions can be considered linear.

A smaller component of the above would be the mass conservation equation which is given by $$ \frac{\partial \rho}{\partial t} + \nabla \cdot(\rho \mathbf{v}) = 0 $$ which is first order in derivative, a pde and is linear in terms of $\rho$.