Classification of partial differential equation

[tsunamiBTP]

I have rather nasty vector partial diff eq which I think has been addressed before; however, when I vistied a few websites (other than Polyarin's site) to classify the equation the notation of these websites is not necessarily what I am accustomed to. I have attached the equation here & was wondering if anyone could point me in the right direction so I can see how others approached solving the problem.

I suspect this has applications in electromagnetism, but it may be applicable elsewhere, MAYBE gas dynamics?

I have a solution but would like to compare mine with others.

Appreciate any feedback

 

[gtoguy97]

.The equation you have is a vector form of the continuity equation for a compressible flow, which is a basic equation used in fluid mechanics and gas dynamics. It states that the net rate of change of mass within an infinitesimal control volume must be equal to the difference between the rate of mass entering the volume and the rate of mass leaving it. It can be written as: \begin{align} \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \textbf{u}) = 0 \end{align}where $\rho$ is the density of the fluid, \textbf{u} is the velocity vector, and $\nabla$ is the gradient operator. This equation is commonly used to model flows such as sound waves, shock waves, and turbulence. You can find more information about it and how to solve it on numerous online resources.