Partial Differential Equations to Describe Electric Current
I don't know if they're &quot;equivalent&quot; to the Navier-Stokes equations, because they describe an essentially 1-dimensional system while Navier-Stokes describe a 3-dimensional system, but we typically describe the propagation of current and voltage signals on a transmission line (a pair of wires) using the  Telegrapher's Equations.
$$\frac{\partial V}{\partial x} = -L\frac{\partial I}{\partial t}$$
$$\frac{\partial I}{\partial x} = -C\frac{\partial V}{\partial t}$$
where \$x\$ is position along the transmission line, \$L\$ is the inductance per unit length of line, and \$C\$ is the capacitance per unit length of line.
I won't say much more because it's probably your homework question to research this problem further, but the main thing we do with these equations is derive the 1-dimensional wave equation from them, showing that signals propagate along transmission lines as waves.