Intuitive explanation
Imagine you're stuck in a traffic jam on a nice 6 lane highway. As it turns out, up ahead they closed all but one lane so everyone has to merge. As you approach and the number of lanes goes down everyone goes a little faster as the total flow rate of cars must be constant. Now once you pass the narrowest point, instead of everyone slowing back down as they expand back out to 6 lanes they go faster!
This happens because the cars in front have nothing holding them back, so they can accelerate. The flow rate of cars must still be conserved, and this is accomplished by everyone increasing their following distance.
This is almost exactly what happens in the case of a converging diverging nozzle. As the air approaches the throat it speeds up and the pressure goes down and the density goes down. Then after it passes the throat the pressure continues to go down and the air continues to speed up because there is nothing pushing back in front and air molecules naturally want to expand.
Energy conservation
The pressure of air flow is constantly decreasing. This pressure gradient is what accelerates the air.
A converging diverging nozzle is placed after a subsonic combustion chamber to take the high temperature, high pressure gas, and transform it into an atmospheric pressure, high velocity gas that will provide thrust through its high momentum.
As Floris points out, this is a energy conserving transformation: taking energy stored as pressure and heat and turning it into kinetic energy.
Without the high pressure, the gas would not go super sonic and the velocity would just go up and back down as is seen in venturis.
Generalizing your mathematics (which are all correct), the 2D stream function automatically satisfies continuity for any 2D case where $\partial\rho / \partial t$ is zero. In flows other than these, you must independently confirm that your results satisfy continuity.
Best Answer
The compressor adds energy to the flow, which in turn increases the pressure of the flow. A simple and approximate way of viewing this problem is by using the Bernoulli equation. Consider, $$P_t = p + \frac{1}{2} \rho \left(u^2 + v^2 + w^2\right) $$ The rotor will add swirl to the flow, which effectively increases the angular momentum of the flow and associated kinetic energy in the tangential velocity component, $\frac{1}{2}\rho v^2$. The stator will remove the imparted swirl from the rotor, but will not add any additional energy to the flow. Thus, the stator can be thought of as converting the kinetic energy of swirl into internal energy of the flow, which is made evident by increase in static pressure of the flow. A conventional velocity and pressure profile across a multistage axial flow compressor looks similar to the following.
More detailed explanations can be obtained in any undergraduate level airbreathing propulsion text. The common analysis encompasses using various reference frames to analyze the true 3-dimensional nature of the flow across a single compressor stage (rotor/stator). These are generally referred to as the throughflow field, cascade field, and secondary flowfield. The analysis also requires velocity diagrams of the velocity components in each subsequent reference frame to which the stage thermodynamic parameters of interest are obtained with the Euler turbine equation.
Also, be careful regarding the shape of a nozzle and its expected role. Depending on the Mach number of the flow entering the nozzle, the shape of the nozzle can have very different effects. A classical results of quasi one-dimensional gas dynamics is the following differential equation termed the area-velocity relation, $$\frac{dA}{A} = (M^2-1) \frac{du}{u} $$ Major results depending on the Mach number $M$ of the gas go as follows:
1.) $M \rightarrow 0$ (incompressible subsonic limit) suggests that $Au$ = constant. Which is the conventional continuity equation for incompressible flow.
2.) $0 \leq M \lt 1$ (subsonic flow), an increase in velocity (+$du$) is associated with a decrease in area (-$dA$). This is also consistent with the conventional continuity equation for incompressible flows. A decrease in area is accompanied by a increase in velocity, and a increase in area is accompanied by a decrease in velocity.
3.) $M \gt 1$ (supersonic flow), an increase in velocity (+$du$) is associated with a increase in area (+$dA$). This is not consistent with the conventional continuity equation for incompressible flows. For instance, in this case, an increase in area is accompanied by an increase in velocity, and a decrease in area is accompanied by a decrease in velocity.
4.) $M = 1$ (sonic flow), yields $dA/A = 0$, which simply means a minimum or maximum condition in area. The only realizable physical solution is the minimum area condition for which one can choke the flow to the sonic condition at the throat (minimum area) of a converging-diverging nozzle.
Based on the above results, the shape of a nozzle can play varying roles and what sometime may appear as a nozzle actually acts a diffuser. A common schematic demonstrating the above is below.