You are of course correct in saying that pictures are very useful. But we must take great care in how we interpret them.
The blue line in your top diagram is not static pressure, but is total pressure.
The area immediately aft of the HP compressor outlet is called the Diffuser. It is a divergent duct and its purpose is to reduce the air velocity to a level that will enable the flame to remain in the primary zone of the combustion chamber, without being blown aft. The flow through the divergent duct of the diffuser is subsonic, so its velocity decreases. This causes its dynamic pressure to decrease and its static pressure to increase. But there is an overall reduction in total pressure because of friction losses. All of this causes the highest static pressure to occur, not at the outlet of the compressor, but at the outlet of the diffuser. It is the highest total pressure which occurs at the outlet of the compressor.
Look at the pressure v. Velocity diagram here. Pressure declines in the burner but velocity goes up almost vertically. Remember F = M x A. Force = Mass times acceleration. The pressure drops due to acceleration and pressure loss across the turbine. Note the gradual pressure drop in the burner section. This is a critical design area. You need pressure drop but you also need a big fire.
In theory the gas turbine employs a constant pressure combustion process. Ideally we would like the static pressure to remain constant. To achieve this we allow the air to expand aft as it is heated. But friction and losses of heat through the engine casing cause some loss of energy and this causes reductions in velocity and total pressure. We do not actually need the pressure drop in the combustion chamber, but we must avoid getting any significant static pressure increase, because this would tend to cause reverse flow.
As the air flows through the turbines a great deal of energy is extracted from it to drive the engine. It is this extraction of energy that causes the reduction in both static pressure and total pressure.
But the real problem with your employment of this diagram in this thread, is that it does not actually address the original question. IE "Why does the air not flow from the high pressure area at the back of the compressor to the lower pressure area at the front?"
maf, in answer to your statement that:
I know how the engine is driven, but was unsure how to explain the self-sustainability of it.
The process of driving the compressor and accessories at any given rpm requires a certain amount of power. This power is provided by the turbines which extract energy from the hot gasses. Self-sustaining rpm is the lowest rpm at which the turbines can provided sufficient power to keep the compressor running at that rpm.
At speeds below self-sustaining rpm, the turbines are unable to provide sufficient power to keep the engine running. So the starter motor must be used to assist the turbines to keep the engine running and also to provide the additional power that is required to accelerate the engine to higher rpm. If you switch off the starter at any rpm lower than self-sustaining, there will be insufficient power available to keep it running, so the engine will run down and stop.
If you switch off the starter at exactly self-sustaining rpm there will be just sufficient power to keep the engine running at that rpm, but not sufficient to enable it to accelerate to higher rpm. This will cause the rpm to stagnate in what is termed a “hung start”. Unless the pilot acts quickly to shut the throttle, the temperature limits of the turbines will be exceeded.
If you switch off the starter at any rpm above self-sustaining, the turbines will provide sufficient power to accelerate the engine to higher rpm. So a standard start sequence involves running the starter until the engine has reached some specified rpm, which is above self-sustaining speed.