# Evolution of pressure in navier stokes

Hello, I haven't studied PDEs much yet, but checked out what the Navier Stokes equations are. I think I understood meaning of the terms in Navier Stokes equations, and what is their purpose in defining the time evolution of velocity of the fluid, but I couldn't see any conditions for the pressure. I would guess the time evolution of the pressure cannot be arbitrary. What equations define pressure in the fluid?

Depends on the model: Newtonian fluid, ideal gas, certain types of fluids, incompressible flow, etc.

The description of the millennium problem (http://www.claymath.org/millennium/Navier-Stokes_Equations/ [Broken]) says, that there we can restrict to incompressible fluids, so I'll stick with it. If have difficulty believing, that the PDEs given in the problem description are the whole truth about the problem, because the PDEs don't even contain the time derivative of the pressure anywhere. Would any pair (u,p) that satisfies the given Navier-Stokes equations really suffice? My instinct says that there must be more conditions to be satisfied for the pressure.

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You can think pressure as the Lagrange Multiplier to the incompressibility constrain on the velocity.

It is useful to consider the pressure gradient as a driving acceleration - forcing function, together with an external body acceleration.

In higher-speed flows, the compressibility effect of the fluid can affect the pressure gradient. In low-speed momentum-driven flows, the pressure gradient term often plays little role in in the dynamics.