How to solve these two mixed PDEs?

Hi, could anyone indicate how to formulate and solve the following mixed PDEs?

\nabla \mathbf{u} + (\nabla \mathbf{u})^{\mathrm{T}} = 0
\frac{\partial P}{\partial t}+\nabla \cdot \frac{\partial \mathbf{u}}{\partial t} + \nabla \cdot \nabla P = 0

I know that if I only have the first equation, then I can use LinearVariationalProblem and LinearVariationalSolver to solve the equation. But I don’t know how to properly formulate the problem for the second equation (and the two together). Especially, I don’t know the time step size beforehand, and the step size may change during the process of solving the two equations.

Thank you very much!