Long-term regularity of two dimensional Navier-Stokes-Poisson equations

This manuscript is devoted to the long-term regularity of the 2-d Navier-Stokes-Poisson system. We allow the initial density to be close to a constant and the potential part of the initial velocity to be small independently of the rescaled viscosity parameter $\varepsilon$ while the rotational part of the initial velocity is assumed to be small compared to $\varepsilon$. We then show that the lifespan of the system $T^{\varepsilon}$ satisfies $T^{\varepsilon}>\varepsilon^{-(1-\vartheta)}$, where the small constant $\vartheta$ is the size of the initial perturbation in some suitable space. The normal form transformation and the classical parabolic energy estimates are the main ingredients of the proof.