We will review several recent results on the existence of global-in-time solutions to the compressible Euler  and the gravitational Euler-Poisson system. The stabilising mechanism is the expansion of the fluid particle trajectories, which generates a strong dispersive effect in the problem. Our focus is on the decisive role of scaling invariances and their interaction with the nonlinearities. To conclude, we will mention several recent results in the opposite direction, with focus on the problem of stellar collapse. These include the existence of finite-time implosion singularities for the gravitational Euler-Poisson system in the supercritical regime and its relativistic analogue - the Einstein-Euler system.