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The Cauchy problem for the Gurevich–Zybin system

We consider the periodic Cauchy problem for the Gurevich–Zybin (GZ) system on the n-dimensional torus. The GZ system models dark matter as a collision-free gas in an expanding universe. We establish local in-time well-posedness for classical solutions in Sobolev spaces. Moreover, the local in-time result is extended to all time under additional assumptions. In particular, we consider three cosmological eras determined by the universal expansion parameter. In each era, global in-time existence is established if the initial density contrast is small and the rate of change of the initial velocity is sufficiently large.