Global existence and time decay rates for the 3D bipolar compressible Navier-Stokes-Poisson system with unequal viscosities

Abstract

We consider the Cauchy problem for the three-dimensional bipolar compressible Navier-Stokes-Poisson system with unequal viscosities. Under the assumption that the H3 norm of the initial data is small but its higher order derivatives can be arbitrarily large, the global existence and uniqueness of smooth solutions are obtained by an ingenious energy method. Moreover, if additionally, the $${\dot H^{- s}}\left({{1 \over 2} \leqslant s < {3 \over 2}} \right)\,\,{\rm{or}}\,\,\dot B_{2,\infty }^{- s}\left({{1 \over 2} < s \leqslant {3 \over 2}} \right)$$ norm of the initial data is small, the optimal decay rates of solutions are also established by a regularity interpolation trick and delicate energy methods.