Global Solutions of the Navier-Stokes Equations for Isentropic Flow with Large External Potential Force

We prove the global-in-time existence of weak solutions to the Navier-Stokes equations of compressible isentropic flow in three space dimensions with adiabatic exponent $γ\ge1$. Initial data and solutions are small in $L^2$ around a non-constant steady state with densities being positive and essentially bounded. No smallness assumption is imposed on the external forces when $γ=1$. A great deal of information about partial regularity and large-time behavior is obtained.

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