Existence of weak solutions to the three-dimensional steady compressible Navier–Stokes equations

We prove the existence of a spatially periodic weak solution to the steady compressible isentropic Navier–Stokes equations in \mathbb{R}^{3} for any specific heat ratio \gamma > 1 . The proof is based on the weighted estimates of both pressure and kinetic energy for the approximate system which result in some higher integrability of the density, and the method of weak convergence.