On the energy inequality for weak solutions to the Navier–Stokes equations of compressible fluids on unbounded domains

We consider the Navier–Stokes equations of compressible isentropic viscous fluids on an unbounded three-dimensional domain with a compact Lipschitz boundary. Under the condition that the total mass of the fluid is finite, we show the existence of globally defined weak solutions satisfying the energy inequality in differential form.