A new construction of weak solutions to compressible Navier–Stokes equations

We prove the existence of the weak solutions to the compressible Navier–Stokes system with barotropic pressure \(p(\varrho )=\varrho ^\gamma \) $p(\varrho )={\varrho }^{\gamma }$ for \(\gamma \ge 9/5\) $\gamma \ge 9/5$ in three space dimension. The novelty of the paper is the approximation scheme that instead of the classical regularization of the continuity equation (based on the viscosity approximation \(\varepsilon \Delta \varrho \) $\epsilon \mathrm{\Delta }\varrho$ ) uses more direct truncation and regularisation of nonlinear terms and the pressure. This scheme is compatible with the Bresch–Jabin compactness criterion for the density. We revisit this criterion and prove, in full rigour, that it can be applied in our approximation at any level.