Global Classical Solution to the Strip Problem of 2D Compressible Navier–Stokes System with Vacuum and Large Initial Data

In this work, we modify the weighted \(L^p\) $L^p$ bounds for elements of the Hilbert space \(\tilde{D}^{1,2}(\Omega )\) ${\tilde{D}}^{1,2}(\mathrm{\Omega })$ . Using this bound, we derive the upper bound for the density, which is the key issue to global solution provided the shear viscosity is a positive constant and the bulk one is \(\lambda = \rho ^{\beta }\) $\lambda ={\rho }^{\beta }$ with \(\beta >4/3\) $\beta >4/3$ . Our results extend the earlier results due to Vaigant-Kazhikhov (Sib Math J 36:1283–1316, 1995) where they required that \(\beta >3\) $\beta >3$ , initial densities is strictly away from vacuum, and that the domain is bounded.