The anelastic approximation and convergence rate for the compressible Navier-Stokes equations with ill-prepared data in bounded domains

We justify the anelastic approximation and establish a convergence rate for local strong solutions to the isentropic compressible Navier-Stokes equations in a three-dimensional bounded domain, with ill-prepared initial data and vorticity-slip boundary conditions. When the Mach and Froude numbers vanish at the same rate $ε$, the system converges to an anelastic system with nonconstant hydrostatic density. Compared with the incompressible limit, the balance between singular pressure and gravity generates a variable-coefficient acoustic operator and additional commutators. To overcome these difficulties, we introduce the pressure perturbation adapted to the hydrostatic balance and establish weighted acoustic and vorticity estimates, yielding uniform-in-$ε$ estimates. We then use the weighted Helmholtz decomposition and time-averaging technique to prove the strong convergence of solenoidal component of the momentum and obtain an error bound controlled by $ε$ and the initial error in the projected momentum. Our analysis reveals the weighted cancellation mechanism that applies to singular limits around a class of nonconstant equilibria.