We are concerned with the isentropic compressible Navier–Stokes system in the two-dimensional torus, with rough data and vacuum; the initial velocity belongs to the Sobolev space \(H^1\) and the initial density is only bounded and nonnegative. Arbitrary regions of vacuum are admissible, and no compatibility condition is required. Under these assumptions and for large enough bulk viscosity, global solutions have been constructed in Danchin and Mucha (Commun Pure Appl Math 76:3437–3492, 2023). The main goal of the paper is to establish that these solutions converge exponentially fast to a constant state, and to specify the convergence rate in terms of the viscosity coefficients. We prove similar exponential decay results for the solutions to the inhomogeneous incompressible Navier–Stokes equations, thereby extending to the torus the recent paper (Danchin et al. in Ann Anal Non Lin IHP, 2023) where bounded domains are considered.
