Incompressible limit of all-time solutions to isentropic Navier-Stokes equations with ill-prepared data in bounded domains

In this paper, we study the incompressible limit of all-time strong solutions to the isentropic compressible Navier-Stokes equations with ill-prepared initial data and slip boundary condition in three-dimensional bounded domains. The uniform estimates with respect to both the Mach number $ϵ\in (0,1]$ and all time $t\in [0,+\infty )$ are derived by establishing a nonlinear integral inequality. In contrast to previous results for well-prepared initial data, the time derivatives of the velocity are unbounded which leads to the loss of strong convergence of the velocity. The novelties of this paper are to establish weighted energy estimates of new-type and to carefully combine the estimates for the fast variables and the slow variables, especially for the highest-order spatial derivatives of the fast variables. The convergence to the global strong solution of incompressible Navier-Stokes equations is shown by applying the Helmoltz decomposition and the strong convergence of the incompressible part of the velocity.