Long time existence of the non-isentropic slightly compressible Navier-Stokes equations with boundary conditions

We investigate the long time existence of smooth solutions to the initial boundary value problem for the non-isentropic slightly compressible Navier-Stokes equations with slip or non-slip boundary conditions on the velocity. We verify that the compressible Navier-Stokes equations with boundary conditions admit a unique smooth solution on the time interval where the smooth solution of the incompressible Navier-Stokes equations exists, when the Mach number is sufficiently small. Moreover, we obtain the uniform convergence of smooth solutions for the compressible system toward those for the corresponding incompressible system on that time interval.