Optimality of the Decay Estimate of Solutions to the Linearised Curl-Free Compressible Navier-Stokes Equations

We discuss optimal estimates of solutions to the compressible Navier-Stokes equations in Besov norms. In particular, we consider the estimate of the curl-free part of the solution to the linearised equations, in the homogeneous case. We prove that our estimate is optimal in the $L^{\mathrm{\infty }}$-norm by showing that the norm is bounded from below by the same decay rate.