Optimal Lp–Lq convergence rates for the compressible Navier–Stokes equations with potential force

In this paper, we are concerned with the optimal Lp–Lq convergence rates for the compressible Navier–Stokes equations with a potential external force in the whole space. Under the smallness assumption on both the initial perturbation and the external force in some Sobolev spaces, the optimal convergence rates of the solution in Lq-norm with 2⩽q⩽6 and its first order derivative in L2-norm are obtained when the initial perturbation is bounded in Lp with 1⩽p<6/5. The proof is based on the energy estimates on the solution to the nonlinear problem and some Lp–Lq estimates on the semigroup generated by the corresponding linearized operator.