Space-time Decay Rate for the Compressible Navier–Stokes–Korteweg System in $${\mathbb {R}}^3$$

Abstract

We investigate the space-time decay rate of solution to the 3D Cauchy problem of the compressible Navier–Stokes–Korteweg system. Based on the previous temporal decay results for this system, it is shown that for any integer \(N\ge 3\), the space-time decay rate of \(k\left( \in \left[ 0, N\right] \right) \)-order spatial derivative of the solution in weighted space \(H^{N-k}_{\gamma }\) is \(t^{-\frac{3}{4}-\frac{k}{2}+\gamma }\). Our methods mainly involve delicate weighted energy estimates and interpolation trick.