Pointwise Space-Time Behavior of a Compressible Navier-Stokes-Korteweg System in Dimension Three

The Cauchy problem of compressible Navier-Stokes-Korteweg system in ℝ is considered here. Due to capillarity effect of material, we obtain the pointwise estimates of the solution in an H-framework, which is different from the previous results for the compressible Navier-Stokes system in an H-framework [24, 25]. Our result mainly relies on two different descriptions of the singularity in the short wave of Green’s function for dealing initial propagation and nonlinear coupling respectively. Our pointwise results demonstrate the generalized Huygens’ principle as the compressible Navier-Stokes system. As a corollary, we have an L estimate of the solution with p > 1, which is a generalization for p ≥ 2 in [33].