Pointwise space-time estimates of compressible Oldroyd-B model

Abstract

In this paper, we investigate pointwise space-time behavior of the compressible Oldroyd-B model in R3 based on Green's function method. We first give the representations of two Green's matrices in the Fourier space: one is a 7×7 matrix for the compressible part and the other is a 6×6 matrix for the incompressible part. Then by using high-medium-low frequency analysis together with real analysis and Fourier analysis we get the pointwise estimates for each entry in these Green's matrices. Finally, we deal with the nonlinear problem and obtain the pointwise space-time description of the solution. It shows that the fluid density ρ, momentum m and the symmetric tensor of constrains T obey the generalized Huygens' principle as the compressible Navier-Stokes equations. As a byproduct, we extend L2-estimate in [47] to Lp-estimate with p>1. Moreover, the decay of T is faster than those of (ρ,m), which also refine the result in [47].