Space-time decay rate of the 3D diffusive and inviscid Oldroyd-B system

Abstract

We investigate the space-time decay rates of solutions to the 3D Cauchy problem of the compressible Oldroyd-B system with diffusive properties and without viscous dissipation. The main novelties of this paper involve two aspects: On the one hand, we prove that the weighted rate of $ k $-th order spatial derivative (where $ 0\leq k\leq3 $) of the global solution $ (\rho, u, \eta, \tau) $ is $ t^{-\frac{3}{4}+\frac{k}{2}+\gamma} $ in the weighted Lebesgue space $ L^2_{\gamma} $. On the other hand, we show that the space-time decay rate of the $ m $-th order spatial derivative (where $ m \in\left [0, 2\right] $) of the extra stress tensor of the field in $ L^2_{\gamma } $ is $ (1+t)^{-\frac{5}{4}-\frac{m}{2}+\gamma} $, which is faster than that of the velocity. The proofs are based on delicate weighted energy methods and interpolation tricks.

Keywords

• oldroyd-b system
• diffusive properties
• without viscous dissipation
• space-time decay rates
• weighted sobolev space