The Cauchy problem for an inviscid Oldroyd-B model in $\mathbb{R}^3$

Abstract

In this paper, we consider the Cauchy problem for an inviscid compressible Oldroyd-B model in three dimensions. The global well posedness of strong solutions and the associated time-decay estimates in Sobolev spaces are established near an equilibrium state. The vanishing of viscosity is the main challenge compared with our previous work [47] where the viscosity coefficients are included and the decay rates for the highest-order derivatives of the solutions seem not optimal. One of the main objectives of this paper is to develop some new dissipative estimates such that the smallness of the initial data and decay rates are independent of the viscosity. In addition, it proves that the decay rates for the highest-order derivatives of the solutions are optimal. Our proof relies on Fourier theory and delicate energy method. This work can be viewed as an extension of [47].