Sharp decay characterization for the incompressible Oldroyd-B model in critical $L^p$ spaces

TL;DR

This paper characterizes the precise decay rates over time for solutions to the incompressible Oldroyd-B model in critical spaces, especially when viscosity and damping are absent, using a new tensor decomposition.

Contribution

It provides the first sharp two-sided decay asymptotics for the Oldroyd-B model without viscosity or damping, introducing a novel tensor decomposition and effective tensor.

Findings

Established necessary and sufficient conditions for decay rates in critical Besov spaces.

Developed a new decomposition of the stress tensor into incompressible and compressible parts.

Proved optimal decay bounds for solutions in the absence of viscosity and damping.

Abstract

This paper establishes a sharp characterization of temporal decay rates for the incompressible Oldroyd-B model in a critical $L^p$ framework, covering the physically relevant and mathematically delicate case where both the fluid viscosity and the stress tensor damping are absent. We prove that an $L^2$-type condition on the low-frequencies part of the initial data $(u_0, \tau_0)$ is almost both necessary and sufficient for obtaining optimal upper and lower bounds on the temporal decay of solutions in critical Besov spaces. A key contribution is a new decomposition of the stress tensor into its incompressible and compressible parts, combined with the introduction of an effective tensor to handle the loss of regularity in the high-frequencies velocity field. This is the first result to reveal such precise two-sided asymptotics for the incompressible Oldroyd-B model without viscosity or damping.