Decay of solutions of diffusive Oldroyd-B system in $\mathbb{R}^2$

TL;DR

This paper proves that solutions to the 2D diffusive Oldroyd-B system decay algebraically over time, leveraging the decay of free energy in the associated Fokker-Planck-Navier-Stokes system, and establishes uniform bounds on velocity norms.

Contribution

It demonstrates algebraic decay rates for strong solutions of the 2D diffusive Oldroyd-B system and links this decay to the free energy dissipation of the underlying Fokker-Planck-Navier-Stokes system.

Findings

Solutions decay algebraically over time.

Free energy of the associated system decays over time.

Velocity norms are uniformly bounded for all time.

Abstract

We show that strong solutions of 2D diffusive Oldroyd-B systems in $\mathbb{R}^2$ decay at an algebraic rate, for a large class of initial data. The main ingredient for the proof is the following fact; an Oldroyd-B system is a macroscopic closure of a Fokker-Planck-Navier-Stokes system, and the free energy of this Fokker-Planck-Navier-Stokes system decays over time. In particular, $\norm{u}_{L^\infty_t L^2_x}$ and $\norm{\nabla_x u}_{L^2_t L^2_x }$ are uniformly bounded for all time.