Large time decay of weak solutions for the 2D Oldroyd-B model of non-Newtonian flows

Abstract This paper is dedicated to investigating large time decay of weak solutions for the 2D Oldroyd-B model of non-Newtonian flows. Due to the inconsistencies of the dissipative effect between Laplacian velocity viscosity and linear stress damping, either the classic Fourier splitting methods or Kato’s methods for both velocity u and stress tensor τ are not available directly. Based on a new observation on the structure of Oldroyd-B model instead of the complex spectral analysis of linear problem, we explore the low frequency effect for weak solutions which allows us to develop the generalized Fourier splitting technique. More precisely, we first examine that the weak solutions with L 2 initial data have the non-uniformly decay ‖ u ( t ) ‖ L 2 ( R 2 ) + ‖ τ ( t ) ‖ L 2 ( R 2 ) → 0 , t → ∞ . Furthermore, we obtain the optimal decay rates of the weak solutions ‖ τ ‖ L 2 ( R 2 ) + ‖ u ‖ L 2 ( R 2 ) ≤ C ( 1 + t ) − 1 2 when ( u 0 , τ 0 ) ∈ L 2 ( R 2 ) ∩ L 1 ( R 2 ) . Our methods here are also available to the time decay issue of the complex fluid flows with partial dissipation.