Global well-posedness and time decay for 2D Oldroyd-B-type fluids in periodic domains with dissipation in the velocity equation only

Abstract

There have been substantial recent developments on the stability problem concerning the Oldroyd-B model of the incompressible non-Newtonian fluids, especially when the system involves only partial dissipation. One particular case is when there is only velocity dissipation, and no damping or dissipation in the equation of the non-Newtonian stress tensor τ. Yi Zhu was able to obtain the global stability for the 3D Oldroyd-B model in the Sobolev setting by employing time-weighted Sobolev spaces (Zhu, 2018). However, her approach can not be extended to the 2D whole space case due to the criticality of the time-weight. This paper presents the global stability and the large-time behavior of solutions to the 2D Oldroyd-B model with only dissipation in a periodic domain. The proof of this result overcomes the difficulty due to the lack of dissipation in τ by exploiting the special wave structure obeyed by the velocity u and P∇⋅τ (the projection of the divergence of τ). The enhanced dissipation in u and P∇⋅τ allows us to gain enough regularity and stabilizing property to control the growth of u and τ. In fact we are also able to show that the H1-norm of ∇u and P∇⋅τ decays exponentially in time.