Global well-posedness of 2D incompressible Oldroyd-B model with only velocity dissipation

Abstract

The small data global well-posedness in Sobolev setting for the incompressible Oldroyd-B equations with only velocity dissipation (namely there is no damping or dissipation of stress tensor) on R2 remains open by now. The approach for the similar problem on R3 becomes invalid due to the fewer decay rates. In this paper, we will give a positive answer to this problem. Namely, we shall focus on the Cauchy problem for the incompressible Oldroyd-B equations with only velocity dissipation on R2 and derive the small data global well-posedness under Sobolev setting. We also give the time decay estimate for the solutions. To achieve this goal, we explore the dissipative structure of system and develop the fractional time-weighted energy framework. Moreover, some delicate commutator and bilinear estimates are proposed to deal with the wildest nonlinear terms.