Global regularity for the 2D Oldroyd-B model with fractional dissipation

This paper deals with the Cauchy problem for the two-dimensional incompressible Oldroyd-B model in the corotational case with fractional dissipation (-Delta)alpha u and (-Delta)beta tau, where 0<alpha,beta<1. Our objective is to establish global regularity of the fractional Oldroyd-B model with minimal amount of dissipation. The proof of the global regularity relies on the introduction of combined quantities, sharp lower bounds for the fractional dissipation, the De Giorgi-Nash estimate and sharp upper bounds for the nonlinearities.