Global existence and stability for the 2D Oldroyd-B model with mixed partial dissipation

This paper focuses on a two-dimensional incompressible Oldroyd-B model with mixed partial dissipation. The goal here is to establish the small data global existence and stability in the Sobolev space H^2(\mathbb R^2). The velocity equation itself, without coupling with the equation of the non-Newtonian stress tensor, is an anisotropic 2D Navier-Stokes whose solutions are not known to be stable in Sobolev spaces due to potential rapid growth in time. By unearthing the hidden wave structure of the system and exploring the smoothing and stabilizing effect of the non-Newtonian stress tensor on the fluid, we are able to solve the desired global existence and stability problem.