Oldroyd-B Model with High Weissenberg Number and Fractional Velocity Dissipation

This paper focuses on a high Weissenberg number Oldroyd-B model of complex fluids with fractional frequency velocity dissipation. Mathematically the fluid velocity u satisfies the Navier–Stokes equations with fractional dissipation (-Δ )^α u while the equation of the non-Newtonian tensor τ involves no diffusion or damping mechanism. The aim here is to solve the small-data global well-posedness and stability problem with the least amount of dissipation and minimal regularity requirement. We are able to establish the desired well-posedness and stability result in a hybrid homogeneous Besov setting for any fractional power in the range 1/2≤α≤ 1 . To deal with the difficulties due to the weak velocity dissipation and the lack of diffusion or damping in the τ -equation, we exploit the coupling and interaction of this Oldroyd-B model to reveal the hidden wave structure and make extensive use of the associated smoothing and stabilizing effect.