Global existence and decay rates of solutions to the Oldroyd-B model with stress tensor diffusion

Abstract

In this paper, we consider the Oldroyd-B model with stress tensor diffusion in Rd with d≥2. We first establish the global well-posedness of solutions to this model for small initial data (u0,τ0)∈(B˙p,1dp−1∩B˙p,1dp+1)d×(B˙p,1dp)d×d with 1≤p<∞. Furthermore, under some additional L2 type conditions on (u0,τ0), but without any more smallness restrictions, we get the L2 decay rates of all derivatives of (u,τ). It is shown that the velocity u decays as fast as the solution to the corresponding homogeneous linear heat equation, and the symmetry τ decays as fast as ∇u. In particular, when d=3 the velocity u admits the decay rate faster than (1+t)−34 in L2.