Navier-Stokes equation with hereditary viscosity

We rigorously study the Navier-Stokes equation with a hereditary viscous term which depends on the past history. Such models arise in the dynamics of non-Newtonian fluids and also as viscoelastic models for the dynamics of turbulence statistics in Newtonian fluids. This problem is mathematically harder than the conventional Navier-Stokes problem due to the lack of certain global estimates. We prove the local solvability theorem using a suitable intermediate m-accretive quantization of the nonlinear term. Finite speed of propagation property of the vorticity field is also proved.