A Nonlocal Parabolic Equation Arising in a Turbulence Model

We consider a class of parabolic partial differential equations with nonlocal diffusion coefficient. The study is motivated by a model of turbulent viscosity for a developed flow in a duct between parallel walls. Under rather more general conditions, we use a rescaling in t to prove existence and well-posedness for weak solutions. Sharp comparison estimates enable us to bound from below the spatial derivative at the wall. The steady state problem is also nonstandard and we show local stability by a spectral continuation argument. Finally numerical results for the steady state solution are compared with available experimental data