Existence, Uniqueness, and Regularity for the Second–Gradient Navier–Stokes Equations in Exterior Domains

We study the well-posedness of the problem ∂u∂t+(Du)u+∇p=νΔu−τΔΔuin ]0,+∞[×Ω, div(u)= 0 in ]0,+∞[×Ω, u(t, x)=∂u/∂n(t,x)=0 on ]0,+∞[×∂Ω, u(0,x)=u_0(x) in Ω, where u:]0,+∞[×Ω→R^n is the velocity field, p:]0,+∞[×Ω→R is the pressure, ν is the kinematical viscosity,τ the so-called hyperviscosity and Ω is a general domain as for existence and uniqueness of the solution, and an exterior domain as for regularity results. Second-Gradient Navier–Stokes Equations in Exterior DomainsThis problem has been physically well motivated in the recent years as the simplest case of an isotropic second-order fluid, i.e. a fluid whose power expended depends on second derivatives of the velocity field.