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 − τΔΔu in ]0,+∞[×Ω, divu = 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. This 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.