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

We study the well-posedness of the problem\r\n⎧\r\n⎪\r\n⎨\r\n⎪\r\n⎩\r\n∂u\r\n∂t\r\n+ (Du)u + ∇p = νΔu − τΔΔu in ]0,+∞[×Ω,\r\ndivu = 0 in ]0,+∞[×Ω,\r\nu(t,x) =\r\n∂u\r\n∂n (t,x) = 0\r\non ]0,+∞[×∂Ω,\r\nu(0,x) = u 0 (x) in Ω,\r\nwhere u :]0,+∞[×Ω → R n is the velocity field, p :]0,+∞[×Ω → R is the pressure,\r\nν is the kinematical viscosity, τ the so-called hyperviscosity and Ω is a general\r\ndomain as for existence and uniqueness of the solution, and an exterior domain as\r\nfor regularity results.\r\nThis problem has been physically well motivated in the recent years as the\r\nsimplest case of an isotropic second-order fluid, i.e. a fluid whose power expended\r\ndepends on second derivatives of the velocity field.