Viscosity versus vorticity stretching: Global well-posedness for a family of Navier–Stokes-alpha-like models

We study global well-posedness and regularity of solutions for a family of incompressible three-dimensional Navier–Stokes-alpha-like models that employ fractional Laplacian operators. This family of equations depends on two parameters, θ1θ1 and θ2θ2, which affect the strength of non-linearity (vorticity stretching) and the degree of viscous smoothing. Varying θ1θ1 and θ2θ2 interpolates between the incompressible Navier–Stokes equations and the incompressible (Lagrangian averaged) Navier–Stokes-αα model. Our main result, which contains previously established results of J.L. Lions and others, provides a relationship between θ1θ1 and θ2θ2 that is sufficient to guarantee global existence, uniqueness and regularity of solutions.