Global existence and regularity for the Lagrangian averaged Navier-Stokes equations with initial data in \(H^{1/2}\). (Q1766147)

The author considers the Lagrangian averaged Navier-Stokes (LANS-\(\alpha\)) equations in a bounded domain in \(\mathbb R^3\) with zero (no-slip) boundary conditions. With periodic boundary conditions in a box, these equations are also known as Camassa-Holm equations. The (LANS-\(\alpha\)) model averages small, computationally unreasonable scales of Navier-Stokes equations smaller than \(\alpha> 0\). First, using the contraction-mapping techniques, the author establishes the existence and uniqueness of local strong regular solutions with initial data in \(H^{1/2}\); then, using some a priori estimate, he concludes that these are global in time regular solutions. The work extends the class of initial data, for which global classical well-posedness can be obtained; moreover, the method developed by the author, in contrast to the previous works, allows to consider a time-dependent external force acting on the fluid.