Singular initial data and uniform global bounds for the hyper-viscous Navier-Stokes equation with periodic boundary conditions. (Q1874494)

The author examines a generalization of incompressible Navier-Stokes equations -- hyperviscous Navier-Stokes equations -- with periodic boundary conditions over a rectangular domain \(\Omega\subset\mathbb{R}^n\). For initial data in \(L^p(\Omega)\), with \(p\) satisfying some condition, the local existence and uniqueness of strong solutions is established. For the case \(p= 2\), the above condition is also sufficient to establish global existence of these unique regular solutions and uniform higher-order bounds. The proof uses energy technique, Gronwall inequality and some ideas of semigroup methods.