A Solvability Criterion for Navier-stokes Equations in High Dimensions

Abstract. We define the Ladyzhenskaya-Lions exponent αl(n) = (2 + n)/4 for Navier-Stokes equations with dissipation −(−∆) in R, for all n ≥ 2. We then prove strong global solvability when α ≥ αl(n), given smooth initial data. If the corresponding Euler equations for n > 2 were to allow uncontrolled growth of the enstrophy 1 2 ‖∇u‖ L2 , then no globally controlled coercive quantity is currently known to exist that can regularize solutions of the Navier-Stokes equations for α < αl(n). The energy is critical under scale transformations only for α = αl(n).