Hydrodynamics at the smallest scales: a solvability criterion for Navier–Stokes equations in high dimensions

Abstract

Strong global solvability is difficult to prove for high-dimensional hydrodynamic systems because of the complex interplay between nonlinearity and scale invariance. We define the Ladyzhenskaya–Lions exponent α l ( n )=(2+ n )/4 for Navier–Stokes equations with dissipation −(− Δ ) α in , for all n ≥2. We review the proof of strong global solvability when α ≥ α l ( n ), given smooth initial data. If the corresponding Euler equations for n >2 were to allow uncontrolled growth of the enstrophy , then no globally controlled coercive quantity is currently known to exist that can regularize solutions of the Navier–Stokes equations for α < α l ( n ). The energy is critical under scale transformations only for α = α l ( n ).

Locations

• Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences - View - PDF