Estimates of the local minimum scale for the incompressible navier-stokes equations

Abstract

We consider solutions to the incompressible Navier-Stokes equations in two and three space dimensions. The minimum scale of a solution to these equations is the length scale characterizing the smallest eddy or the sharpest shear layer. We derive estimates for the minimum scale for local regions in space. We show that the minimum scale of solutions in a sub-domain is proportional to the square root of the viscosity divided by the square root of the local maximum velocity gradient. We also prove that the solution is analytic in the sub-domain, with the radius of convergence of Taylor series proportional to the local minimum scale. © 1995, Taylor & Francis Group, LLC. All rights reserved.