Bounds on solutions to the boundary-free incompressible Navier-Stokes initial value problem

Abstract

The integral formulation of the Navier-Stokes initial value problem for boundary-free, incompressible fluid flow is used to establish Volterra integral inequalities on the physical velocity field u μ (x, t). Standard comparison theorems for Volterra equations are then applied to obtain weakly singular, nonlinear Volterra equations of the second kind for upper bounds on ¦ u μ( x, t)¦ . With local existence and uniqueness guaranteed and smoothness of solutions characterized by results on Volterra equations, solutions for bounds may be obtained by standard analytical and numerical methods. A specific example is considered. Existence and uniqueness of local solutions u μ (x, t) to the Navier-Stokes problem is then guaranteed within the radius of convergence of bounds on ¦ u μ( x, t)¦ , thereby precluding local breakdown phenomena. In addition, bounds on ¦ u μ( x, t)¦ are used to obtain improved duration times for convergence of local iteration solutions for u μ (x, t). Finally, a new technique for establishing sufficient conditions for global existence, based on successive application of Volterra comparison theorems, is indicated.