NOTE ON MONOTONICITY IN SINGULAR VOLTERRA INTEGRAL EQUATIONS

Observing the monotonic type for a class of singular Volterra integral equations we get a short proof of the singular Gronwall inequality in a completed setting with upper bounds as usual and additional lower bounds. Moreover, the solutions to linear singular Volterra integral equations admit norm bounds which (under an obvious restriction) depend in a monotone increasing way on the prescribed data. We use this observation to solve a nonlinear problem: In terms of linear singular Volterra equations we formulate an (seemingly new) iterative approximation scheme to mild Navier-Stokes solutions. The monotonicity of the bounds mentioned above leads to the proof of convergence and error estimates to our scheme inside a scale of Banach spaces locally in time.