On an auto-controlled global existence scheme of the incompressible Navier Stokes equation

TL;DR

This paper introduces a novel global scheme for the incompressible Navier-Stokes equations using a damping potential via time dilation, providing bounds on solutions based on data regularity.

Contribution

It presents an alternative analytical scheme that ensures global existence and bounds without relying on external control functions.

Findings

Establishes global upper bounds for solutions and derivatives.

Demonstrates scheme's effectiveness under regularity conditions.

Provides an analytical framework for Navier-Stokes regularity analysis.

Abstract

We propose a global scheme for the incompressible Navier Stokes equation, where at each time step a damping potential term is introduced via a time dilation transformation of the equation itself. This leads a global upper bounds of the value function and its spatial derivatives. The regularity is limited only by the regularity of the viscosity coefficient function and by the regularity and polynomial decay of the data. On an analytical level the scheme proposed is an alternative to schemes with external control functions.