Tamed 3D Navier-Stokes Equation: Existence, Uniqueness and Regularity

TL;DR

This paper introduces a tamed 3D Navier-Stokes equation, proving existence and uniqueness of smooth solutions, and offers a new approach to constructing suitable weak solutions for the classical equation.

Contribution

It establishes the existence and uniqueness of smooth solutions to a tamed 3D Navier-Stokes equation and provides a novel method for constructing suitable weak solutions.

Findings

Proved existence and uniqueness of smooth solutions.

Connected tame solutions to classical Navier-Stokes solutions.

Provided a new construction method for weak solutions.

Abstract

In this paper, we prove the existence and uniqueness of a smooth solution to a tamed 3D Navier-Stokes equation in the whole space. In particular, if there exists a bounded smooth solution to the classical 3D Navier-Stokes equation, then this solution satisfies our tamed equation. Moreover, using this renormalized equation we can give a new construction for a suitable weak solution of the classical 3D Navier-Stokes equation introduced in [Scheffer: Hausdorff measure and the Navier-Stokes equations. Comm. Math. Phys., 1977] and [Caffarelli, Kohn, Nirenberg: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math., 1982].