Functional Reformulation of the Continuity Equation in Gases with Constant Density and its Application to the Existence Problem of Smooth Solutions to the Navier Stokes System

TL;DR

This paper introduces a new functional reformulation of the incompressible Navier-Stokes equations based on the energy equation and ideal gas law, providing a novel framework for analyzing existence and regularity of solutions.

Contribution

It develops a rigorous reformulation of the Navier-Stokes equations using a pressure-based functional, enabling new approaches to existence, regularity, and singularity analysis.

Findings

Defines a pressure-based functional norm to bound viscous dissipation

Replaces classical regularity criteria with the new functional

Establishes a comprehensive framework for local existence and uniqueness

Abstract

We propose a rigorous reformulation of the incompressible Navier Stokes equations, starting from the energy equation and the ideal gas law. This reformulation allows the definition of a functional over the pressure field, which is used to bound the viscous dissipation term. It is shown that this norm can replace classical regularity criteria and serves as the foundation for a complete functional framework that includes local existence, singularity control, variational formulation, and uniqueness conditions.