The Divergence-Free Radiant Transform

TL;DR

This paper introduces the Divergence-Free Radiant Transform (DFRT), a spectral method tailored for divergence-free vector fields, with foundational properties, algebraic structure, and applications to incompressible fluid dynamics.

Contribution

The paper develops the rigorous mathematical foundation of DFRT, including basis construction, transform properties, and its application to Navier-Stokes equations, connecting to cohomological frameworks.

Findings

DFRT basis functions satisfy divergence-free condition.

Derived modal evolution equations for Navier-Stokes in DFRT coordinates.

Showed entropy-maximizing energy distribution leads to exponential decay.

Abstract

This paper presents the rigorous mathematical construction and foundational properties of the Divergence-Free Radiant Transform (DFRT), a spectral transform specifically designed for divergence-free vector fields, with applications in incompressible fluid dynamics and other solenoidal systems. The DFRT basis functions are constructed using a curl-based formulation that ensures the divergence-free condition is satisfied identically. We define the forward and inverse transforms, prove the Parseval identity, and establish the completeness of the basis. The DFRT coefficient space is equipped with an algebraic structure via a spectral coboundary operator, defined using Wigner 3j and 6j symbols to encode angular momentum coupling. This cohomological structure, and its connection to the Geometric Refinement Transform (GRT), is developed in a companion paper using a bigraded cohomology framework. We derive a modal evolution equation for the incompressible Navier-Stokes equations in DFRT coordinates and introduce a persistent regularity class based on cohomological constraints. Finally, we present a variational argument showing that an entropy-maximizing energy distribution leads to exponential decay, offering a new perspective on regularity and singularity prevention in incompressible flows.