The Energy Balance Relation for Weak solutions of the Density-Dependent Navier-Stokes Equations

TL;DR

This paper establishes the energy balance relation for weak solutions of the inhomogeneous Navier-Stokes equations with density-dependent effects, under specific Besov space regularity conditions.

Contribution

It introduces a density-dependent version of the classical Kármán-Howarth-Monin relation and extends energy balance results to weak solutions with certain Besov space regularity.

Findings

Energy balance holds for weak solutions with velocity, density, and pressure in Besov spaces of smoothness 1/3.

Derived a density-dependent Kármán-Howarth-Monin relation.

Identified regularity conditions ensuring energy conservation in inhomogeneous fluids.

Abstract

We consider the incompressible inhomogeneous Navier-Stokes equations with constant viscosity coefficient and density which is bounded and bounded away from zero. We show that the energy balance relation for this system holds for weak solutions if the velocity, density, and pressure belong to a range Besov spaces of smoothness $1/3$. A density-dependent version of the classical K\'arm\'an-Howarth-Monin relation is derived.