Estimates of a possible gap related to the energy equality for a class of non-Newtonian fluids

TL;DR

This work addresses the 3D initial-value problem for power-law fluids with shear-dependent viscosity in a spatially periodic domain and proves the existence of a weak solution that satisfies an energy equality for $v_0 olinebreakin J^2_{per}( olinebreak Omega)$ and $p olinebreakin ( frac{9}{5},2)$. Employing a Faedo–Galerkin approach, the authors obtain uniform a priori bounds and establish strong gradient convergence, culminating in an energy-gap analysis that yields two equivalent expressions for the dissipation gap $M(s,t)$. The gap is shown to be determined entirely by energy-related quantities, mirroring the Navier–Stokes energy equality, and the results are achieved via carefully constructed weight functions and a detailed limit passage. The findings provide a rigorous link between non-Newtonian fluid models and the classical energy balance, clarifying when and how an energy-dissipation gap arises due to weak regularity or turbulent-like effects.

Abstract

The paper is concerned with the 3D-initial value problem for power-law fluids with shear dependent viscosity in a spatially periodic domain. The goal is the construction of a weak solution enjoying an energy equality. The results hold assuming an initial data $v_0\in J^2(Ω)$ and for $p\in \left(\frac 95,2\right)$. It is interesting to observe that the result is in complete agreement with the one known for the Navier-Stokes equations. Further, in both cases, the additional dissipation, which measures the possible gap with the classical energy equality, is only expressed in terms of energy quantities.

Theorems & Definitions (10)

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• proof : Proof of Theorem \ref{['mainT']}