[Paper Review] Energy conservation via a combination of velocity and its gradient in the Navier-Stokes system

Yan Qing Wang, Yulin Ye|arXiv (Cornell University)|Jun 2, 2021

Navier-Stokes equation solutions24 references4 citations

TL;DR

This paper establishes a unified energy conservation criterion for incompressible and compressible Navier-Stokes equations by combining velocity and its spatial gradient in Lebesgue spaces. It proves that if velocity and its gradient satisfy certain integrability conditions, energy equality holds, extending and unifying prior results for both incompressible and compressible flows.

ABSTRACT

In the spirit of recent work \cite{[NNT]},it is shown that $v\in L^{\frac{2p}{p-1}}(0,T; L^{\frac{2q}{q-1}}(\mathbb{T}^{3})) $ and $ abla v\in L^{p}(0,T; L^{q}(\mathbb{T}^{3})) $ imply the energy equality in homogeneous incompressible Navier-Stokes equations and together with bounded density with positive lower bound yields the energy conservation in the general compressible Navier-Stokes equations. This unifies the known energy conservation criteria via the velocity and its gradient in incompressible Navier-Stokes equations. This also helps us to extend the conditions via the velocity or gradient of the velocity for energy equality from the incompressible fluid to compressible flow and improves the recent results due to Nguyen-Nguyen-Tang \cite[Nonlinearity 32 (2019)]{[NNT]} and Liang \cite[Proc. Roy. Soc. Edinburgh Sect. A (2020)]{[Liang]}.

Motivation & Objective

• To unify existing energy conservation criteria for incompressible Navier-Stokes equations based on velocity and its gradient.
• To extend these criteria to the compressible Navier-Stokes equations under bounded density with positive lower bound.
• To improve recent results by Nguyen-Nguyen-Tang (2019) and Liang (2020) on energy equality conditions.
• To establish a general framework for energy conservation using joint regularity of velocity and its gradient in Lebesgue spaces.

Proposed method

• Uses Lebesgue space integrability conditions: $ v \in L^{\frac{2p}{p-1}}(0,T; L^{\frac{2q}{q-1}}(\mathbb{T}^3)) $ and $ \nabla v \in L^p(0,T; L^q(\mathbb{T}^3)) $.
• Applies functional analytic techniques to derive energy equality from these integrability assumptions.
• Extends the incompressible framework to compressible flows by incorporating bounded density with positive lower bound.
• Employs interpolation and embedding theorems to relate velocity and gradient regularity to energy conservation.
• Relies on the structure of the Navier-Stokes equations in periodic domains $ \mathbb{T}^3 $ to ensure mathematical closure.
• Combines results from prior works (e.g., [NNT], [Liang]) into a single, coherent criterion.

Experimental results

Research questions

• RQ1 Can a single condition combining velocity and its gradient regularity unify existing energy conservation criteria in incompressible Navier-Stokes equations?
• RQ2 How can the energy conservation criterion be extended from incompressible to compressible Navier-Stokes flows?
• RQ3 What integrability conditions on velocity and its gradient ensure energy equality in the compressible case with bounded density?
• RQ4 In what way does the joint regularity of velocity and its gradient improve upon prior individual criteria?
• RQ5 How does the inclusion of bounded density with positive lower bound affect energy conservation in compressible flows?

Key findings

• Energy equality holds in the incompressible Navier-Stokes equations if $ v \in L^{\frac{2p}{p-1}}(0,T; L^{\frac{2q}{q-1}}(\mathbb{T}^3)) $ and $ \nabla v \in L^p(0,T; L^q(\mathbb{T}^3)) $ for suitable $ p, q $.
• The same condition, combined with bounded density having a positive lower bound, ensures energy conservation in the compressible Navier-Stokes equations.
• The proposed criterion unifies and generalizes previous results by Nguyen-Nguyen-Tang and Liang for energy equality.
• The framework extends the applicability of energy conservation criteria from incompressible to compressible fluid flows.
• The result demonstrates that joint regularity of velocity and its gradient is sufficient for energy conservation, even in the presence of compressibility.