[Paper Review] A new proof to the energy conservation for the Navier-Stokes equations

Cheng Yu|arXiv (Cornell University)|Apr 19, 2016

Navier-Stokes equation solutions7 references11 citations

TL;DR

This paper presents a new proof of energy conservation for weak solutions of the incompressible Navier-Stokes equations in a periodic domain, leveraging a lemma by Lions on the convergence of mollified divergence terms. The key result establishes energy equality under the condition $\frac{1}{r} + \frac{1}{s} \leq \frac{1}{2}$, $s \geq 4$, improving upon prior results by avoiding dimensional dependence and offering a streamlined analytical approach via mollification and functional estimates.

ABSTRACT

In this paper we give a new proof to the energy conservation for the weak solutions of the incompressible Navier-Stokes equations. This result was first proved by Shinbrot. The new proof relies on a lemma introduced by Lions.

Motivation & Objective

• To provide a new, streamlined proof of energy conservation for weak solutions of the incompressible Navier-Stokes equations in a periodic domain.
• To establish energy equality under the condition $\frac{1}{r} + \frac{1}{s} \leq \frac{1}{2}$ with $s \geq 4$, independent of spatial dimension $d$.
• To utilize Lions’ lemma on the convergence of mollified divergence terms to handle the nonlinear convective term $\text{div}({\bf u} \otimes {\bf u})$ in the weak formulation.
• To improve upon Shinbrot’s original result by simplifying the analytical framework through mollification and interpolation estimates.
• To extend the applicability of energy conservation criteria to a broader class of weak solutions with minimal integrability assumptions.

Proposed method

• Mollify the velocity field $\bf u$ using a standard $C^\infty_0$ mollifier $\eta_\varepsilon$ to define $\overline{\bf u} = \bf u * \eta_\varepsilon$.
• Test the weak formulation of the Navier-Stokes equations with $\overline{\bf u}$, leading to an energy identity involving $\overline{\bf u}$ and its time derivative.
• Decompose the nonlinear term $\overline{\text{div}({\bf u} \otimes {\bf u})}$ into three remainder terms $R_1$, $R_2$, and $R_3$ to isolate singularities from nonlinearity.
• Apply Lions’ lemma to control $R_1 = \overline{\text{div}({\bf u} \otimes {\bf u})} - \text{div}({\bf u} \otimes \overline{{\bf u}})$ in $L^{\frac{2p}{p+2}}(0,T; L^{\frac{2q}{q+2}}(\Omega))$ with convergence to zero as $\varepsilon \to 0$.
• Use interpolation and H"older's inequality to bound $R_2 = \text{div}({\bf u} \otimes \overline{{\bf u}}) - \text{div}(\overline{{\bf u}} \otimes \overline{{\bf u}})$, showing it vanishes as $\varepsilon \to 0$ under $p,q \geq 4$.
• Leverage the fact that $R_3 = \text{div}(\overline{{\bf u}} \otimes \overline{{\bf u}})$ integrates to zero due to $\text{div}(\overline{{\bf u}}) = 0$, simplifying the energy identity in the limit.

Experimental results

Research questions

• RQ1 Under what minimal integrability conditions on $\bf u$ does energy conservation hold for weak solutions of the incompressible Navier-Stokes equations?
• RQ2 Can the energy equality be recovered for weak solutions without assuming classical regularity, using only weak convergence and mollification techniques?
• RQ3 How does the use of Lions’ lemma on divergence terms improve the analytical treatment of the convective nonlinearity in the energy identity?
• RQ4 Is it possible to remove dimensional dependence in energy conservation criteria for weak solutions, as seen in Serrin-type conditions?
• RQ5 Can the proof framework be adapted to other equations, such as the Euler or compressible Navier-Stokes equations, with similar energy conservation results?

Key findings

• Energy conservation holds for any weak solution $\bf u \in L^\infty(0,T; L^2(\Omega)) \cap L^2(0,T; H^1(\Omega))$ satisfying $\bf u \in L^r(0,T; L^s(\Omega))$ with $\frac{1}{r} + \frac{1}{s} \leq \frac{1}{2}$ and $s \geq 4$.
• The proof establishes the energy equality $\int_\Omega |{\bf u}(t,x)|^2 dx + 2\mu \int_0^T \int_\Omega |\nabla{\bf u}|^2 dx dt = \int_\Omega |{\bf u}_0|^2 dx$ for all $t \in [0,T]$.
• The remainder term $R_1$ arising from the difference between $\overline{\text{div}({\bf u} \otimes {\bf u})}$ and $\text{div}({\bf u} \otimes \overline{{\bf u}})$ converges to zero in $L^{\frac{2p}{p+2}}(0,T; L^{\frac{2q}{q+2}}(\Omega))$ as $\varepsilon \to 0$.
• The term $R_2$, related to the difference between $\bf u \otimes \overline{{\bf u}}$ and $\overline{{\bf u}} \otimes \overline{{\bf u}}$, vanishes in the limit due to $L^p$-$L^q$ bounds and convergence of $\bf u - \overline{{\bf u}}$.
• The condition $\frac{1}{r} + \frac{1}{s} \leq \frac{1}{2}$ with $s \geq 4$ is optimal in the sense that it generalizes Shinbrot’s result without dependence on dimension $d$.
• The method avoids the need for classical regularity or higher integrability by relying on mollification and Lions’ lemma, enabling a direct proof of energy equality under minimal assumptions.