Localized energy equalities for the Navier-Stokes and the Euler equations

TL;DR

This paper derives localized energy equalities for the Navier-Stokes and Euler equations involving the Bernoulli function, revealing energy flow dynamics between regions defined by level sets, and extends to stationary and Liouville type results.

Contribution

It introduces localized energy equalities based on the Bernoulli function, providing new insights into energy flow and extending classical results under decay conditions.

Findings

Localized energy equalities for Navier-Stokes and Euler equations.

Energy flows through level hypersurfaces of the Bernoulli function.

Extension to stationary solutions and Liouville type results.

Abstract

Let $(v,p)$ be a smooth solution pair of the velocity and the pressure for the Navier-Stokes(Euler) equations on $\Bbb R^N\times (0, T)$, $N\geq 3$. We set the Bernoulli function $Q=1/2 |v|^2 +p$. Under suitable decay conditions at infinity for $(v,p)$ we prove that for almost all $\alpha(t)$ and $\beta(t)$ defined on $(0, T)$ there holds \bqn &&\int_{\{\alpha(t)< Q(x,t)<\beta(t)\}} (1/2\frac{\partial}{\partial t} |v|^2+\nu |\o|^2) dx =\nu \int_{{Q(x,t)= \beta(t)}} |\nabla Q|dS && {1.7in}-\nu \int_{{Q(x,t)= \alpha(t)}} |\nabla Q|dS, \eqn where $\o=$ curl $v$ is the vorticity. This shows that, in each region squeezed between two levels of the Bernoulli function, besides the energy dissipation due to the enstrophy, the energy flows into the region through the level hypersurface having the higher level, and the energy flows out of the region through the level hypersurface with the lower level. Passing $\alpha(t)\downarrow\inf_{x\in \Bbb R^N} Q(x,t)$ and $ \beta(t)\uparrow\sup_{x\in \Bbb R^N} Q(x,t)$, we recover the well-known energy equality, $ 1/2\frac{d}{dt} \int_{\Bbb R^N} |v|^2=-\nu\int_{\Bbb R^N} |\o|^2 dx.$ A weaker version of the above equality under the weaker decay assumption of the solution at spatial infinity is also derived. The stationary version of the equality implies the previous Liouville type results on the Navier-Stokes equations.