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

Research paper by **Dongho Chae**

Indexed on: **24 Oct '12**Published on: **24 Oct '12**Published in: **Mathematics - Analysis of PDEs**

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#### 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.