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

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.