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arXiv 1209.4432math.AP
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Navier-Stokes方程和Euler方程的局部能量方程

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

Dongho Chae

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12 pages
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摘要：

设$(v,p)$是Navier-Stokes（Euler）方程在$\Bbb R^N\times (0, T)$，$N\geq 3$上的速度和压力的光滑解对。我们设置伯努利函数$Q=1/2 |v|^2 +p$。在适当的无穷远衰变条件下，我们证明了对几乎所有定义在$(0, T)$上的$α(t)$和$β(t)$，存在\bqn && \int_{\{α(t)< Q(x,t)<β(t)\}}（1/2 \frac{\partial}{\partial t} |v| ^2 + v| ø| ^2）dx =ν \int_{Q(x,t)= β(t)}| \nabla Q| dS && {1.7 in}-ν \int_{Q(x,t)= α(t)}| \nabla Q|其中$ø=$旋度$v$是涡度。这表明，在伯努利函数的两个能级之间的每个压缩区域中，除了由于拟能引起的能量耗散之外，能量通过具有较高能级的能级超曲面流入该区域，并且能量通过具有较低能级的能级超曲面流出该区域。通过$α(t)\downarrow\inf_{x\in \Bbb R^N} Q(x,t)$和$ β(t)\uparrow\sup_{x\in \Bbb R^N} Q(x,t)$，我们恢复了著名的能量等式，$ 1/2\frac{d}{dt} \int_{\Bbb R^N} |v|^2=-ν\int_{\Bbb R^N} |ø|^2 dx.$上述等式的一个较弱的版本下，较弱的衰减假设的解决方案在空间无穷远。该等式的定常形式包含了以前关于Navier-Stokes方程的Liouville型结果.

英文摘要：

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 $α(t)$ and $β(t)$ defined on $(0, T)$ there holds \bqn &&\int_{\{α(t)< Q(x,t)<β(t)\}} (1/2\frac{\partial}{\partial t} |v|^2+ν|ø|^2) dx =ν\int_{Q(x,t)= β(t)} |\nabla Q|dS && {1.7in}-ν\int_{Q(x,t)= α(t)} |\nabla Q|dS, \eqn where $ø=$ 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 $α(t)\downarrow\inf_{x\in \Bbb R^N} Q(x,t)$ and $ β(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=-ν\int_{\Bbb R^N} |ø|^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.

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