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三维周期Navier-斯托克斯方程Leray-Hopf解的能量不等式的加强

A strengthening of the energy inequality for the Leray-Hopf solutions of the 3D periodic Navier-Stokes equations

Radu Dascaliuc

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摘要：

本文建立了三维$Ω$周期Navier-斯托克斯方程的Leray-Hopf解的不等式：对于所有的$t_0$ Leray-Hopf点，\[ϕ(|u(t)|^2)-ϕ(|u(t_0)|^2)\le 2\int_{t_0}^{t}ϕ'(|u(τ)|^2) [-ν|A^{1/2}u(τ)|^2+(g(τ),u(τ))]\,dτ\]，$t\ge t_0$和$ϕ:\mathbb{R}_{+}\to\mathbb{R}$是导数有界的绝对连续非减函数.对于所有$ξ>0$，使用$ϕ'(ξ)\ge0$的%。这里，$(\cdot,\cdot)$和$|\cdot|$对应地是$Ω$上的$L^2$内积和$L^2$范数，并且$A$是Stokes算子。

英文摘要：

In present note we establish the following inequality for the the Leray-Hopf solutions of the 3-D $Ω$-periodic Navier-Stokes Equations: \[ϕ(|u(t)|^2)-ϕ(|u(t_0)|^2)\le 2\int_{t_0}^{t}ϕ'(|u(τ)|^2) [-ν|A^{1/2}u(τ)|^2+(g(τ),u(τ))]\,dτ\] for all $t_0$ Leray-Hopf points, $t\ge t_0$, and $ϕ:\mathbb{R}_{+}\to\mathbb{R}$ is an absolutely continouos non-decreasing function with bounded derivative. %with $ϕ'(ξ)\ge0$ for all $ξ>0$. Here $(\cdot,\cdot)$ and $|\cdot|$ is correspondingly the $L^2$ inner product and the $L^2$ norm on $Ω$, and $A$ is the Stokes operator.

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