The inf-sup condition is one of the essential tools in the analysis of the Stokes equations and especially in numerical analysis. In its usual form, the condition states that for every pressure $p\in L^2(\Omega)\setminus \mathbb{R}$, (i.e. with mean value zero) a velocity $u\in H^1_0(\Omega)^d$ can be found, so that $(div\,u,p)=\|p\|^2$ and $\|\nabla u\|\le c \|p\|$ applies, where $c>0$ does not depend on $u$ and $p$. However, if we consider domains that have a Neumann-type outflow condition on part of the boundary $\Gamma_N\subset\partial\Omega$, the inf-sup condition cannot be used in this form, since the pressure here comes from $L^2(\Omega)$ and does not necessarily have zero mean value. In this note, we derive the inf-sup condition for the case of outflow boundaries.

翻译：inf-sup条件是分析Stokes方程，特别是数值分析中的基本工具之一。在其通常形式中，该条件表明：对于每个压力$p\in L^2(\Omega)\setminus \mathbb{R}$（即均值为零），可以找到一个速度$u\in H^1_0(\Omega)^d$，使得$(div\,u,p)=\|p\|^2$且$\|\nabla u\|\le c \|p\|$成立，其中$c>0$与$u$和$p$无关。然而，若考虑边界部分$\Gamma_N\subset\partial\Omega$上具有Neumann型外流条件的区域，则inf-sup条件无法以此形式使用，因为此处的压力来自$L^2(\Omega)$且不一定具有零均值。本文推导了外流边界情形下的inf-sup条件。