Let $I$-identity operator, $\Pi\_N$ is the orthogonal projection in $L\_2$ onto subspace by the first $N$ eigenfunctions of the Stokes operator in $\Omega$, $\alpha\_j$ denotes the increasing sequence of the eigenvalues for the Stokes operator, $c>0$ is the some constant not depending on $N$. We know that $\Pi\_N v = 0$.


How to derive the following inequality using the Poincare inequality? Which form of inequality is used here?
$|v|\_{L\_2}=|(I-\Pi\_N)v|\_{L\_2}\le c\alpha\_N^{-1/2}|v|\_{H^1}$.



Constant in Poincaré Inequality
This is a fairly standard stuff. Suppose that the Stokes operator $A=-\Delta$ is defined on smooth divergence-free vector fields $u$ which satisfy the standard no-slip boundary condition $$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.u\right|\_{\partial\Omega}=0.\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(1)$$
Let $H$ be the closure in $L^2$-norm of the space of smooth divergence-free vector fields with compact support:
$$H=\overline{\{u=(u\_1,\dots,u\_d)\in (C\_0^\infty(\Omega))^d:\ \mbox{div}\ u=0\}}^{L^2}$$ 


It is well known that $A$ gives rise to a self-adjoint operator with compact inverse on $H$ provided that $\Omega\subset \mathbb R^d$ is a bounded domain with Lipschitz boundary (see, e.g., Chapter 1 of [*Navier-Stokes Equations*](http://books.google.com.ua/books?id=NAF6W1oN9pcC&printsec=frontcover&dq=temam+navier+stokes+equations&source=bl&ots=CtHDY81gFQ&sig=_BjQkwIPRzUSgaM4jzzTF67ipLI&hl=en&ei=BIpxTeO0H4-WhQenzujPDg&sa=X&oi=book_result&ct=result&resnum=1&ved) by Temam). Moreover, one can show that $D(A^{1/2})\subset (H^1(\Omega))^d$ and that, for any $u\in D(A^{1/2})$,
$$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\|A^{1/2}u\|\_{L^2}^2=\|\nabla u\|\_{L^2}^2.\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad(2)$$


Now, let $\{e\_k\}\_{k\in\mathbb N}$ be the orthonormal basis in $H$ which consists of eigenvectors of $A$. The required estimate is implied by (2) and the trivial inequality
$$\|(I-\Pi\_N)v\|\_{L^2}^2=\sum\limits\_{k\geq N} |(v,e\_k)|^2\leq \frac{1}{\lambda\_N}\sum\limits\_{k\geq N}\lambda\_k |(v,e\_k)|^2=\frac{1}{\lambda\_N}\|A^{1/2}v\|^2\_{L^2}\qquad\qquad$$
which holds true for any $v\in D(A^{1/2})$ such that $\Pi\_N v =0$.




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The similar estimates remain valid if (1) is replaced with the Navier or periodic boundary conditions.


