Recall (11) in [KT01]: $$u(x,t) = S(t)u_0 - (V\grad\Pi N(u))(x,t),\hspace{6pt} N(u)=u\otimes u.$$ By definition, $S(t)u_0$ belongs to the space $X$ defined by (3) in [KT01], let $B(u,u)=-V\grad\Pi N(u)$, if we know $B: X\times X\rightarrow X$ is bilinear operator, then the following lemma gives us a solution to (11). The following formulation of the fixed point theorem is from [CP96]. **Lemma 9 (Fixed point theorem):** Let $X$ be a Banach space with norm $\norm{\cdot}$, and $B:X\times X\rightarrow X$ a bilinear operator such that for $x,y\in X$, we have $$\norm{B(x,y)} \leq \gamma \norm{x}\norm{y}.$$ Then for all $y\in X$ such that $\norm{y} < \f{1}{4\gamma}$, the sequence defined by $x^0=0$ and $x^{j+1}=y+B(x^j,x^j)$ converges in $X$, to a solution of $$x = y + B(x,x)$$ which is the only one such that $\norm{x}\leq \f{ 1- \sqrt{1 - 4\gamma\norm{y}}}{2\gamma}$. **//Proof://** WLOG, we assume $\gamma = 1$. Let $F(x)= y+B(x,x)$, then for all $x_1,x_2\in X$, we have $$\begin{split} \abs{ F(x_1) - F(x_2)} &\leq \abs{ B(x_1,x_1) - B(x_2,x_2)} ~\\ &\leq \abs{B(x_1, x_1-x_2) + B( x_1-x_2, x_2)} ~\\ &\leq ( \norm{x_1} + \norm{x_2} )\norm{x_1-x_2}. \end{split}$$ By definition of $x^j$, we have $\norm{x^{j+1}} \leq \norm{y} + \norm{x^j}^2$. A simple sequence lemma: Let $b_0=0;b_{j+1}=\alpha + b_j^2;\alpha>0$, then if $\alpha<1/4$, then $b_j$ converges and $b_j\rightarrow (1-\sqrt{1-4\alpha})/2$, $(j\rightarrow \infty)$ and $b_j<1/2$ for all $j$. Therefore, by comparing $\norm{x_j}$ and $b_j$, we can conclude that $\norm{x_j} < (1-\sqrt{1-4\alpha})/2$ for all $j$, where $\alpha=\norm{y}$. Hence, we now can find that $$\abs{ F(x^j) - F(x^k)} \leq (1-\sqrt{1-4\alpha})\norm{ x^j-x^k },\hspace{6pt}\forall\,j,k.$$ Then by contraction principle, we obtain $x^j\rightarrow x$ for some $x\in X$ such that $F(x)=x$. Uniqueness and norm estimate follow directly. === References [KT01] H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math. **157**, 22-35 (2001) [CP96] M. Cannone and F. Planchon. Self-similar solutions for Navier-Stokes equations in $\mathbb{R}^3$. Commun. Partial Differ. Equations **21**, 179–193 (1996).