Exercise 7.6 ( page 152 and page 157) In the book The Three-Dimensional Navier-Stokes Equations, James C. Robinson, Jose L. Rorigo, Witold Sadowski. Cambridge University press:
" Let the domain be bounded and smooth in the space, and $\frac{1}{2}\frac{d}{dt}\|A^{s/2}u_n\|_2^2+\|A^{(s+1)/2}u_n\|_2^2\le|\langle P_n[(u_n\cdot\nabla )u_n)], A^su_n\rangle|$. Prove that
$\frac{d}{dt}\|A^{s/2}u_n\|_2^2+\|A^{(s+1)/2}u_n\|_2^2\le c\|u_n\|^2_{H^s}\|A^{s/2}u_n\|_2^2$"
Please help me to prove this exercise.
I try to perform three following steps, but I am not successful at Step 2:
Step 1: $|\langle P_n[(u_n\cdot\nabla )u_n)], A^su_n\rangle|=|\langle A^{s/2}P_n[(u_n\cdot\nabla )u_n)], A^{s/2}u_n\rangle|$
Step 2: $|\langle A^{s/2}P_n[(u_n\cdot\nabla )u_n)], A^{s/2}u_n\rangle|\le c\| A^{s/2}(u_n\cdot\nabla )u_n)\|_2\|A^{s/2}u_n\|_2 $
Step 3 ( using Young inequality): $\| A^{s/2}(u_n\cdot\nabla )u_n)\|_2\|A^{s/2}u_n\|_2\le c\|u_n\|^2_{H^s}\|A^{s/2}u_n\|_2^2+ \|A^{(s+1)/2}u_n\|_2^2$
However, I can not prove Step 2$.
Thank you very much.
Your step 2 won't work because $A$ and $P_n$ don't commute: you have to distinguish between the case $s=2k$ or $s=2k+1$, $k \in \mathbb{N}$.
For instance, if $s=2k$, you shall write by Cauchy-Schwarz
$$|\langle A^{s/2}P_n[(u_n\cdot\nabla )u_n)], A^{s/2}u_n\rangle|\le \| A^{s/2} P_n(u_n\cdot\nabla )u_n)\|_2\|A^{s/2}u_n\|_2,$$
so that
$$|\langle A^{s/2}P_n[(u_n\cdot\nabla )u_n)], A^{s/2}u_n\rangle|\le c\| P_n(u_n\cdot\nabla )u_n)\|_{H^s}\|A^{s/2}u_n\|_2,$$
by using standard elliptic regularity estimates for the Stokes operator $A$.