1)I want to perform Fourier transform w.r.t space for Navier stokes Equation in $\mathbb{R}^3$, but I have difficulty dealing with the nonlinear term $u\cdot\nabla u$. Can anyone help with that?
2)Tao performs the Fourier transform in $\mathbb{R}^3/\mathbb{Z}^3$, but I couldn't see why it is like that
The Fourier transform over the torus finds the coefficients $\mathcal{T}[u](k) = \hat{u}(k)$ in the Fourier series
$$u(x) = \sum_{k \in \mathbb{Z}^3}\hat{u}(k) e^{-2\pi i k \cdot x}$$
The transform of a product of terms like $u_1 \frac{\partial u_2}{\partial x_1}$ is a convolution of the transforms, that is
$$\mathcal{T}\left[u_1 \frac{\partial u_2}{\partial x_1}\right](k)= \sum_{n \in \mathbb{Z}^3}[\hat{u}_1(k-n)i]\cdot n_1\hat{u}_2(n), $$ where we used the fact that $\mathcal{T}\left(\partial u_2/\partial x_1\right)(k)=ik_1\hat{u}_2(k)$, here $k=(k_1,k_2,k_3)$.