What is the Fourier transform of $(u\cdot \nabla u)$, $u$ is two dimensional vector? Here is my attempt: \begin{array} &\widehat{(u\cdot \nabla u})&= \left(\begin{matrix} \widehat{u_1 \partial_1 u_1} +\widehat{u_2 \partial_2 u_1}\\ \widehat{u_1 \partial_1 u_2} +\widehat{u_2 \partial_2 u_2}\end{matrix}\right)\\ &= \left(\begin{matrix} i\xi_1 \int \widehat{u_1}(\xi-y) \widehat{u_1}(y)dy + i\xi_2 \int \widehat{u_2}(\xi-y) \widehat{u_2}(y)dy\\ i\xi_1 \int \widehat{u_1}(\xi-y) \widehat{u_2}(y)dy + i\xi_2 \int \widehat{u_2}(\xi-y)\widehat{u_2}(y)dy \end{matrix}\right)\\ &= \left(\begin{matrix} \widehat{u_1} \widehat{u_1} & \widehat{u_2} \widehat{u_2}\\ \widehat{u_1} \widehat{u_2} & \widehat{u_2} \widehat{u_2} \end{matrix}\right) \left(\begin{matrix} i\xi_1 \\ i\xi_2 \end{matrix}\right)\\ & = i\xi(\widehat{u\otimes u}) \end{array} In the second step, I have used the fact that the Fourier transform of the product of two functions is the convolution.
isn't it straightforward? Did I miss anything? Any help or source would be appreciated.\
Additionally, can we deal similarly with $(\widehat{u\cdot \nabla \theta})$, where $u$ is a vector and $\theta$ is a scalar quantity?
