Fourier transform of high dimensional PDE

Question

Suppose that $$\partial_t f = \nabla_x \cdot (D\nabla_x f + Cxf),\quad x \in \mathbb{R}^d,$$ in which $D$ is a fixed positive semi-definite matrix and $C$ is a fixed matrix of dimension $d \times d.$ In many papers I saw that authors claimed that the Fourier transform $\hat{f}(\xi,t)$ will satisfy $$\partial_t \hat{f} = -\xi^{\intercal}D\xi \hat{f} – \xi^{\intercal}C\nabla_{\xi} \hat{f},$$ with no details of justifications. I believe the details should not be that easy because of the form of the PDE. So I hope someone can help with the computations. (I have no issue applying high-dimensional Fourier transform to the classical linear PDEs like heat/wave/Laplace equations, but here I will do not know how to start…) Thank you very much!

Answer 1

The first term is really a slightly souped-up version of the Laplacian, which you say you know how to work with when taking Fourier transforms. This is really nothing more than a certain case of the interchange of multiplication and differentiation that the Fourier transform gives us.

As to the question in your comment, the answer in the question you linked to has the Fourier transform written as "evaluated at a point," but this is just a matter of using dummy variables to distinguish the "spatial variables" $(x,y)$ and the "momentum variables" $(u,v)$. I don't think there is any particular mathematical significance to writing $\mathscr F(\partial^2_{xy}f)$ versus $\mathscr F(\partial^2_{xy}f)(u,v)$ except for the advantage of clarity in this case because $\mathscr F(\partial^2_{xy} f)(u,v) = -4\pi^2uv\mathscr F(f)(u,v)$, emphasizing where the $u$ and $v$ go.

In the example you have here, the analogous remark would be that it's a matter of your personal preference if you want to write $$ \partial_t \hat{f}(\xi,t) = -\xi^{\intercal}D\xi \hat{f}(\xi,t) - \xi^{\intercal}C\nabla_{\xi} \hat{f}(\xi,t) $$ or $$ \partial_t \hat{f} = -\xi^{\intercal}D\xi \hat{f} - \xi^{\intercal}C\nabla_{\xi} \hat{f}. $$