Let $\mathcal{H} = L^2(\mathbb{T}^3; \mathbb{R}^3)$ be the Hilbert space of $2\pi$-periodic, square-integrable, vector-valued function $\textbf{u}: \mathbb{T}^3 \to \mathbb{R}^3$, with the inner product$$\langle \textbf{u}, \textbf{v}\rangle = \int_{\mathbb{T}^3} \textbf{u}(\textbf{x}) \cdot \textbf{v}(\textbf{x})\,d\textbf{x}.$$We define subspaces $\mathcal{V}$ and $\mathcal{W}$ of $\mathcal{H}$ by\begin{align*} \mathcal{V} & = \{\textbf{v} \in C^\infty(\mathbb{T}^3; \mathbb{R}^3) \mid \nabla \cdot \textbf{v} = 0\},\\ \mathcal{W} & = \{\textbf{w} \in C^\infty(\mathbb{T}^3; \mathbb{R}^3) \mid \textbf{w} = \nabla \varphi \text{ for some }\varphi: \mathbb{T}^3 \to \mathbb{R}\}.\end{align*}Now, I can show that $\mathcal{H} = \mathcal{M} \oplus \mathcal{N}$ the orthogonal direct sum of $\mathcal{M} = \overline{\mathcal{V}}$ and $\mathcal{B} = \overline{\mathcal{W}}$.
Let $P$ be the orthogonal projection onto $\mathcal{M}$. The velocity $\textbf{v}(\textbf{x}, t) \in \mathbb{R}^3$ and pressure $p(\textbf{x}, t) \in \mathbb{R}$ of an incompressible, viscous fluid satisfy the equations$$\textbf{v}_t + \textbf{v} \cdot \nabla \textbf{v} + \nabla p = \nu \Delta \textbf{v}, \quad \nabla \cdot \textbf{v} = 0.$$Question. How do I see that the velocity $\textbf{v}$ satisfies the nonlocal equation$$\textbf{v}_t + P[\textbf{v} \cdot \nabla \textbf{v}] = \nu \Delta\textbf{v}?$$
Rearrange the equation and apply the projection on both sides:
$$v \cdot \nabla v = v\Delta v - v_{t} - \nabla p \Rightarrow P[v \cdot \nabla v] = P[v\Delta v - v_{t}] - P[\nabla p] = P[v\Delta v - v_{t}] $$
since $\nabla p \in \bar{W}$. For $v \in C^{\infty}(\mathbb{T}^{3}, \mathbb{R}^{3})$, it is not hard to see
$$\nabla \cdot (v\Delta v - v_{t}) = v\Delta(\nabla \cdot v) - (\nabla \cdot v)_{t} = 0 \Rightarrow v\Delta v - v_{t} \in \bar{V}$$
Thus,
$$P[v \cdot \nabla v] = v\Delta v - v_{t} \Rightarrow v_{t} + P[v \cdot \nabla v] = v\Delta v$$
Note that the explicit projection is called the Leray projection: https://en.wikipedia.org/wiki/Leray_projection