I am currently studuying the Navier-Stokes equation, more precisely, The Leray's Theorem (1933) that gives existence of weak solutions. In order to study it, our teacher gave us a problem to solve and in it there is a question about the Leray's Projector :
$$ P: u \in L^2(R^d) \mapsto P(u) \in K$$
where $ K = \{ u \in L^2(R^d) : \operatorname{div}(u) = 0 \text{ in } \mathcal{D}'(R^d)\}$
My problem is that, even if I think I have understood the Distribution Theory for scalar valued-functions, I don't know of we make the similar generalization from vector-valued functions. Can someone try to explain me that ? And how we then define the Divergence and other differential Operators ?
Thank you very much.
A $\mathbb{R}^d$-valued distribution $u$ can be treated as $d$ scalar-valued distributions, $u_1, \ldots, u_d \in \mathcal{D}'(\mathbb{R}^d)$.
The divergence is then simply $\operatorname{div}u = \nabla\cdot u = \partial_1 u_1 + \cdots + \partial_d u_d \in \mathcal{D}'$.