How to get $\nabla p=\nabla (-\Delta)^{-1} \nabla \cdot (u\cdot \nabla u)$ from the Navier-Stokes equations?

Question

The incompressible Navier-Stokes equations are given by:

$$u_t + u \cdot \nabla u -\Delta u + \nabla p=0,\qquad \text{div } u=0,$$ where $u$ is the velocity field and $p$ is the pressure.

Here is a fact that I don't know why it's true from the above equations: $$\nabla p=\nabla (-\Delta)^{-1} \nabla \cdot (u\cdot \nabla u).$$

Thanks!

Answer 1

Formally, you have the so called pressure poisson equation: $$ -\Delta p=\nabla\cdot((u\cdot\nabla)u)\tag{1} $$

Now applying the inverse of $-\Delta$ on both sides, you get $$ p=(-\Delta)^{-1}\nabla\cdot((u\cdot\nabla)u) $$ and thus $$ \nabla p=\nabla(-\Delta)^{-1}\nabla\cdot((u\cdot\nabla)u) $$

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To see how to get (1), consider $$ \nabla\cdot(u_t+(u\cdot\nabla)u)=\nabla\cdot(\Delta u-\nabla p) $$ and use the divergence-free condition for $u$. On the LHS, you have $$ \partial_t(\nabla\cdot u)+\nabla\cdot((u\cdot\nabla)u)=0+\nabla\cdot((u\cdot\nabla)u) $$ On the RHS, you have $$ \Delta(\nabla\cdot u)-\Delta p=-\Delta p. $$