The Euler equations are given as $$ \pmb{u}\_t +\pmb{u}\cdot D\pmb{u} = Dp$$ $$div\mbox{ }\pmb{u} = 0$$


Where $$u = [u\_1,u\_2,\ldots u\_n]^T$$


Now I want to rewrite these same equations but with a new combination of variables $$\pmb{v} = [u\_1,u\_2,\ldots u\_n,p]^T $$


I want to do this just for the sake of getting some mathematical advantage and not for any physical inference. Is there any such formulation of equation already done? If not I'd like to know if it can be done? In this new combination, the equations will be only in terms of $\pmb{v}$ and some constants and no appearance of $p$ explicitly.



Writing Euler's equations in a different combination of variables? without explicit appearance of the variable $p$
The construction of adding $p$ as an additional element to the vector $\mathbf u$ only hides it in the vector $\mathbf v$, without actually eliminating it from the Euler equation. This can be easily done with a vector $\mathbf e=(0,0,\ldots 0,0,1)$, and a diagonal matrix $\Delta={\rm diag}\,(1,1,,\ldots 1,1,0)$ so that $\Delta\cdot\mathbf v=(\mathbf u,0)$ and $\mathbf e\cdot\mathbf v=p$, but such a construction seems rather pointless:  

$$\partial\_t (\Delta\cdot\mathbf v)+(\Delta\cdot\mathbf v)D(\Delta\cdot\mathbf v)=D(\mathbf e\cdot\mathbf v).$$


To really eliminate the pressure from the Euler equation you can use the [Leray projection](https://en.wikipedia.org/wiki/Leray_projection), which maps a vector field to its zero-divergence part: If $\mathbf v=\mathbf w+\nabla\phi$ with $\text{div}\,\mathbf w=0$, then the Leray projection is $\mathbb P(\mathbf v)=\mathbf w$. Formally, this construction can be written as
$$\mathbb P(\mathbf u) = \mathbf u - \nabla \Delta^{-1} (\nabla \cdot \mathbf u).$$
The Euler equation then reads
$$\frac{\partial}{\partial t}\mathbf u+\mathbb P[\mathbf u\cdot\nabla\mathbf u]=0.$$
More explicitly, this can be written as (see for example [page 6 of these notes](https://www.math.ucdavis.edu/~hunter/notes/euler.pdf)):
$$\frac{\partial}{\partial t}\mathbf u(\mathbf x,t)+\mathbf u(\mathbf x,t)\cdot\nabla\mathbf u(\mathbf x,t)=\frac{1}{C\_d}\int\_{\mathbb R^d}\frac{\mathbf x- \mathbf y}{|\mathbf x-\mathbf y|^d}\sum\_{i,j}\frac{\partial u\_i(\mathbf y,t)}{\partial y\_j}\frac{\partial u\_j(\mathbf y,t)}{\partial y\_i}\,d\mathbf y,$$
with $C\_d$ the surface area of the $d$-dimensional unit sphere.


