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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.

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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, 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): $$\frac{\partial}{\partial t}\mathbf u(\mathbf x,t)+\mathbf u(\mathbf x,t)\cdot\nabla\mathbf u(\mathbf x,t)=\frac{1}{C_d}\int_{\mathbb R^d}\frac{\mathbf x- \mathbf y}{|\mathbf x-\mathbf y|^d}\sum_{i,j}\frac{\partial u_i(\mathbf y,t)}{\partial y_j}\frac{\partial u_j(\mathbf y,t)}{\partial y_i}\,d\mathbf y,$$ with $C_d$ the surface area of the $d$-dimensional unit sphere.

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  • $\begingroup$ Request you to give me the equations in terms of $\pmb{e}$ and $\pmb{v}$ as you have mentioned that $p = \pmb{e}\cdot\pmb{v}$, eliminating both $\pmb{u}$ and $p$. $\endgroup$
    – user102868
    Commented Sep 20, 2020 at 2:55
  • $\begingroup$ What is $\Delta^{-1}$. Should I simplify $ \mathbb{P}[ \pmb{u}\cdot\nabla\pmb{u}]$ by substituting $\pmb{u}\cdot\nabla \pmb{u}$ in $\mathbb{P}(\pmb{u}) = \pmb{u} - \nabla\Delta^{-1}(\nabla\cdot\pmb{u})$? I dont want $\mathbb{P}$ appearing in my final equation. $\endgroup$
    – user102868
    Commented Sep 20, 2020 at 3:16
  • $\begingroup$ Or is there any direct formula or Fourier multiplier for $\mathbb{P}[\pmb{u}\cdot \nabla \pmb{u}]$? $\endgroup$
    – user102868
    Commented Sep 20, 2020 at 3:28
  • $\begingroup$ I added the requested equation that does not contain $p$ explicitly, but really, that is just hiding the pressure in the notation; the Leray projection is the way to go if you want to truly eliminate the pressure. $\endgroup$
    – Carlo Beenakker
    Commented Sep 20, 2020 at 7:11
  • $\begingroup$ Thank you very much for the help with that equation @CarloBeenakker. In your method, you have a term that is a Leray projection of a vector field, but how will you do furthur analysis without explicitly having a expression for the Leray projection in terms of $\pmb{u}$. Just utilize the fact that the divergence of the Leray projection is 0 and thats it? there has to be some way to express it in terms of $\pmb{u}$ ? $\endgroup$
    – user102868
    Commented Sep 20, 2020 at 7:36

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