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I've been working on this for months and can't find a good answer.

I'm looking at the incompressible Euler equations in cylindrical coordinates ($r$, $\theta$, $z$), and I am looking at the non linear terms' impact on the momentum conservation equation, notably on the regularity on the axis.

I am using the Fourier transform to describe the $\theta$ dependence of the velocity:

$\overrightarrow{u} = \sum\limits_{m=-N}^{N} \overrightarrow{u_m} e^{im\theta}$

Classically, the regularity of the solution in velocity is expressed with derivatives up to a certain order being null, or explicitly giving the $r$ dependence. If one considers the change of variable $u^+ = u^r + i u^\theta$ and $u^- = u^r - i u^\theta$, one can find some requirements for the regularity of the velocity. Different sources agree on the following properties to be imposed on each terms ($u^+_m$,$u^-_m$, $u^z_m$) of the velocity, for $m>1$:

$u^+_m = r^{m+1} \hat{u}^+_m$

$u^-_m = r^{m-1} \hat{u}^-_m$

$u^z_m = r^{m} \hat{u}^z_m$

However, if one injects this form of the velocity in the momentum conservation equation, one get, for the $z$ equation for example, dividing by $r^m$:

$\frac{\partial \hat{u}^z_m}{\partial t} + \sum\limits_{m_1,m_2\\ m_1 + m_2=m}\left(\frac{r \hat{u}^+_{m_1}}{2} \frac{\partial \hat{u}^z_{m_2}}{\partial r} + \frac{\hat{u}^-_{m_1}}{2r}\frac{\partial \hat{u}^z_{m_2}}{\partial r} + m_2 \hat{u}^+_{m_1}\hat{u}^z_{m_2} + \hat{u}^z_{m_1}\frac{\partial \hat{u}^z_{m_2}}{\partial z} \right)= -\frac{1}{r^m}\frac{\partial p}{\partial z} $

(The non linear terms look different normally with the negative $m_1$ or $m_2$ values, but the point I'm trying to make still holds.)

This equation is the problem I have. One can see that the terms imposed by the non linear terms are not as regular as $u^z$ should be. More precisely all the terms $\frac{\hat{u}^-_{m_1}}{2r}\frac{\partial \hat{u}^z_{m_2}}{\partial r}$ are singular on the axis. This problem is also present in the other equations of the momentum conservation equation. I have tried to improve this by adding some properties to the velocity, but since those properties also have to be verified, it does not help.

My questions are therefore:

-Is there any way to impose the regularity of the non linear terms on the axis ?

-Are there other conditions for the velocity on the axis that could improve this behavior ?

-Have I made any false assumptions or mistakes ?

Any thoughts and ideas are welcomed !

Edit: I have come across a paper (https://www.semanticscholar.org/paper/Formulation-of-a-Galerkin-spectral-element-fourier-Blackburn-Sherwin/b8fcada2f591251942b9ebc8f46f68b5855e7046) that advocates to put the problematic terms to zero. But the argumentation is not clear to me...

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