I have following sets of PDES:
$\frac{\partial \omega_1'}{\partial t}+u_2'\frac{\partial \omega_1'}{\partial x_2}+u_3'\frac{\partial \omega_1'}{\partial x_3}= \nu(\frac{\partial^2 \omega_1'}{\partial x_2^2}+\frac{\partial^2 \omega_1'}{\partial x_3^2})$
I want to assume a Beltrami flow, which basically neglects the non-linear terms in this PDE to zero, hence the PDE simplifies to:
$\frac{\partial \omega_1'}{\partial t}=\nu(\frac{\partial^2 \omega_1'}{\partial x_2^2}+\frac{\partial^2 \omega_1'}{\partial x_3^2})$
The functions $u_2'$ and $u_3$' can be obtained as:
$\frac{\partial \omega_1'}{\partial x_2}=\frac{\partial^2 u_3'}{ \partial x_2^2}+\frac{\partial^2 u_3'}{\partial x_3^2} \\ \frac{\partial \omega_1'}{\partial x_3}=-\frac{\partial^2 u_2'}{ \partial x_2^2}-\frac{\partial^2 u_2'}{\partial x_3^2}$.
I found the solution for $\omega_1'$ using the separation of variables as:
$\omega_1'=\sum_{j=0}^{p}\sum_{k=0}^{n}(C_{1,k} cos(\sqrt{a_k}x_2)+C_{2,k} sin(\sqrt{a_k}x_2))(C_{3,j} cos(\sqrt{b_j}x_3)+C_{4,j} sin(\sqrt{b_j}x_3))e^{-\nu(a_k+b_j)t}$ Thus $u_2'$ and $u_3'$ can be easily found using the above-mentioned relation. However in order to fulfill the boundary conditions as
$u_2'(x_2=0)=0 \\ u_2'(x_2=L_2)=0\\ u_3'(x_2=0)=0\\ u_3'(x_2=L_2)=0$ I need to look at the whole sum of the solution of $\omega_1'$, therefore adding each soltuin for each $a_k$, thus superpositioning the solutions. Each solution individually solves
$u_2'\frac{\partial \omega_1'}{\partial x_2}+u_3'\frac{\partial \omega_1'}{\partial x_3}=0$.
However once I take a sum of each solution this does not work any more.
The question therefore is, weather there is a technique for THIS non linear PDE to combine single solutions correctly?
I would highly appreciate any kind of help.
I know there are attempts to project superposition principles for non-linear PDE systems, although the cases in which these work are very special. I am no expert in this field, but I may refer you to the following article, which deals with non-linear superposition in terms of certain equations (Navier-Stokes is included!): https://ieeexplore.ieee.org/document/4342629
In any case, I hope experts can help you out, your approach surely is very exciting and it's nice to see people deal with special solutions of Navier-Stokes!