I am looking for feedback on this attempt to prove a conserved quantity for incompressible Navier-Stokes
$$(\mathbf{u} \cdot \nabla) \mathbf{u} + \frac{\partial \mathbf{u}}{\partial t} = \nu \Delta \mathbf{u} - \nabla p$$ $$\nabla \cdot \mathbf{u}=0$$
where $\mathbf{u}=(u,v,w)$ No dimensional constants are given. They are not required to state initial conditions. There are no boundary conditions given. Initial conditions are $\mathbf{u}(x,y,z,0)=\mathbf{u}_0(x,y,z)$
The PDE has scaling invariance; it is preserved by the transformation
$$(x^\prime,y^\prime,z^\prime)=k(x,y,z)$$ $$(u^\prime,v^\prime,w^\prime)=\frac{1}{k}(u,v,w)$$ $$t^\prime=k^2t$$ $$p^\prime=\frac{1}{k^2}p$$
This is the finite transformation of the scaling group; it is a Lie group; its infinitesimal operator need not be listed here.
This means the solutions $\mathbf{u}(x,y,z,t)$ and $p(x,y,z,t)$ are homogeneous functions. Scaling is taken to mean "multiply the argument by the number $k$" and as a result "the function is multiplied by $k$ raised to a certain power". Then, the velocity is homogeneous in $x,y,z$ of degree 1 and homogeneous in $t$ of degree -1; the pressure is homogeneous in $x,y,z$ of degree 2 and homogeneous in $t$ of degree -2 ("pressure" is $p/\rho$, where $\rho=\mbox{const}$ is the fluid density)
To ensure the proper scaling properties of $\mathbf{u}$ and $p$, we have
$$x \frac{\partial \mathbf{u}}{\partial x} + y \frac{\partial \mathbf{u}}{\partial y}+z \frac{\partial \mathbf{u}}{\partial z}=\mathbf{u}$$ $$t \frac{\partial \mathbf{u}}{\partial t} = -\mathbf{u}$$ $$x \frac{\partial p}{\partial x}+y \frac{\partial p}{\partial y}+z \frac{\partial p}{\partial z}=2p$$ $$t \frac{\partial p}{\partial t} = -2p$$
This gives (use a sym algebra package)
$$\boxed{\mathbf{u}=\frac{x}{t+C}\mathbf{F}\left( \frac{y}{x}, \frac{z}{x} \right)}$$ $$\boxed{p=\frac{x^2}{(t+C)^2}F\left( \frac{y}{x}, \frac{z}{x} \right)}$$
Here the "$t+C$" comes from time-translational invariance of the PDE; $C=$const. $\mathbf{F}$ and $F$ are arbitrary functions.
Transform the solution under scaling:
$$\mathbf{u}^\prime=\frac{x^\prime}{t^\prime+C^\prime}\mathbf{F}\left( \frac{y^\prime}{x^\prime}, \frac{z^\prime}{x^\prime} \right)=\frac{kx}{k^2(t+C)}\mathbf{F}\left( \frac{ky}{kx}, \frac{kz}{kx} \right)=\frac{1}{k}\mathbf{u}$$
$$p^\prime=\frac{x^{\prime 2}}{(t^\prime+C^\prime)^2}F \left( \frac{y^\prime}{x^\prime}, \frac{z^\prime}{x^\prime} \right)=\frac{k^2x^2}{k^4(t+C)^2} F \left( \frac{ky}{kx}, \frac{kz}{kx} \right)=\frac{1}{k^2}p,$$
it transforms correctly as expected.
Because we see the separation of variables, quantities such as
$$\frac{u}{v-w}$$ $$\frac{w^2}{p}$$
etc. are conserved (and scale-invariant), because the time dependence cancels out.
Could you kindly provide feedback whether this claim is true. It would be very helpful if specific mistakes (if any) in the proof are pointed out.
Answers should be about the 3-dimensional equation and its solutions; and all variables should be clearly stated as well as their relationship with the Cartesian coordinates $x,y,z$ and the time $t$.
Also, all conditions stated in the problem must be followed, particularly about the initial, boundary conditions and any dimensional constants.