The "real butterfly effect"

There are some issues with the post on the website. I cannot understand how to pass form Navier-Stokes equations, which are PDEs, to some ODE system (Lorenz' equations?) which seem to be the subject of the discussion (also looking at the formulation of the question by Prof.Legolasov). Unfortunately I do not have access to the paper.

From a pure mathematical viewpoint everything is clear, so that the problem must concern the physical modeling: some breakdown of the mathematical description should take place at some level for physical reasons, but without reading the paper it is difficult to discuss it.

Here is a brief description of the mathematical scenario regarding the dependence on initial data.

Every (autonomous) ODE system as Lorenz' one can be written as $$\frac{dx}{dt} = F(x(t))\tag{1}$$ where $x\in M$, $M$ being some $C^k$ manifold and $F$ a $C^k$ vector field on $M$ with $k\geq 1$.

Now it is a standard result of ODE theory that, if $x=x(t|x_0)$ is the maximal solution of (1) with initial condition $x(0)= x_0\in M$, thus defined in an open interval $I_{x_0}\ni 0$, then

(1) the set $$D:= \left\{(t,x_0) \in \mathbb{R}\:\left|\: t \in I_{x_0}, x_0 \in M \right.\right\}$$ is open in $\mathbb{R}\times M$

(2) the map $$\Phi : D\ni (t,x_0) \mapsto \Phi_t(x_0):= x(t,x_0) \in M$$ is jointly $C^k$ (and $C^{k+1}$ in the variable $t$).

Hence, in the full domain of the solution there is a $C^k$ (hence continuous) dependence form initial data. In particular, when fixing some $T$, the map $$M_T \ni x \mapsto \Phi_T(x)\:,$$ where $M_T = \{x \in M \:|\: (x,T) \in D\}$, is necessarily continuous.

If instead we are dealing with proper PDEs, Navier-Stokes equations precisely, things are much more delicate and, as is very well known. Just the proof of an existence and uniqueness theorem for given initial data (which are now functions) is problematic. Continuous dependence form initial date is even more problematic.

Regarding Prof.Legolasov's suggestion I have some problems with it.

(a) Are we referring to Lorenz' ODE system? That is non Hamiltonian as the manifold has odd dimension, so what is $H$?

(b) Are we instead using some non-symplectic Poisson structure?

(c) Next, even referring to $L^2(M)$ (what measure?) over a Poisson manifold $M$, $$-i\{H, \cdot\} : C^\infty_c(M) \to L^2(M)$$ is symmetric for instance in $\mathbb{R}^{2n}$ referring to the natural Euclidean structure and the standard symplectic structure in orthonormal Cartesian coordinates, but it is not necessarily essentially selfadjoint (essentially selfadjointness of PDE on Riemannian manifolds is a delicate issue and general results are known for elliptic operators and $-i\{H, \cdot\}$ is not elliptic in general [what Riemannian metric in general when we are endowed with a Poisson structure only?]).

(d) Finally, even if we produce a unitary strongly continuous group generated by some selfadjoint extension of $-i\{H, \cdot\}$, I cannos see the relation with the continuous dependence from the initial data of the ODE associated to $H$.