bonvoyage36

It's not a good phrasing to express the point, because "solution is invariant under operation O" has an established meaning, that the solution does no change after the operation. What you mean can be properly phrased as "equations are time-reversal invariant".

I know what you meant; I've just tried to point out an error in your sentence which pops up sometimes, which may have mislead others. It's all about the time reversal invariance of evolution equations, not solutions.

In derivations of the Navier Stokes equations from reversible particle models, the former get their irreversibility from some approximation, e.g. a transition to a less detailed state and a simpler evolution equation for it is made. Often the actual microstate is replaced by some probabilistic description, such as probability density, or some kind of implied average.

equations can be time-symmetric, or invariant re time reversal. What you're describing is equations being invariant re time reversal.

Strictly speaking, naturally on its own, it doesn't. Detailed equations remain reversible. Even for very big N, typical isolated classical mechanical systems are reversible. However, typical initial conditions imply transitions to equilibrium, or very long stay in it. The reversed process (ending in Poincare return) will happen eventually, but the time is so incredibly long, it can't be verified.