# Reynolds decomposition of non-linear dynamics

Can we apply the Reynolds decomposition, $$u(x,t)=U(x)+u'(x,t),$$ to any strongly non-linear dynamics problem, where the final state is dependent on the initial condition like Lorenz's equation or Burger's equation or Kurmoto-Sivashinsky equation? Even the non-linear dynamics problems have steady states, so can we apply the Reynolds decomposition to them?

Are there any laws which decides whether the Reynolds decomposition can be applied to any physical system or not?

• For a steady state, probably yes. The question is whether it'll often be as useful as in fluid dynamics. Jul 6, 2018 at 8:16

Assuming the standard definition of the Reynolds decomposition, you can always do it if the expectation value of $u(x,t)$ is well-defined over the time interval you care about. In other words, you can do it as long as this is well-defined:
$$\mathbb{E}_t[u(x,t)] = \frac{1}{t_b - t_a}\int_{t_a}^{t_b} u(x,t)dt = U(x)$$
A little bonus: if you think your function is square-integrable over that time interval, you can go further and create a generalized Fourier series in time defining each "moment" of your solution. In the best case, this might generate a series of ODEs describing a recurrence relation for each moment, letting you pull out a series solution that agrees with $u(x,t)$ almost everywhere.