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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?

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

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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:

$$\mathbb{E}_t[u(x,t)] = \frac{1}{t_b - t_a}\int_{t_a}^{t_b} u(x,t)dt = U(x)$$

Sufficient (but not necessary) conditions for this are if both the time interval is finite and the function is bounded on that time interval.

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.

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  • $\begingroup$ +1 The average is defined over an ensemble (even in fluid dynamics), and equating the ensemble average to time average is an additional hypothesis, with a statistically steady state being a necessary condition for making that hypothesis. However ensemble average is applicable even to unsteady states and therefore is more general. So ensemble average (but not time average) is applicable to any system at all. $\endgroup$
    – Deep
    Jul 7, 2018 at 10:56

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