Oscillations, amplitude, wavelengths in the context of nonperiodic solutions I'm not sure if I will be able to phrase my question so that it's not too vague but here goes.
I notice that in the PDE literature, say, when one talks about solutions to Navier Stokes, authors "heuristically" discuss situations where "viscosity dominates" or "transport dominates," etc etc. And the heuristic argument involves some scaling arguments talking about typical amplitudes, wavelengths, frequencies, etc of solutions to NSE. My question is, in such a context, what exactly is one talking about when discussing these properties which are (in the strict sense) properties of periodic functions and not general functions?
 A: It is common in the analysis of PDEs to consider the situation where the data or solution is "concentrated at a frequency scale." Such functions are also known as wave packets. For example, one might consider a function $\phi_N:\mathbb{R}^d\to\mathbb{C}$ which is defined by taking its Fourier transform to be
$$
\widehat{\phi_N}(\xi) = \eta(N^{-1}\xi),
$$
where $\eta:\mathbb{R}^d\to[0,1]$ is a smooth radially symmetric function which is $1$ on $[1,2]$ and $0$ when $|\xi|<\frac{9}{10}$ or $|\xi| > \frac{21}{10}$. Then $\widehat{\phi_N}(\xi)$ is supported at frequencies $\xi$ that are comparable in size to $N$, in the sense that there exist constants $C_1,C_2$ independent of $N$ such that $\xi\in\operatorname{supp}\widehat{\phi_N}$ if and only if $C_1 N \leq |\xi| \leq C_2N$. The corresponding function $\phi_N$ is then said to be a wavepacket at frequency scale $N$, or just frequency $N$ for short; equivalently it is a wavepacket at spatial scale $\frac{1}{N}$. If you are acquainted with Littlewood-Paley theory then this will be very familiar.
$\phi_N$ is not literally periodic, but it responds to calculus operations much like a periodic function with frequency $N$ would. For instance, differentiating $\phi_N$ will bring out a factor of $N$ for all purposes involving norm estimates, analogously to how $\sin(Nx)$ behaves under differentiation. $\phi_N$ should be understood as a function whose variations, or oscillations, occur on a spatial scale of $\frac{1}{N}$.
Note that these latter properties of $\phi_N$ are rather generic; they hold for any function whose Fourier transform is essentially concentrated at frequencies $\xi\sim N$. One might thus consider a solution to the PDE in question whose spatial dependence is of the form $\psi_N(x)$, where $\psi_N$ is some function concentrated at frequencies $\xi\sim N$, and whose time dependence is given by the amplitude $A(t)$. Thus, the solution is essentially of the form $A(t)\psi_N(x)$. Assuming that this description of the solution roughly holds at least in some time interval, one can try to use this description to understand the basic scaling properties of the PDE.
This type of analysis is entirely heuristic. In most PDE there is no reason for the solution to remain of the form $A(t)\psi_N(x)$, or for any such expression to be a literal solution to the PDE. As soon as there is a nonlinear effect, different frequency components of the solution will interact with each other and therefore we expect the spatial profile to evolve in time. However, in PDE with terms that are not outrageously singular, since $\psi_N$ is negligible in frequency space outside of its support, one could hope that these frequency interactions are negligible at least for a short time, and therefore there may be a timescale on which this description of the solution persists. This timescale is thus essentially the timescale on which nonlinear effects (in the form of frequency interactions) do not yet significantly impact the evolution. On this timescale one can pretend that the solution really is described by $A(t)\psi_N(x)$ in the sense that it scales in the same way, and use this to obtain a heuristic understanding of aspects of the local-in-time evolution.
Incidentally, this method of scaling analysis is widespread and not limited to Navier-Stokes; it comes up just about anytime one comes across a new evolution equation on $\mathbb{R}^d$ or $\mathbb{T}^d$ for the first time. In the case of $\mathbb{T}^d$, one still often prefers to work with these ensembles of frequencies (at a particular scale) rather than a single frequency, as frequency ensembles are more generic than single frequencies (and often single frequencies cannot even hope to be a solution).
