Long wavelength instability: Linear Vs nonlinear phenomenon

Question

I am looking into stability for certain nonlinear PDE on $\mathbb{R}$ around a specific steady solution, $f_0(x)$. The nonlinear Cauchy PDE is given by:

$\dfrac{\partial f(x,t)}{\partial t}=\mathbf{N}(f(x,t))$, where $\mathbf{N}$ is a nonlinear second order operator.

Now, I have been able to establish that $f_0(x)$ is exponentially linearly stable for all initial perturbations that have frequency greater than some constant $c$. This has been shown by converting the problem into Fourier domain, in which the **linearized PDE (around $f_0(x)$) decouples into (infinite) system of ODEs, one for each frequency.**

However, the decoupling doesn't hold for the nonlinear PDE. So an initial perturbation which is linearly stable may excite modes of frequency lower than $c$ due to coupling in the nonlinear equation. Hence, we cannot naively claim that "linear stability => nonlinear stability" in this case.

So I am looking for examples where such a situation has been studied. Probably in fluid mechanics or other physical phenomenon ? I am hoping to either prove or disprove the notion of nonlinear stability for the system I am dealing with.

This post imported from StackExchange MathOverflow at 2017-08-29 09:30 (UTC), posted by SE-user mystupid_acct

Comments

The answer would obviously depend on what $N$ is. You can easily cook up (finite) systems of ODEs with either behavior. In particular, what do you know about the linear stability of $f_0$ for frequency $< c$?

This post imported from StackExchange MathOverflow at 2017-08-29 09:30 (UTC), posted by SE-user Willie Wong

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@WillieWong The modes with frequency <c are exponentially unstable. I am basically looking for references where a PDE case has been explored under similar circumstances.

This post imported from StackExchange MathOverflow at 2017-08-29 09:30 (UTC), posted by SE-user mystupid_acct

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The best you can expect then is that there exists something like a center-stable-manifold. In this paper we studied a situation where there is one unstable mode. The higher frequencies are all in the continuous spectrum, and while we don't have exponential decay (linear stability), we have dispersion (which gives a weak form of stability). And we can prove codimension-1 stability in this setting.

This post imported from StackExchange MathOverflow at 2017-08-29 09:30 (UTC), posted by SE-user Willie Wong

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For the general theory, since you do have linearly stable frequencies, probably what you want to look at are the literature on the construction of invariant manifolds for infinite dimensional dynamical systems. S.N. Chow and Peter Bates are two names that come up a lot in that field, among others.

This post imported from StackExchange MathOverflow at 2017-08-29 09:30 (UTC), posted by SE-user Willie Wong

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Something like that was done for the Vlasov equation (multidimensional) by Mouhot and Villani, see "On Landau damping", Acta Mathematica, 2011, V. 207, Issue 1, pp 29–201 link.springer.com/article/10.1007/s11511-011-0068-9, or the same article in arXiv.

This post imported from StackExchange MathOverflow at 2017-08-29 09:30 (UTC), posted by SE-user Andrew

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Oh. I was hoping it wouldn't take fields medal level work to do what I want to do.

This post imported from StackExchange MathOverflow at 2017-08-29 09:30 (UTC), posted by SE-user mystupid_acct

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In the context of relativity I recommend to take a look at Christodoulou & Klainerman (1994): "The Global Nonlinear Stability of the Minkowski Space".

Answer 1

If there are some frequencies unstable for the linearized ODE, then the linear approximation is generally unstable, isn't it? Any infinitesimal (i.e., linear) initial perturbation contains all frequencies, so the low frequency part of the linear solution will give a growing linear solution. No non-linear coupling is necessary for that - the linear solution is already divergent. Correct me if I am wrong.