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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.

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  • $\begingroup$ 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 $< c$? $\endgroup$ – Willie Wong Aug 15 '17 at 14:26
  • $\begingroup$ @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. $\endgroup$ – mystupid_acct Aug 15 '17 at 14:48
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    $\begingroup$ 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. $\endgroup$ – Willie Wong Aug 16 '17 at 17:57
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    $\begingroup$ 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. $\endgroup$ – Willie Wong Aug 16 '17 at 18:06
  • $\begingroup$ 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. $\endgroup$ – Andrew Aug 18 '17 at 10:40
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One well-studied (and more-or-less canonical) example of long-wave instability/short-wave stability is the Kuramoto–Sivashinsky (KS) equation, which arises as an asymptotic description of flame fronts and thin films:

$$u_t + uu_x + u_{xx} + u_{xxxx} = 0$$

Wavenumbers $|k| < 1$ are linearly unstable (at $u=0$) as a result of the backward second-order diffusion, but wavenumbers $|k| > 1$ are stabilized by the forward forth-order diffusion.

The KS equation has inertial manifolds (with finite-dimensional dynamics in the limit $t\to \infty$) and complicated nonlinear dynamics, including spatiotemporal chaos.

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