I am looking into stability for certain nonlinear PDE on R around a specific steady solution, f0(x). The nonlinear Cauchy PDE is given by:
∂f(x,t)∂t=N(f(x,t)), where N is a nonlinear second order operator.
Now, I have been able to establish that f0(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 f0(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