Question about the loss of hyperbolicity in nonlinear PDE: when complex eigenvalues appear, what is the effect on flow? I understand that we do not have general results on existence in this case, but is it only the mathematical tools that are lacking where can we show physical phenomena of instability?
There are hyperbolic and non-hyperbolic systems. At hyperbolic systems it is possible in some cases to make exponential stabilization. The non-hyperbolic systems are characterized with power law behavior. Applications are , for example, in plasma confinement at tokamaks for equilibrium and nonequilibrium case.
see this article
https://apps.dtic.mil/dtic/tr/fulltext/u2/a344449.pdf
I suppose you are considering some special cases where the PDE can have locally such an effect, for example as happens for the governing equation in transonic steady flows.
The appearence of complex eigenvalues means that there are no longer characteristic lines in the physical plane but only in the mathematical complex plane. This is the case of elliptic equations and the physics associated to such a mathematical character is generally of an equilibrium state, that is a perturbation in a point is propagated, istantaneoulsy, in isotropic way along all the directions.
- Badges
- Science topic
- Similar topics
- Mathematics