On Nonlinear Stabilization of Linearly Unstable Maps

Thierry Gallay (1) , Benjamin Texier (2) , Kevin Zumbrun (3)

1 IF - Institut Fourier
2 IMJ - Institut de Mathématiques de Jussieu
3 Mathematics Department

We examine the phenomenon of nonlinear stabilization, exhibiting a variety of related examples and counterexamples. For Gâteaux differentiable maps, we discuss a mechanism of nonlinear stabilization, in finite and infinite dimensions, which applies in particular to hyperbolic partial differential equations, and, for Fréchet differentiable maps with linearized operators that are normal, we give a sharp criterion for nonlinear exponential instability at the linear rate. These results highlight the fundamental open question whether Fréchet differentiability is sufficient for linear exponential instability to imply nonlinear exponential instability, at possibly slower rate.

Domaines: Equations aux dérivées partielles [math.AP]

Thierry Gallay, Benjamin Texier, Kevin Zumbrun. On Nonlinear Stabilization of Linearly Unstable Maps. Journal of Nonlinear Science, 2017, 27 (5), pp.1641-1666. ⟨hal-01987362⟩