Pathological set with loss of regularity for nonlinear Schrödinger equations
Rémi Carles
Université de Rennes, Rennes, FranceLouise Gassot
Université de Rennes, Rennes, France
Abstract
We consider the mass-supercritical, defocusing, nonlinear Schrödinger equation. We prove loss of regularity in arbitrarily short times for regularized initial data belonging to a dense set of any fixed Sobolev space for which the nonlinearity is supercritical. The proof relies on the construction of initial data as a superposition of disjoint bubbles at different scales. We get an approximate solution with a time of existence bounded from below, provided by the compressible Euler equation, which enjoys zero speed of propagation. Introducing suitable renormalized modulated energy functionals, we prove spatially localized estimates which make it possible to obtain the loss of regularity.
Cite this article
Rémi Carles, Louise Gassot, Pathological set with loss of regularity for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré C Anal. Non Linéaire (2024), published online first
DOI 10.4171/AIHPC/126