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Singular evolution of Gaussian measures under energy-critical and supercritical nonlinear Schrödinger equations
Abstract
We consider a family of nonlinear Schrödinger equations with odd-power nonlinearities and fractional dispersion in an energy-critical/supercritical regime. For a family of Gaussian measures supported on high-regularity spaces, we show that the flow immediately evolves these measures to ones that are mutually singular with respect to their initial distributions. In the cubic case, the threshold for singularity exactly corresponds to the energy scaling for the equation. In particular, we establish singular evolution of Gaussian measures under the flow of the energy-critical cubic NLS on the 4-dimensional torus.
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BibTeX
@misc{coe2026singular,
title = {{Singular evolution of Gaussian measures under energy-critical and supercritical nonlinear Schrödinger equations}},
author = {James Coe},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03065},
eprint = {2611.03065},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03065},
}
RIS
TY - UNPB
TI - Singular evolution of Gaussian measures under energy-critical and supercritical nonlinear Schrödinger equations
AU - James Coe
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03065
UR - https://arxiv.org/abs/2611.03065
AB - We consider a family of nonlinear Schrödinger equations with odd-power nonlinearities and fractional dispersion in an energy-critical/supercritical regime. For a family of Gaussian measures supported on high-regularity spaces, we show that the flow immediately evolves these measures to ones that are mutually singular with respect to their initial distributions. In the cubic case, the threshold for singularity exactly corresponds to the energy scaling for the equation. In particular, we establish singular evolution of Gaussian measures under the flow of the energy-critical cubic NLS on the 4-dimensional torus.
ER -