PreprintCC BY 4.0arXiv
Existence and Regularity of Stable Resonant Spectral Submanifolds and Linearization Maps
Abstract
We prove the existence of Cr*,α-regular sub-stable invariant manifolds and conjugacies for analytic maps near a fixed point in the presence of resonances, assuming an invertible semisimple linearization. The regularity order r and the admissible Hölder exponents α depend on the resonant indices and the relative rates of contraction. Our approach uses functional expansions based on logarithmic polynomials, which allow the invariance equation to be solved explicitly and algorithmically to arbitrary order. The existence results are obtained via a fixed-point argument in a suitable Banach space. We further present several analytic examples that illustrate the theorem and demonstrate the loss of regularity caused by resonances.
Related
- Global Existence of classical solutions to 3D nonlinear Klein-Gordon equations with low-regularity initial data
- Existence of Weak Solutions and Higher-Order Regularity for a Heat-Wave Fluid-Structure Interaction System on a Periodic Strip
- Existence, Uniqueness, and Pathwise Regularity for Multidimensional Semilinear SPDEs with Locally Lipschitz Coefficients and Rough Initial Data
- Maximal Regularity and Existence for Superquadratic Parabolic Hamilton--Jacobi Equations: The Endpoint case
- Doubly critical mixed-order quasilinear equations: existence, multiplicity and regularity
BibTeX
@misc{kogelbauer2026existence,
title = {{Existence and Regularity of Stable Resonant Spectral Submanifolds and Linearization Maps}},
author = {Florian Kogelbauer and Rafael de la Llave},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.18236},
eprint = {2609.18236},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18236},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Existence and Regularity of Stable Resonant Spectral Submanifolds and Linearization Maps
AU - Florian Kogelbauer
AU - Rafael de la Llave
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2609.18236
UR - https://arxiv.org/abs/2609.18236
AB - We prove the existence of C^r_*,α-regular sub-stable invariant manifolds and conjugacies for analytic maps near a fixed point in the presence of resonances, assuming an invertible semisimple linearization. The regularity order r and the admissible Hölder exponents α depend on the resonant indices and the relative rates of contraction. Our approach uses functional expansions based on logarithmic polynomials, which allow the invariance equation to be solved explicitly and algorithmically to arbitrary order. The existence results are obtained via a fixed-point argument in a suitable Banach space. We further present several analytic examples that illustrate the theorem and demonstrate the loss of regularity caused by resonances.
ER -