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Ill-Posedness of the Euler Equations Linearized around Homogeneous Steady States
Abstract
Let L2m(ℝ2) be the space of square-integrable functions on ℝ2 with m-fold rotational symmetry. Let ω̅(r,θ) = r−αf(θ) be a homogeneous steady state of the two-dimensional incompressible Euler equations with (m· l)-fold rotational symmetry. If f(θ) is a constant function we say that ω̅ is a radial power-law vortex. We prove that the incompressible Euler equations in vorticity form, linearized around any homogeneous steady state ω̅ that is not a radial power-law vortex, are ill-posed on L2m(ℝ2) for any m≥ 2, any l≥ 1 and α∈ (0,1).
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BibTeX
@misc{coiculescu2026illposedness,
title = {{Ill-Posedness of the Euler Equations Linearized around Homogeneous Steady States}},
author = {Matei P. Coiculescu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06313},
eprint = {2609.06313},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06313},
}
RIS
TY - UNPB
TI - Ill-Posedness of the Euler Equations Linearized around Homogeneous Steady States
AU - Matei P. Coiculescu
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06313
UR - https://arxiv.org/abs/2609.06313
AB - Let L^2_m(ℝ^2) be the space of square-integrable functions on ℝ^2 with m-fold rotational symmetry. Let ω̅(r,θ) = r^−αf(θ) be a homogeneous steady state of the two-dimensional incompressible Euler equations with (m· l)-fold rotational symmetry. If f(θ) is a constant function we say that ω̅ is a radial power-law vortex. We prove that the incompressible Euler equations in vorticity form, linearized around any homogeneous steady state ω̅ that is not a radial power-law vortex, are ill-posed on L^2_m(ℝ^2) for any m≥ 2, any l≥ 1 and α∈ (0,1).
ER -