Hadamard well-posedness and nonuniform dependence for the incompressible free-boundary ideal MHD equations
Abstract
We prove local Hadamard well-posedness for the three-dimensional incompressible free-boundary ideal magnetohydrodynamic equations without surface tension under the Taylor sign condition. For every 5/2<s<4, the result holds on bounded fluid domains whose closures are diffeomorphic to the closed ball, with the velocity, magnetic field, and boundary in the standard Hs spaces. No additional regularity in the magnetic direction is required. The proof rests on a fourth order geometric energy inequality whose coefficients are controlled by these Hs norms, the fixed geometric bounds, and the Taylor lower bound. A harmonic extension of the boundary normal provides the coefficient control needed at this regularity. We also prove that, near every admissible initial state and at each positive time in a common local existence interval, the Eulerian solution map is not uniformly continuous on any nonempty open subset of a sufficiently small data neighborhood. The construction combines localized magnetic oscillations prescribed at the observation time with time reversal and Galilean translations.
Related
- Local Hadamard well-posedness of the 2D primitive equations without thermal diffusion under strict stratification
- A well-posedness theory for the 2D inhomogeneous incompressible Euler equations and a class of global-in-time solutions
- Local well-posedness of the two-dimensional Incompressible Schrödinger flow
- Swirl ill-posedness for incompressible fluids in (sub)critical spaces
- Hadamard Local Well-Posedness for Compressible Liquids with a Free Surface
BibTeX
@misc{hao2026hadamard,
title = {{Hadamard well-posedness and nonuniform dependence for the incompressible free-boundary ideal MHD equations}},
author = {Chengchun Hao and Siqi Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26268},
eprint = {2610.26268},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26268},
}
RIS
TY - UNPB
TI - Hadamard well-posedness and nonuniform dependence for the incompressible free-boundary ideal MHD equations
AU - Chengchun Hao
AU - Siqi Yang
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26268
UR - https://arxiv.org/abs/2610.26268
AB - We prove local Hadamard well-posedness for the three-dimensional incompressible free-boundary ideal magnetohydrodynamic equations without surface tension under the Taylor sign condition. For every 5/2<s<4, the result holds on bounded fluid domains whose closures are diffeomorphic to the closed ball, with the velocity, magnetic field, and boundary in the standard H^s spaces. No additional regularity in the magnetic direction is required. The proof rests on a fourth order geometric energy inequality whose coefficients are controlled by these H^s norms, the fixed geometric bounds, and the Taylor lower bound. A harmonic extension of the boundary normal provides the coefficient control needed at this regularity. We also prove that, near every admissible initial state and at each positive time in a common local existence interval, the Eulerian solution map is not uniformly continuous on any nonempty open subset of a sufficiently small data neighborhood. The construction combines localized magnetic oscillations prescribed at the observation time with time reversal and Galilean translations.
ER -