Charge-free well-posedness and charge-driven norm inflation for the Euler-Maxwell system
Abstract
We study the two-dimensional incompressible Euler-Maxwell system under the normal geometry in which the velocity and electric field are planar and the magnetic field is normal to the plane. We identify a structural distinction between the projected, charge-free formulation and the unprojected system. For the projected system, we establish a local well-posedness theory for initial velocities in H1(ℝ2) with bounded initial vorticity and electromagnetic data in H3/2(ℝ2), under a strict sub-luminal condition on the velocity. The main ingredient is an endpoint time-like trace estimate for half-waves, which controls the magnetic gradient in Lt2 along every fluid trajectory. This closes the Yudovich vorticity estimate without requiring an Eulerian Lipschitz bound on the electromagnetic field. For the unprojected system, the longitudinal electric mode produces an additional vorticity source term containing the charge density div E. In this case, we construct smooth, compactly supported radial electromagnetic data converging to zero in H3/2(ℝ2), with zero initial velocity, whose unique global smooth solution exhibits a vorticity norm inflation in L∞x. Thus, at the same electromagnetic Sobolev regularity, the Gauss constraint separates endpoint regularity propagation from charge-driven vorticity norm inflation.
Related
- Blow-up, critical norm inflation, and global well-posedness for the Geng--Xue system
- Global well-posedness and Decay estimates for solutions of the Navier--Stokes--Maxwell system
- The Vlasov-Maxwell point charge system : study of the spherically symmetric case
- Global well-posedness and Newtonian limit of the compressible upper-convected Maxwell model in analytic spaces
- Well-posedness, regularity and stability for linear Maxwell's equations with interface damping
BibTeX
@misc{houamed2026chargefree,
title = {{Charge-free well-posedness and charge-driven norm inflation for the Euler-Maxwell system}},
author = {Haroune Houamed},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24481},
eprint = {2609.24481},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24481},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Charge-free well-posedness and charge-driven norm inflation for the Euler-Maxwell system
AU - Haroune Houamed
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24481
UR - https://arxiv.org/abs/2609.24481
AB - We study the two-dimensional incompressible Euler-Maxwell system under the normal geometry in which the velocity and electric field are planar and the magnetic field is normal to the plane. We identify a structural distinction between the projected, charge-free formulation and the unprojected system. For the projected system, we establish a local well-posedness theory for initial velocities in H^1(ℝ^2) with bounded initial vorticity and electromagnetic data in H^3/2(ℝ^2), under a strict sub-luminal condition on the velocity. The main ingredient is an endpoint time-like trace estimate for half-waves, which controls the magnetic gradient in L_t^2 along every fluid trajectory. This closes the Yudovich vorticity estimate without requiring an Eulerian Lipschitz bound on the electromagnetic field. For the unprojected system, the longitudinal electric mode produces an additional vorticity source term containing the charge density div E. In this case, we construct smooth, compactly supported radial electromagnetic data converging to zero in H^3/2(ℝ^2), with zero initial velocity, whose unique global smooth solution exhibits a vorticity norm inflation in L^∞_x. Thus, at the same electromagnetic Sobolev regularity, the Gauss constraint separates endpoint regularity propagation from charge-driven vorticity norm inflation.
ER -