Optimal Time-Decay of global solutions to the Navier-Stokes-Maxwell system
Abstract
We prove the existence and uniqueness of global-in-time solutions to the Navier--Stokes--Maxwell (NSM) system for small initial data in the critical Fujita--Kato space Ḣd/2−1(ℝd), in any dimension d≥ 3. Under the additional assumption that the initial data belong to the Besov space Ḃ−d/22,∞(ℝd), we also establish the optimal decay rate t−s/2−d/4 at infinity of the solution in Ḣs(ℝd), for any s∈(−d/2,d/2−1]. This is achieved by constructing a Lyapunov functional that is equivalent to the pointwise energy on the Fourier side, and by combining it with an adaptation of the Fourier splitting method to establish its optimal decay rate. All the analysis carried out here - from the global existence theory to the study of the large-time behavior - is performed within a framework that is uniform with respect to the speed of light c∈(0,∞). In particular, in the non-relativistic limit c→∞, this allows us to recover the same results for the corresponding limiting magnetohydrodynamic system.
Related
- Global finite-energy weak solutions to the Navier--Stokes--Maxwell in 1D
- Global well-posedness and Decay estimates for solutions of the Navier--Stokes--Maxwell system
- Global weak solutions with vacuum for isentropic compressible Navier-Stokes and Navier-Stokes-Poisson equations in the unit disk under Dirichlet boundary conditions
- Global existence of smooth solutions to the Navier-Stokes equations for initial data with infinite energy
- Global weak solutions of the Navier-Stokes-Korteweg equations in one dimension
BibTeX
@misc{houamed2026optimal,
title = {{Optimal Time-Decay of global solutions to the Navier-Stokes-Maxwell system}},
author = {Haroune Houamed and Slim Ibrahim and Belkacem Said-Houari},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15095},
eprint = {2609.15095},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15095},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal Time-Decay of global solutions to the Navier-Stokes-Maxwell system
AU - Haroune Houamed
AU - Slim Ibrahim
AU - Belkacem Said-Houari
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15095
UR - https://arxiv.org/abs/2609.15095
AB - We prove the existence and uniqueness of global-in-time solutions to the Navier--Stokes--Maxwell (NSM) system for small initial data in the critical Fujita--Kato space Ḣ^d/2−1(ℝ^d), in any dimension d≥ 3. Under the additional assumption that the initial data belong to the Besov space Ḃ^−d/2_2,∞(ℝ^d), we also establish the optimal decay rate t^−s/2−d/4 at infinity of the solution in Ḣ^s(ℝ^d), for any s∈(−d/2,d/2−1]. This is achieved by constructing a Lyapunov functional that is equivalent to the pointwise energy on the Fourier side, and by combining it with an adaptation of the Fourier splitting method to establish its optimal decay rate. All the analysis carried out here - from the global existence theory to the study of the large-time behavior - is performed within a framework that is uniform with respect to the speed of light c∈(0,∞). In particular, in the non-relativistic limit c→∞, this allows us to recover the same results for the corresponding limiting magnetohydrodynamic system.
ER -