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arXiv · 2609.33241

Global well-posedness and Decay estimates for solutions of the Navier--Stokes--Maxwell system

Belkacem Said-Houari
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Abstract

We investigate the global solvability and the large-time asymptotic behavior of solutions to the Navier--Stokes--Maxwell system in two and three space dimensions. The system describes the interaction between a viscous incompressible fluid and an electromagnetic field through the Lorenz force and Ohm's law. The system exhibits a parabolic--hyperbolic coupling in which the dissipative Navier--Stokes system interacts with the hyperbolic Maxwell system. The Maxwell system is damped through Ohm's law with the Maxwell correction. For sufficiently small initial data with suitable Sobolev regularity, we establish the global well-posedness of strong solutions and derive optimal decay rates for the solution and its higher-order spatial derivatives. In addition, we show that the electric field decays faster by the extra factor $(1+t)^{-1/2}$ compared to the velocity and magnetic components. One of the key ingredients of our analysis is a detailed study of the linearized system, which reveals refined decay properties of the electromagnetic components. These linear estimates play a crucial role in the analysis of the two-dimensional case, allowing us to overcome the logarithmic loss arising from the borderline decay estimates. For the nonlinear system, our approach is based on a time-weighted energy method, specifically designed to capture the distinct dissipative properties of the fluid and electromagnetic components. A suitable compensating functional is introduced to exploit the coupling between the electric and magnetic fields and thereby recover the missing dissipation of the magnetic field. Combined with careful analysis of the nonlinear terms, these estimates yield our desired result.

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Belkacem Said-Houari. 2026-09-27. Global well-posedness and Decay estimates for solutions of the Navier--Stokes--Maxwell system. https://arxiv.org/abs/2609.33241

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