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Nonlinear stability for the 2D incompressible MHD system with fractional dissipation in the horizontal direction

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Abstract

This paper focuses on a 2D magnetohydrodynamic system with fractional horizontal dissipation in the domain \(\Omega = \mathbb T\times \mathbb R\) with \(\mathbb T=[0,1]\) being a periodic box. The goal here is to understand the stability problem on perturbations near any fixed magnetic field \(A=(A_1, A_2)\), where \(A_1, A_2 \in \mathbb {R}\). Due to the lack of vertical dissipation, this stability problem is difficult. This paper solves the desired stability problem by simultaneously exploiting two smoothing and stabilizing mechanisms: the enhanced dissipation due to the coupling between the velocity and the magnetic fields, and the strong Poincaré type inequalities for the oscillation part of the solution, namely the difference between the solution and its horizontal average. In addition, the oscillation part of the solution is shown to converge exponentially to zero in \(H^2\) as \(t\rightarrow \infty \). As a consequence, the solution converges to its horizontal average asymptotically.

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Acknowledgements

Feng was partially supported by an AMS-Simons Travel Grant. Wang was partially supported by an AMS-Simons Travel Grant. Wu was partially supported by the National Science Foundation of the United States (DMS 2104682) and the AT &T Foundation at Oklahoma State University. The authors would like thank the anonymous reviewers for their extremely helpful comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Niagara University, 5795 Lewiston Rd, Lewiston, NY, 14109, USA

    Wen Feng

  2. Department of Mathematics, University of Arizona, Tucson, AZ, 85721, USA

    Weinan Wang

  3. Department of Mathematics, Oklahoma State University, Stillwater, OK, 74078, USA

    Jiahong Wu

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  1. Wen Feng
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Feng, W., Wang, W. & Wu, J. Nonlinear stability for the 2D incompressible MHD system with fractional dissipation in the horizontal direction. J. Evol. Equ. 23, 32 (2023). https://doi.org/10.1007/s00028-023-00886-y

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