Stability and large-time behavior on 3D incompressible MHD equations with partial dissipation near a background magnetic field

Lin, Huan; Wu, Jiahong; Zhu, Yan

doi:10.48550/arxiv.2210.16600

Cited by 1 publication

(8 citation statements)

References 23 publications

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“…This is mainly due to the divergence-free condition and the anisotropic structure. The results obtained improve the previous ones due to 25].…”

supporting

confidence: 73%

“…Remark 1.1. Theorem 1.1 improves the resuts obtained in [24,25]. In [24], Lin-Wu-Zhu studied the global stability under the smallness condition (u 0 , b 0 ) H 4 ≤ ε.…”

Section: Introductionsupporting

confidence: 58%

“…This is mainly due to the divergence-free condition and the anisotropic structure. The results obtained improve the previous ones due to 25].Recently, the stability of the incompressible MHD equations with partial dissipation has attracted more and more attention, since it mathematically reveals the stability mechanism of a background magnetic field, compared with the Navier-Stokes equations. For 2D MHD equations with partial dissipation, one can refer to [7, 13, 18-20, 22, 29] and the references therein for interest.…”

mentioning

confidence: 50%

“…Remark 1.2. The decay rates stated in Theorem 1.2 are sharper than those derived in [25]. Indeed, Lin-Wu-Zhu [25] showed that for any δ ∈ (0, 1),…”

Section: Introductionmentioning

confidence: 85%

“…In [24], Lin-Wu-Zhu studied the global stability under the smallness condition (u 0 , b 0 ) H 4 ≤ ε. This result was extended and improved by [25], where the following conditions were technically needed for the global H 3 stability:…”

Section: Introductionmentioning

confidence: 99%

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The Global Well-Posedness and Decay Estimates for the 3D Incompressible MHD Equations With Vertical Dissipation in a Strip

Lin

Suo

2023

International Mathematics Research Notices

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The three-dimensional incompressible magnetohydrodynamic (MHD) system with only vertical dissipation arises in the study of reconnecting plasmas. When the spatial domain is the whole space $\mathbb R^3$, the small data global well-posedness remains an extremely challenging open problem. The one-directional dissipation is simply not sufficient to control the nonlinearity in $\mathbb R^3$. This paper solves this open problem when the spatial domain is the strip $\Omega := \mathbb R^2\times [0,1]$ with Dirichlet boundary conditions. By invoking suitable Poincaré type inequalities and designing a multi-step scheme to separate the estimates of the horizontal and the vertical derivatives, we are able to establish the global well-posedness in the Sobolev setting $H^3$ as long as the initial horizontal derivatives are small. We impose no smallness condition on the vertical derivatives of the initial data. Furthermore, the $H^3$-norm of the solution is shown to decay exponentially in time. This exponential decay is surprising for a system with no horizontal dissipation. This large-time behavior reflects the smoothing and stabilizing phenomenon due to the interaction within the MHD system and with the boundary.

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“…This is mainly due to the divergence-free condition and the anisotropic structure. The results obtained improve the previous ones due to 25].…”

supporting

confidence: 73%

“…Remark 1.1. Theorem 1.1 improves the resuts obtained in [24,25]. In [24], Lin-Wu-Zhu studied the global stability under the smallness condition (u 0 , b 0 ) H 4 ≤ ε.…”

Section: Introductionsupporting

confidence: 58%

mentioning

confidence: 50%

“…Remark 1.2. The decay rates stated in Theorem 1.2 are sharper than those derived in [25]. Indeed, Lin-Wu-Zhu [25] showed that for any δ ∈ (0, 1),…”

Section: Introductionmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

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