Large-time behavior of solutions to the 2D generalized magneto-micropolar equations

Yana Guo; Chaoying Li; Yana Guo; Chaoying Li

doi:10.3934/era.2025333

Electronic Research Archive

2025, Volume 33, Issue 12: 7551-7569. doi: 10.3934/era.2025333

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Theory article

Large-time behavior of solutions to the 2D generalized magneto-micropolar equations

Yana Guo ¹,
Chaoying Li ^{2
,
,}

1.
School of Mathematics and Statistics, Shangqiu Normal University, Shangqiu 476000, China
2.
Aliyun School Big Data School, Changzhou University, Changzhou 213164, China

Received: 24 October 2025 Revised: 27 November 2025 Accepted: 05 December 2025 Published: 17 December 2025

This work provides rigorous verification of the super-smoothing effect of higher-order fractional Laplacian dissipation. Shang and Zhao (2017) have proved the global regularity of classical solutions of the 2D incompressible magneto-micropolar equations with linear velocity damping $ u $, microrotational dissipation $ (-\Delta)\omega $, and magnetic diffusion $ (-\Delta)^\beta b, \beta > 1 $. This paper is devoted to further investigating the large-time behavior of global smooth solutions of the system with $ 1 < \beta\leq\frac{3}{2} $. We apply the negative Sobolev space to overcome the difficulty caused by fractional-order dissipation and establish $ \int_{0}^{t}\|\nabla b(\tau)\|_{L^{2}}d\tau\leq C $. Furthermore, by fully exploiting the special structure of the system and combining the properties of a heat operator with the generalized Fourier splitting methods, we obtain the decay rates of the solutions and their first-order derivatives for $ 1\leq p\leq\frac{2}{\beta} $.
- magneto-micropolar equations,
- decay rates,
- fractional magnetic dissipation,
- linear velocity damping,
- Hardy-Littlewood-Sobolev inequality
Citation: Yana Guo, Chaoying Li. Large-time behavior of solutions to the 2D generalized magneto-micropolar equations[J]. Electronic Research Archive, 2025, 33(12): 7551-7569. doi: 10.3934/era.2025333

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Abstract

This work provides rigorous verification of the super-smoothing effect of higher-order fractional Laplacian dissipation. Shang and Zhao (2017) have proved the global regularity of classical solutions of the 2D incompressible magneto-micropolar equations with linear velocity damping $ u $, microrotational dissipation $ (-\Delta)\omega $, and magnetic diffusion $ (-\Delta)^\beta b, \beta > 1 $. This paper is devoted to further investigating the large-time behavior of global smooth solutions of the system with $ 1 < \beta\leq\frac{3}{2} $. We apply the negative Sobolev space to overcome the difficulty caused by fractional-order dissipation and establish $ \int_{0}^{t}\|\nabla b(\tau)\|_{L^{2}}d\tau\leq C $. Furthermore, by fully exploiting the special structure of the system and combining the properties of a heat operator with the generalized Fourier splitting methods, we obtain the decay rates of the solutions and their first-order derivatives for $ 1\leq p\leq\frac{2}{\beta} $.

References

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