Space-time decay rate for the 3D compressible quantum magnetohydrodynamic model

Siyi Luo; Yinghui Zhang; Siyi Luo; Yinghui Zhang

doi:10.3934/era.2025189

Electronic Research Archive

2025, Volume 33, Issue 7: 4184-4204. doi: 10.3934/era.2025189

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Research article

Space-time decay rate for the 3D compressible quantum magnetohydrodynamic model

Siyi Luo ,
Yinghui Zhang ^,

School of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, China

Received: 14 March 2025 Revised: 18 June 2025 Accepted: 01 July 2025 Published: 10 July 2025

This paper investigates the space-time decay properties of solutions to the three-dimensional compressible quantum magnetohydrodynamic (QMHD) model. By employing weighted Sobolev space techniques, we establish the optimal decay rate for the $ k $-th order spatial derivatives of solutions with $ k \in [0, 4] $, which concides with the heat equation. Specifically, we prove that the decay rate in the weighted space $ H^2_{\gamma}(\mathbb{R}^3) $ is given by $ t^{-\frac{3}{4} - \frac{k}{2} + \gamma} $ for the spatial derivatives of order $ k $. The key contribution lies in developing a unified framework that connects the weighted energy estimates with time decay analysis, which enables us to simultaneously capture both the spatial regularity and temporal decay characteristics of the solution. This result generalizes the previous decay estimates and provides a new description of the solution's asymptotic behavior in quantum magnetohydrodynamic systems.
- compressible vQMHD model,
- weighted estimates,
- space-time decay rates
Citation: Siyi Luo, Yinghui Zhang. Space-time decay rate for the 3D compressible quantum magnetohydrodynamic model[J]. Electronic Research Archive, 2025, 33(7): 4184-4204. doi: 10.3934/era.2025189

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Abstract

This paper investigates the space-time decay properties of solutions to the three-dimensional compressible quantum magnetohydrodynamic (QMHD) model. By employing weighted Sobolev space techniques, we establish the optimal decay rate for the $ k $-th order spatial derivatives of solutions with $ k \in [0, 4] $, which concides with the heat equation. Specifically, we prove that the decay rate in the weighted space $ H^2_{\gamma}(\mathbb{R}^3) $ is given by $ t^{-\frac{3}{4} - \frac{k}{2} + \gamma} $ for the spatial derivatives of order $ k $. The key contribution lies in developing a unified framework that connects the weighted energy estimates with time decay analysis, which enables us to simultaneously capture both the spatial regularity and temporal decay characteristics of the solution. This result generalizes the previous decay estimates and provides a new description of the solution's asymptotic behavior in quantum magnetohydrodynamic systems.

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