Forward self-similar solutions to the MHD equations: Existence and pointwise estimates

Yifan Yang

doi:10.3934/dcds.2025124

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2026, Volume 47: 406-456. Doi: 10.3934/dcds.2025124

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Forward self-similar solutions to the MHD equations: Existence and pointwise estimates

Yifan Yang

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, China

Received: March 2025

Early access: August 8, 2025

Published online: August 8, 2025

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Abstract

In this paper, we study the forward self-similar solutions to the three-dimensional Magnetohydrodynamic equations (MHD equations) in the whole space. By employing the Leray-Schauder theorem and blow-up argument, we construct a global-time forward self-similar solutions, which is smooth in $ \mathbb{R}^{3}\times(0,\infty) $. Furthermore, by investigating the regularity of the weak solutions to the corresponding Leray system in the weighted Sobolev space, we can derive the pointwise estimate for the forward self-similar solution.

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Mathematics Subject Classification: Primary: 76W05, 76B03; Secondary: 35A01, 35Q35.

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