On the splash singularity for the free-boundary problem of the viscous and non-resistive incompressible magnetohydrodynamic equations in 3D

Guangyi Hong; Tao Luo; Zhonghao Zhao

doi:10.1016/j.jde.2024.11.026

On the splash singularity for the free-boundary problem of the viscous and non-resistive incompressible magnetohydrodynamic equations in 3D

Guangyi Hong, Tao Luo, Zhonghao Zhao^*

^*Corresponding author for this work

Department of Mathematics

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

1 Citation (Scopus)

Abstract

In this paper, the existence of finite-time splash singularity is proved for the free-boundary problem of the viscous and non-resistive incompressible magnetohydrodynamic (MHD) equations in R³, based on a construction of a sequence of initial data alongside delicate estimates of the solutions. The result and analysis in this paper generalize those by Coutand and Shkoller in [14, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 2019] from the viscous surface waves to the viscous conducting fluids with magnetic effects for which non-trivial magnetic fields may present on the free boundary. The arguments in this paper also hold for any space dimension d ≥ 2. © 2024 Elsevier Inc.

Original language	English
Pages (from-to)	40-80
Journal	Journal of Differential Equations
Volume	419
Online published	28 Nov 2024
DOIs	https://doi.org/10.1016/j.jde.2024.11.026
Publication status	Published - 25 Feb 2025

Funding

The authors are deeply grateful to the referee for his/her careful reading of the manuscript and the numerous very helpful suggestions which have helped to clarify some important points and improve the presentation of the paper greatly. The work of Guangyi Hong was partially supported by the National Natural Science Foundation (grant Nos. 12201221 and 12371203), the Guangdong Basic and Applied Basic Research Foundation (grant Nos. 2021A1515111038 and 2024A1515012306), and the Guangzhou Municipal Science and Technology Project (grant No. 2024A04J3788). Luo's research is supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. 11307420).

Research Keywords

Finite-time singularity
Free-boundary problem
Interface singularity
Magnetohydrodynamic equations

RGC Funding Information

RGC-funded

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title = "On the splash singularity for the free-boundary problem of the viscous and non-resistive incompressible magnetohydrodynamic equations in 3D",

abstract = "In this paper, the existence of finite-time splash singularity is proved for the free-boundary problem of the viscous and non-resistive incompressible magnetohydrodynamic (MHD) equations in R3, based on a construction of a sequence of initial data alongside delicate estimates of the solutions. The result and analysis in this paper generalize those by Coutand and Shkoller in [14, Ann. Inst. H. Poincar{\'e} C Anal. Non Lin{\'e}aire, 2019] from the viscous surface waves to the viscous conducting fluids with magnetic effects for which non-trivial magnetic fields may present on the free boundary. The arguments in this paper also hold for any space dimension d ≥ 2. {\textcopyright} 2024 Elsevier Inc.",

keywords = "Finite-time singularity, Free-boundary problem, Interface singularity, Magnetohydrodynamic equations",

author = "Guangyi Hong and Tao Luo and Zhonghao Zhao",

year = "2025",

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doi = "10.1016/j.jde.2024.11.026",

language = "English",

volume = "419",

pages = "40--80",

journal = "Journal of Differential Equations",

issn = "0022-0396",

publisher = "Academic Press Inc.",

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TY - JOUR

T1 - On the splash singularity for the free-boundary problem of the viscous and non-resistive incompressible magnetohydrodynamic equations in 3D

AU - Hong, Guangyi

AU - Luo, Tao

AU - Zhao, Zhonghao

PY - 2025/2/25

Y1 - 2025/2/25

N2 - In this paper, the existence of finite-time splash singularity is proved for the free-boundary problem of the viscous and non-resistive incompressible magnetohydrodynamic (MHD) equations in R3, based on a construction of a sequence of initial data alongside delicate estimates of the solutions. The result and analysis in this paper generalize those by Coutand and Shkoller in [14, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 2019] from the viscous surface waves to the viscous conducting fluids with magnetic effects for which non-trivial magnetic fields may present on the free boundary. The arguments in this paper also hold for any space dimension d ≥ 2. © 2024 Elsevier Inc.

AB - In this paper, the existence of finite-time splash singularity is proved for the free-boundary problem of the viscous and non-resistive incompressible magnetohydrodynamic (MHD) equations in R3, based on a construction of a sequence of initial data alongside delicate estimates of the solutions. The result and analysis in this paper generalize those by Coutand and Shkoller in [14, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 2019] from the viscous surface waves to the viscous conducting fluids with magnetic effects for which non-trivial magnetic fields may present on the free boundary. The arguments in this paper also hold for any space dimension d ≥ 2. © 2024 Elsevier Inc.

KW - Finite-time singularity

KW - Free-boundary problem

KW - Interface singularity

KW - Magnetohydrodynamic equations

UR - http://www.scopus.com/inward/record.url?scp=85210141899&partnerID=8YFLogxK

UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-85210141899&origin=recordpage

U2 - 10.1016/j.jde.2024.11.026

DO - 10.1016/j.jde.2024.11.026

M3 - RGC 21 - Publication in refereed journal

SN - 0022-0396

VL - 419

SP - 40

EP - 80

JO - Journal of Differential Equations

JF - Journal of Differential Equations

ER -

On the splash singularity for the free-boundary problem of the viscous and non-resistive incompressible magnetohydrodynamic equations in 3D

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GRF: Long Time Well-Posedness and Dynamics for the Plasma-Vacuum Interface Problem in Magnetohydrodynamics (MHD) Nonlinear Partial Differential Equations.

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On the splash singularity for the free-boundary problem of the viscous and non-resistive incompressible magnetohydrodynamic equations in 3D

Abstract

Funding

Research Keywords

RGC Funding Information

Access Output

Other Access Options

Permanent Link

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GRF: Long Time Well-Posedness and Dynamics for the Plasma-Vacuum Interface Problem in Magnetohydrodynamics (MHD) Nonlinear Partial Differential Equations.

Cite this