Global solution of 3D axially symmetric nonhomogeneous incompressible MHD equations

Su, Wenhuo; Guo, Zhenhua; Yang, Ganshan

doi:10.1016/j.jde.2017.08.035

Cited by 5 publications

(2 citation statements)

References 21 publications

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“…Among these references, the work in [26] by He and Xin (and the work in [62] by Zhou, independently) should be noted, in which they established two well-known Ladyzhenskaya-Prodi-Serrin (LPS, for short) regularity criteria using only the information of the fluid velocity. Concerning the small perturbation solutions in the axially symmetric case, the 3D ideal resistive MHD equations were investigated by Lei [39] and the 3D non-homogeneous incompressible MHD equations were considered by Su et al [54]. For the 3D non-resistive viscous MHD fluids, it is evident that a strong enough magnetic field can effectively prevent the formation of singularities and even inhibit flow instabilities from occurring [30][31][32].…”

Section: Introductionmentioning

confidence: 99%

The horizontal magnetic primitive equations approximation of the anisotropic MHD equations in a thin 3D domain

Zhang,

Liu

2024

Nonlinearity

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In this paper, we give a rigorous justification for the derivation of the primitive equations with only horizontal viscosity and magnetic diffusivity (PEHM) as the small aspect ratio limit of the incompressible three-dimensional scaled horizontal viscous magnetohydrodynamics (SHMHD) equations. Choosing an aspect ratio parameter ε ∈ ( 0 , ∞ ) , we consider the case that if the orders of the horizontal and vertical viscous coefficients µ and ν are μ = O ( 1 ) and ν = O ( ε α ) , and the orders of magnetic diffusion coefficients κ and σ are κ = O ( 1 ) and σ = O ( ε α ) , with α > 2, then the limiting system is the PEHM as ɛ goes to zero. For H 1 -initial data, we prove that the global weak solutions of the SHMHD equations converge strongly to the local-in-time strong solutions of the PEHM, as ɛ tends to zero. For H 1 -initial data with additional regularity ( ∂ z A ~ 0 , ∂ z B ~ 0 ) ∈ L p ( Ω ) ( 2 < p < ∞ ) , we slightly improve the well-posed result in Cao et al (2017 J. Funct. Anal. 272 4606–41) to extend the local-in-time strong convergences to the global-in-time one. For H 2 -initial data, we show that the global-in-time strong solutions of the SHMHD equations converge strongly to the global-in-time strong solutions of the PEHM, as ɛ goes to zero. Moreover, the rate of convergence is of the order O ( ε γ / 2 ) , where γ = min { 2 , α − 2 } with α ∈ ( 2 , ∞ ) . It should be noted that in contrast to the case α > 2, the case α = 2 has been investigated by Du and Li in (2022 arXiv:2208.01985), in which they consider the primitive equations with magnetic field (PEM) and the rate of global-in-time convergences is of the order O ( ε ) .

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Section: Introductionmentioning

confidence: 99%

The horizontal magnetic primitive equations approximation of the anisotropic MHD equations in a thin 3D domain

Zhang,

Liu

2024

Nonlinearity

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“…For axially symmetric case of 3D incompressible MHD system in the whole space, Lei ([29]) proved the global well-posedness of axially symmetric solutions to the system in the cylindrical coordinates (r, z, t) when some of the components of solutions are zero. Recently, Su, Guo and Yang ( [43]) established a family of exact solutions with large initial data of finite energy, and without the restrictions that some components of the solutions were supposed to be zero. For 3D compressible MHD system in the cylindrical coordinates (r, t), Li, Li and Ou ( [30]) proved the global existence and uniqueness of axially symmetric classical solutions to the vacuum free boundary problem of the MHD system away from the symmetry axis with large initial data, moreover, they also showed the expanding rate of the free boundary.…”

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Global wellposedness of vacuum free boundary problem of isentropic compressible magnetohydrodynamic equations with axisymmetry

Li¹,

Ou²

2022

DCDS-B

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In this paper, we prove the global existence of the strong solutions to the vacuum free boundary problem of isentropic compressible magnetohydrodynamic equations with small initial data and axial symmetry, where the solutions are independent of the axial variable and the angular variable. The solutions capture the precise physical behavior that the sound speed is C 1/2 -Hölder continuous across the vacuum boundary provided that the adiabatic exponent γ ∈ (1, 2). The main difficulties of this problem lie in the singularity at the symmetry axis, the degeneracy of the system near the free boundary and the strong coupling of the magnetic field and the velocity. We overcome the obstacles by constructing some new weighted nonlinear functionals (involving both lower-order and higher-order derivatives) and establishing the uniformin-time weighted energy estimates of solutions by delicate analysis, in which the balance of pressure and self-gravitation, and the dissipation of velocity are crucial.

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Existence and Stability of Cylindrical Symmetric Static Solutions to the Landau–Lifshitz Equation for Multidirectional Ferromagnets

Zhang

Yang

2022

Appl Math Optim

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Global solution of 3D axially symmetric nonhomogeneous incompressible MHD equations

Cited by 5 publications

References 21 publications

The horizontal magnetic primitive equations approximation of the anisotropic MHD equations in a thin 3D domain

The horizontal magnetic primitive equations approximation of the anisotropic MHD equations in a thin 3D domain

Global wellposedness of vacuum free boundary problem of isentropic compressible magnetohydrodynamic equations with axisymmetry

Existence and Stability of Cylindrical Symmetric Static Solutions to the Landau–Lifshitz Equation for Multidirectional Ferromagnets

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