A regularity result for the incompressible magnetohydrodynamics equations with free surface boundary

Luo, Chenyun; Zhang, Junyan

doi:10.1088/1361-6544/ab60d9

Cited by 18 publications

(12 citation statements)

References 47 publications

(74 reference statements)

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“…Then the first author and Wang [25] proved the LWP. The second and the third authors [39] proved the minimal regularity H 5 2 +ε estimates for a small fluid domain. For the plasma-vacuum model under Rayleigh-Taylor sign condition, Hao [27] proved the a priori estimates when J = 0 and the first author [22,23] proved the LWP for axi-symmetric case.…”

Section: Review Of Previous Resultsmentioning

confidence: 93%

Local Well-posedness of the Free-Boundary Incompressible Magnetohydrodynamics with Surface Tension

Gu¹,

Luo²,

Zhang³

2021

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We prove the local well-posedness of the 3D free-boundary incompressible ideal magnetohydrodynamics (MHD) equations with surface tension, which describe the motion of a perfect conducting fluid in an electromagnetic field. We adapt the tangential smoothing method developed in [13] to generate an approximate problem with artificial viscosity indexed by κ > 0 whose solution converges to that of the MHD equations as κ → 0. This paper is the continuation of the second and third authors' previous work [40] in which the a priori energy estimate for incompressible free-boundary MHD with surface tension is established. However, the existence is not a trivial consequence of the a priori estimate as it cannot be adapted directly to the approximate problem.

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Section: Review Of Previous Resultsmentioning

confidence: 93%

Local Well-posedness of the Free-Boundary Incompressible Magnetohydrodynamics with Surface Tension

Gu¹,

Luo²,

Zhang³

2021

Preprint

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“…Hao-Luo [30] also proved the LWP for the linearized problem when the fluid region is diffeomorphic to a ball and of large curvature. Luo-Zhang [44] proved the low regularity a priori estimates when the fluid domain is small. We also mention that Lee [36,37] obtained a local solution via the vanishing viscosity-resistivity limit.…”

Section: Free-boundary Mhd Equations: Incompressible Casementioning

confidence: 99%

Anisotropic Regularity of the Free-Boundary Problem in Compressible Ideal Magnetohydrodynamics

Lindblad¹,

Zhang²

2021

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We consider 3D free-boundary compressible ideal magnetohydrodynamic (MHD) system under the Rayleigh-Taylor sign condition. It describes the motion of a free-surface perfect conducting fluid in an electro-magnetic field. The local wellposedness was recently proved by Trakhinin and Wang [66] by using Nash-Moser iteration. In this paper, we prove the a priori estimates without loss of regularity for the free-boundary compressible MHD system in Lagrangian coordinates in anisotropic Sobolev space, with more regularity tangential to the boundary than in the normal direction. It is based on modified Alinhac good unknowns, which take into account the covariance under the change of coordinates to avoid the derivative loss; full utilization of the cancellation structures of MHD system, to turn normal derivatives into tangential ones; and delicate analysis in anisotropic Sobolev spaces. Our method is also completely applicable to compressible Euler equations and thus yields an alternative estimate for compressible Euler equations without the analysis of div-curl decomposition or the wave equation in Lindblad-Luo [42], that do not work for compressible MHD. To the best of our knowledge, we establish the first result on the energy estimates without loss of regularity for the free-boundary problem of compressible ideal MHD. * in the second equation of (1.17), we getIn the energy estimates, we need to commute ∇ A with ∂ I * and then integrate by parts. However, the commutator [∂ I * , A li ]∂ l f contains the following terms whose L 2 (Ω)-norms cannot be controlled in the anisotropic Sobolev space), which cannot be controlled even in the standard Sobolev spaces when i 0 = 0;where f = Q or v i and I ′ is a multi-index with I ′ = 1. To overcome such difficulty, we can use the ideas of the Alinhac good unknown method, i.e., we can rewrite ∂ I * (∇ A Q) and ∂ I * (∇ A • v) in terms of the sum of the covariant derivative part and the commutator part satisfying(1.25)

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“…This is known to be the reference domain. Using a partition of unity, e.g., [7,26], a general domain can also be treated with the same tools we shall present. However, choosing Ω as above allows us to focus on the real issues of the problem without being distracted by the cumbersomeness of the partition of unity.…”

Section: Reformulation In Lagrangian Coordinatesmentioning

confidence: 99%

“…For the mathematical studies of free-boundary incompressible MHD system without surface tension, Hao-Luo [15,17] proved the a priori estimates and the linearized LWP by generalizing [4,23], and the first author and Wang [13] proved the LWP for the nonlinear problem. See also Lee [21,22] for an alternative proof by using the vanishing viscosity-resistivity method, Sun-Wang-Zhang [36] for the incompressible MHD current-vortex sheets, Sun-Wang-Zhang [37] and the first author [9,10] for the plasma-vacuum interface model, and the second and the third authors [26] for the minial regularity estimates in a small fluid domain. For the compressible ideal MHD, we refer to [2,38,41,30,39,25] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Zero Surface Tension Limit of the Free-Boundary Problem in Incompressible Magnetohydrodynamics

Luo

Zhang

2021

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We show that the solution of the free-boundary incompressible ideal magnetohydrodynamic (MHD) equations with surface tension converges to that of the free-boundary incompressible ideal MHD equations without surface tension given the Rayleigh-Taylor sign condition holds true initially. This result is a continuation of the authors' previous works [13,27,12]. Our proof is based on the combination of the techniques developed in our previous works [13,27,12], Alinhac good unknowns, and a crucial anti-symmetric structure on the boundary.

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A regularity result for the incompressible magnetohydrodynamics equations with free surface boundary

Cited by 18 publications

References 47 publications

Local Well-posedness of the Free-Boundary Incompressible Magnetohydrodynamics with Surface Tension

Local Well-posedness of the Free-Boundary Incompressible Magnetohydrodynamics with Surface Tension

Anisotropic Regularity of the Free-Boundary Problem in Compressible Ideal Magnetohydrodynamics

Zero Surface Tension Limit of the Free-Boundary Problem in Incompressible Magnetohydrodynamics

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