Well-posedness for the Incompressible Hall-MHD Equations in Low Regularity Spaces

Wu, Xing; Yu, Yanghai; Tang, Yanbin

doi:10.1007/s00009-018-1096-x

Cited by 19 publications

(12 citation statements)

References 22 publications

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“…Compared with the previous local well-posedness results in Sobolev spaces (see e.g. [5,9,10,25]), our result lowers the regularity of the initial magnetic field B 0 to belonging to H 3 2 (R 3 ). Remark 1.2.…”

Section: Introductioncontrasting

confidence: 66%

“…Serrin type blow-up criteria for smooth solutions were proved by Chae and Lee in [6]. The local well-posedness of (1.1) in H s (R 3 ) × H s (R 3 ) with s > 3 2 was proved by Dai [9] and with 3 2 < s ≤ 5 2 was obtained in [25]. Very recently, Dai [10] proved that (1.1) is locally well-posed in H s−1+σ (R 3 ) × H s (R 3 ) with s > 3 2 and σ > 0.…”

Section: Introductionmentioning

confidence: 95%

“…Let us explain why we can tackle the magetic field B in H 3 2 (R 3 ). In [5,9,10,25], the authors derived the uniform bounds for approximate solutions by establishing the inequality of the type:…”

Section: Introductionmentioning

confidence: 99%

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Well-posedness for the incompressible Hall-MHD system with initial magnetic field belonging to $H^{\frac{3}{2}}(\mathbb{R}^3)$

Zhang¹

2021

Preprint

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In this paper, we first prove the local well-posedness of strong solutions to the incompressible Hall-MHD system with initial data (uIn particular, if the viscosity coefficient is equal to the resistivity coefficient, we can reduce σ to 0 with the aid of the new formulation of the Hall-MHD system observed by [11]. Moreover, we establish the global well-posedness in H 1 2 +σ (R 3 ) × H 3 2 (R 3 ) with σ ∈ (0, 2) for small initial data and get the optimal time-decay rates of solutions. Our results improves some previous works.

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Section: Introductioncontrasting

confidence: 66%

Section: Introductionmentioning

confidence: 95%

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Well-posedness for the incompressible Hall-MHD system with initial magnetic field belonging to $H^{\frac{3}{2}}(\mathbb{R}^3)$

Zhang¹

2021

Preprint

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“…Recently, there are many researches on the standard Hall-MHD equations with −∆u and −∆b , concerning global weak solutions [1], local and global (small) strong solutions [2,3], and the large time behavior of weak and strong solutions [4,5,6,7]. For the system (1.1), Chae, Wan and Wu [8] proved the local existence and uniqueness of the solution to the Hall-MHD equations with only a fractional Laplacian magnetic diffusion…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The continuous dependence for the Hall-MHD equations with fractional magnetic diffusion

Zhong

2018

Applicable Analysis

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“…In [2], it has been observed that the possible blow-up of smooth solutions may be controlled in terms of the velocity only. More well-posedness results of strong solutions for less regular data in Sobolev or Besov spaces, have been established in [5], [20] and [21], and the convergence to the MHD system with no Hall-term for h → 0 has been addressed in [24]. Examples of smooth data with arbitrarily large L ∞ norms giving rise to global unique solutions have been exhibited very recently in [15].…”

Section: Introductionmentioning

confidence: 99%

On the well-posedness of the Hall-magnetohydrodynamics system in critical spaces

Danchin

Tan

2020

Communications in Partial Differential Equations

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We investigate the existence and uniqueness issues of the 3D incompressible Hall-magnetohydrodynamic system supplemented with initial velocity u 0 and magnetic field B 0 in critical regularity spaces.In the case where u 0 , B 0 and the current J 0 := ∇ × B 0 belong to the homo-and are small enough, we establish a global result and the conservation of higher regularity. If the viscosity is equal to the magnetic resistivity, then we obtain the global well-posedness provided u 0 , B 0 and J 0 are small enough in the larger Besov spaceḂ 1 2 2,r , r ≥ 1. If r = 1, then we also establish the local existence for large data, and exhibit continuation criteria for solutions with critical regularity.Our results rely on an extended formulation of the Hall-MHD system, that has some similarities with the incompressible Navier-Stokes equations.

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Well-posedness for the Incompressible Hall-MHD Equations in Low Regularity Spaces

Cited by 19 publications

References 22 publications

Well-posedness for the incompressible Hall-MHD system with initial magnetic field belonging to $H^{\frac{3}{2}}(\mathbb{R}^3)$

Well-posedness for the incompressible Hall-MHD system with initial magnetic field belonging to $H^{\frac{3}{2}}(\mathbb{R}^3)$

The continuous dependence for the Hall-MHD equations with fractional magnetic diffusion

On the well-posedness of the Hall-magnetohydrodynamics system in critical spaces

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