On the temporal decay for the Hall-magnetohydrodynamic equations
Dongho Chae, Maria Schonbek
我们建立了霍尔磁流体动力学方程的弱解的时间衰减估计。有了这些估计在手，我们得到代数时间衰减高阶Sobolev规范的小初始数据的解决方案。
We establish temporal decay estimates for weak solutions to the Hall-magnetohydrodynamic equations. With these estimates in hand we obtain algebraic time decay for higher order Sobolev norms of small initial data solutions.
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Regularity of the $3D$ stationary Hall magnetohydrodynamic equations on the plane
在$ \Bbb R^2$上研究了三维定态Hall磁流体动力学方程组弱解的正则性。证明了每个弱解都是光滑的。此外，我们证明了霍尔方程的Liouville型定理。
On partial regularity for the $3D$ non-stationary Hall magnetohydrodynamics equations on the plane
在$ \Bbb R^2$上研究了三维非定常Hall磁流体力学方程组弱解的部分正则性。特别是我们证明了一个弱解的存在性，其可能的奇点集的时空豪斯多夫维数至多为2。
In this paper, we establish the global existence to the three-dimensional incompressible Hall-MHD equations for a class of large initial data, whose $L^{\infty}$ norms can be arbitrarily large.
In this paper, the global smooth solution of Cauchy's problem of incompressible, resistive, viscous Hall-magnetohydrodynamics (Hall-MHD) is studied. By exploring the nonlinear structure of Hall-MHD equations, a class of large initial data is constructed, which can be arbitrarily large in $H^3(\mathbb{R}^3)$. Our result may also be considered as the extension of work of Lei-Lin-Zhou from the second-order semi-linear equations to the second-order quasilinear equations, because the Hall term elevates the Hall-MHD system to the quasilinear level.
We prove the global well-posedness of the Cauchy problem to the 3D incompressible Hall-magnetohydrodynamic system supplemented with initial data in critical Besov spaces, which generalize the result in [10]. Meanwhile , we analyze the long-time behavior of the solutions and get some decay estimates. Finally, a stability theorem for global solutions is established.
On partial regularity for the steady Hall magnetohydrodynamics system
研究了区域$Ω\subset \Bbb R^3$中定常Hall磁流体动力学方程组适当弱解的部分正则性。特别地，我们证明了适当的弱解的可能奇点集的豪斯多夫维数至多为1。此外，在$Ω=\Bbb R^3$的情况下，我们证明了可能的奇点集是紧的。
We discover cancellations upon $H^{2}(\mathbb{R}^{n})$-estimate of the Hall term for $n \in \{2,3\}$. As its consequence, first, we derive a regularity criterion for the 3-dimensional Hall-magnetohydrodynamics system in terms of only horizontal components of velocity and magnetic fields. Second, we prove the global regularity of the $2\frac{1}{2}$-dimensional electron magnetohydrodynamics system with magnetic diffusion $(-Δ)^{\frac{3}{2}} (b_{1}, b_{2}, 0) + (-Δ)^α (0, 0, b_{3})$ for $α> \frac{1}{2}$. Lastly, we extend this result to the $2\frac{1}{2}$-dimensional Hall-magnetohydrodynamics sys…
We are concerned with the global existence of finite energy weak solutions to 3D density-dependent magnetohydrodynamics (MHD) system with Hall-effect set in a general smooth bounded domain. The perfectly conducting wall boundary condition is imposed on the magnetic field. Due to the degeneracy of Hall-effect term (a tri-linear term) in vacuum, we assumed initial density lies in the bounded function space and having a positive lower bound. Particularly, these bounds are preserved by the density transport equation which helps yield a satisfying compactness argument of the magnetic field.
In this paper, we establish the global existence to the three-dimensional incompressible Hall-MHD equations for a class of large initial data, whose $L^{\infty}$ norms can be arbitrarily large. In addition , we give an example to show that such a large initial value does exist. Our idea is splitting the generalized heat equations from Hall-MHD system to generate a small quantity for large time $t$.
In this paper, we are concerned with the energy and magnetic helicity conservation of weak solutions for both the electron and Hall magnetohydrodynamic equations. Various sufficient criteria to ensure the energy and magnetic helicity conservation in Onsager's critical spaces $\underline{B}^α_{p,VMO}$ and $B^α_{p,c(\mathbb{N})}$ in these systems are established. Moreover, for the E-MHD equations, we observe that the conservation criteria of energy and magnetic helicity to the E-MHD equations correspond to the helicity and energy to the ideal incompressible Euler equations, respectively.
In this paper, the Cauchy's problem for fractional MHD system with the Hall and ion-slip effects is considered. By exploring the structure of semilinear and quasilinear terms, we prove the global existence of solutions for a class of large initial data. Both the velocity and magnetic fields could be arbitrarily large in $H^3(\mathbb{R}^3)$.
On the temporal estimates for the incompressible Navier-Stokes equations and the Hall-magnetohydrodynamic equations
本文研究了不可压缩纳维-斯托克斯方程和霍尔磁流体动力学方程解的衰减率，通过改进的傅里叶分裂方法和伪测度空间中的初始数据，提高了弱解的衰减率。
On the double Beltrami states in Hall magnetohydrodynamics
本文研究 Hall MHD 方程中的双 Beltrami 态，通过变分方法在双螺旋度守恒约束下将其分类并刻画为能量极小化子，进而分析黏性与电阻性情形下时间依赖双 Beltrami 态的精确形式与稳定性。
Existence, uniqueness and optimal decay rates for the 3D compressible Hall-magnetohydrodynamic system
本文研究了三维可压缩霍尔磁流体动力学系统的柯西问题。我们首先在临界Besov空间中建立了当初值接近稳定平衡态时系统强解的唯一全局可解性。此外，在一个适当的附加条件下，只涉及低频率的数据和$L^{2}$ -临界正则性框架，我们表现出最佳的时间衰减率构造的整体解决方案。证明依赖于应用傅里叶分析的混合抛物-双曲系统，并在一个细化的时间加权能量泛函。
Whether or not the solution to the $2\frac{1}{2}$-dimensional Hall-magnetohydrodynamics system starting from smooth initial data preserves its regularity for all time remains a challenging open problem. Although the research direction on component reduction of regularity criterion for Navier-Stokes equations and magnetohydrodynamics system have caught much attention recently, the Hall term has presented much difficulty. In this manuscript we discover a certain cancellation within the Hall term and obtain various new regularity criterion: first, in terms of a gradient of only the third componen…
We are concerned with the 3D incompressible Hall-magnetohydro-dynamic system (Hall-MHD). Our first aim is to provide the reader with an elementary proof of a global well-posedness result for small data with critical Sobolev regularity, in the spirit of Fujita-Kato's theorem [10] for the Navier-Stokes equations. Next, we investigate the long-time asymptotics of global solutions of the Hall-MHD system that are in the Fujita-Kato regularity class. A weak-strong uniqueness statement is also proven. Finally, we consider the so-called 2 1/2 D flows for the Hall-MHD system, and prove the global exist…
On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics
本文建立了不可压电阻性霍尔磁流体动力学方程组经典解的最优爆破准则。我们还证明了小初值下的两个整体时间经典解的存在性结果，其小性条件分别由适当的Sobolev范数和Besov范数给出.虽然索伯列夫空间形式已经是\cite{Chae-Degond-Liu}中相应结果的改进，但通过Besov空间估计实现了尺度性质方面的最优性。能量估计在$\dot{B}^s_{2,1}$范数意义下的特殊性质对这一结果至关重要。与通常的MHD相反，$2\frac12$维Hall MHD的整体适定性是完全开放的。
Beale--Kato--Majda-type continuation criteria for Hall- and electron-magnetohydrodynamics
该研究针对带电阻的电子MHD和Hall-MHD，分别建立了类似Beale--Kato--Majda结果的无量纲延拓准则，明确了对应解可延拓的积分条件。
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 $J_0:=\nabla\times B_0$ belong to the homogeneous Besov space $\dot B^{\frac 3p-1}_{p,1},$ $\:1\leq p<\infty,$ 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 …
On the regularity of magneto-vorticity field and the global existence for the Hall magnetohydrodynamic equations
本文研究了不可压缩粘性电阻磁流体动力学方程的正则性及全局解的存在性，通过分析磁-涡流场的正则性条件，得出了全局解存在的条件，并证明了二维情况下磁流体动力学方程的全局适定性。
On illposedness of the Hall and electron magnetohydrodynamic equations without resistivity on the whole space
本研究证明全空间上无电阻霍尔和电子磁流体力学方程对紧支撑数据具有强不适定性，通过构造退化波包并利用广义能量估计等方法实现。
Low Regularity Norm Inflation of Axisymmetric Hall Magnetohydrodynamics in Sobolev Space
本文研究轴对称霍尔磁流体动力学，证明其光滑解在$1<σ<7/2$时于对应Sobolev空间中存在范数膨胀，且若光滑解唯一则该结论可推广至一般霍尔磁流体动力学。
Well-posedness for Hall-magnetohydrodynamics
本文证明了不可压缩、有阻、有粘或无粘的Hall-MHD模型在大数据情况下局部光滑解的存在性和在小数据情况下全局光滑解的存在性。我们还证明了一个Liouville定理的固定的解决方案。
Determining wavenumbers for Hall and electron magnetohydrodynamics turbulence
本文研究了霍尔和电子磁流体动力学湍流中确定波数的数学框架，证明了弱解中时间依赖确定波数的存在，并建立了与现象学耗散波数的比较。
Four-fifths laws in electron and Hall magnetohydrodynamic fluids: Energy, Magnetic helicity and Generalized helicity
该研究聚焦电子与霍尔磁流体系统，借鉴Eyink纵向结构函数及经典MHD方程的最新成果，推导出能量、磁螺旋度和广义螺旋度对应的五分之四型守恒量定律。
We consider the magnetohydrodynamics system with Hall effect accompanied with initial data in supercritical Sobolev space. Via an appropriate randomization of the supercritical initial data, both local and small data global well-posedness for the system are obtained almost surely in critical Sobolev space.
Stochastic magnetohydrodynamics system: cross and magnetic helicity in ideal case; non-uniqueness up to Lions' exponents from prescribed initial data
本文研究随机驱动的三维磁流体力学系统，在理想情形下分析交叉螺旋度与磁螺旋度的演化规律，并利用凸积分技术证明从规定初值出发直至 Lions 指数时解的非唯一性。
We derive dyadic models for the magnetohydrodynamics with Hall effect by including the intermittency dimension as a parameter. For such dyadic models, existence of global weak solutions is established. In addition, local strong solution is obtained; while global strong solution is obtained in the case of high intermittency dimension. Moreover, we show that positive solution with large initial data develops blow-up in finite time provided the intermittency dimension is lower than a threshold.
Cauchy problem for 3D incompressible Hall-magnetohydrodynamics (Hall-MHD) system with fractional Laplacians is studied. First, global well-posedness of small-energy solutions with general initial data in $H^s$, $s>\frac{5}{2}$, is proved. Second, a special class of large-energy initial data is constructed, with which the Cauchy problem is globally well-posed. The proofs rely upon a new global bound of energy estimates involving Littlewood-Paley decomposition and Sobolev inequalities, which enables one to overcome the $\frac{1}{2}$-order derivative loss of the magnetic field.
In this article, we prove various illposedness results for the Cauchy problem for the incompressible Hall- and electron-magnetohydrodynamic (MHD) equations without resistivity. These PDEs are fluid descriptions of plasmas, where the effect of collisions is neglected (no resistivity), while the motion of the electrons relative to the ions (Hall current term) is taken into account. The Hall current term endows the magnetic field equation with a quasilinear dispersive character, which is key to our mechanism for illposedness. Perhaps the most striking conclusion of this article is that the Cauchy…
Local well-posedness for the Hall-MHD equations with fractional magnetic diffusion
霍尔磁流体动力学（Hall-MHD）方程是从动力学模型严格推导出来的，在天体物理学和天体物理学中有着广泛的应用。研究了由分数阶Laplacian算子$(-Δ)^α$给出的具有磁扩散的Hall-MHD方程经典解的局部适定性.由于霍尔-MHD方程中的霍尔项的存在，标准能量估计似乎表明，我们需要$α\ge 1$，以获得当地的适定性。本文突破了这一障碍，证明了分数阶Hall-MHD方程对任何$α>\frac12$都是局部适定的.这里的方法充分利用了耗散的平滑效应，并通过Besov空间技术建立了Sobolev范数的局部界。本文所提出的方法也适用于其它偏微分方程的类似情况。
On product estimates in Besov-Morrey spaces and stationary solutions for the Hall-MHD and Navier-Stokes systems
本文在尺度不变的Besov-Morrey空间中研究三维稳态不可压缩电阻粘性Hall-MHD系统的适定性，通过推导乘积估计处理非线性项，并推广到Navier-Stokes方程。
On analyticity and temporal decay rates of solutions to the viscous resistive Hall-MHD system
我们解决的解析性和大时间衰减率的强解的霍尔MHD方程。通过Gevrey估计，我们证明了具有$r>\f 52$的$H^r(\mathbb{R}^3)$的强解在$t>0$之后立即变为解析解，并且解析半径随时间的增长类似于$\sqrt{t}$.还获得了高阶导数衰减的上界和下界，这扩展了Chae和Schonbek的先前工作（J.微分方程255（2013），3971- 3982）。
The equations of extended magnetohydrodynamics
本文研究扩展磁流体力学（XMHD）的柯西问题，将其改写为含零阶伪微分算子的适定对称双曲-抛物系统，在最小索伯列夫正则性下得到局部时间解，并阐释惯性波的产生与传播。
We study the Lie point symmetries and the similarity transformations for the partial differential equations of the nonlinear one-dimensional magnetohydrodynamic system with the Hall term known as HMHD system. For this 1+1 system of partial differential equations we find that is invariant under the action of a seventh dimensional Lie algebra. Furthermore, the one-dimensional optimal system is derived while the Lie invariants are applied for the derivation of similarity transformations. We present different kinds of oscillating solutions.
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