Han Liu
Using Littlewood-Paley theory, one formulates the determining wavenumbers for the Hall-MHD system, defined for each individual solution $(u,b)$. It is shown that the long time behaviour of strong solutions is almost finite dimensional as the wavenumbers are bounded in certain average senses.
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This paper is devoted to the algorithmic development of inverse elastic scattering problems. We focus on reconstructing the locations and shapes of elastic scatterers with known dictionary data for the nearly incompressible materials. The scatterers include non-penetrable rigid obstacles and penetrable mediums, and we use time-harmonic elastic point signals as the incident input waves. The scattered waves are collected in a relatively small backscattering aperture on a bounded surface. A two-stage algorithm is proposed for the reconstruction and only two incident waves of different wavenumbers…
We consider the electron magnetohydrodynamics (MHD) with static background ion flow. A special situation of $B(x,y,t)=\nabla\times (a\vec e_z)+b \vec e_z$ with scalar-valued functions $a(x,y,t)$ and $b(x,y,t)$ was studied numerically in the physics paper [7]. The authors concluded from numerical simulations that there is no evidence of dissipation cutoff for the electron MHD. In this paper we show the existence of determining wavenumber for the electron MHD, and establish a regularity condition only on the low modes of the solution. Our results suggest that the conclusion of the physics paper …
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。
Energy cascade for cross-shear length scales in free-shear three-dimensional incompressible viscous flows
本文研究自由剪切流中能量级联现象，通过不可压缩牛顿流体方程分析，揭示高斯-低斯波数分解下的能量预算关系及级联机制。
We demonstrate that the solutions to the Cauchy problem for the three dimensional incompressible magneto-hydrodynamics (MHD) system can develop diferent types of norm inflations in $\dot{B}_{\infty}^{-1, \infty}$. Particularly the magnetic field can develop norm inflation in short time even when the velocity remains small and vice verse. Efforts are made to present a very expository development of the inginious construction of Bourgain and Pavlović for Navier-Stokes equation.
We prove the existence of both local and global smooth solutions to the Cauchy problem in $\R^3$ for the incompressible magnetohydrodynamics (MHD) system. We also prove that the solution to the incompressible MHD system can be obtained as the incompressible limit of the corresponding compressible system. We apply methods in extracting weak dissipative structure which were suggested by Lei-Liu-Zhou.
In this paper, we study the well-posedness of boundary layer problems for the inhomogeneous incompressible magnetohydrodynamics(MHD) equations, which are derived from the two dimensional density-dependent incompressible MHD equations.Under the assumption that initial tangential magnetic field is not zero and density is a small perturbation of the outer constant flow in supernorm,the local-in-time existence and uniqueness of inhomogeneous incompressible MHD boundary layer equation are established in weighted Conormal Sobolev spaces by energy method. As a byproduct, the local-in-time well-posedn…
We demonstrate that the solutions to the Cauchy problem for the three dimensional incompressible magneto-hydrodynamics (MHD) system can develop diferent types of norm inflations in $\dot{B}_{\infty}^{-1, \infty}$. Particularly the magnetic field can develop norm inflation in short time even when the velocity remains small and vice verse. Efforts are made to present a very expository development of the inginious construction of Bourgain and Pavlović for Navier-Stokes equations.
This paper examines the global (in time) regularity of classical solutions to the 2D incompressible magnetohydrodynamics (MHD) equations with only magnetic diffusion. Here the magnetic diffusion is given by the fractional Laplacian operator $(-Δ)^β$. We establish the global regularity for the case when $β>1$. This result significantly improves previous work which requires $β>\frac32$ and brings us closer to the resolution of the well-known global regularity problem on the 2D MHD equations with standard Laplacian magnetic diffusion, namely the case when $β=1$.
This paper concerns with the explicit blowup phenomenon for 3D incompressible MHD equations in R^3. More precisely, we find two family of explicit blowup solutions for 3D incompressible MHD equations in R^3. One family of solutions admit the smooth initial data, and the initial data of another family of solutions are not smooth. The energy of those solutions is infinite. Moreover, our results tell us that the blowup phenomenon of 3D incompressible MHD can only take place in the velocity field of the fluid, but no blowup for the magnetic field.
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 [17,32,16]. Our proof is based on the combination of the techniques developed in our previous works [17,32,16], Alinhac good unknowns, and a crucial anti-symmetric structure on the boundary.
We consider the three-dimensional incompressible free-boundary magnetohydrodynamics (MHD) equations in a bounded domain with surface tension on the boundary. We establish a priori estimate for solutions in the Lagrangian coordinates with $H^{3.5}$ regularity. To the best of our knowledge, this is the first result focusing on the incompressible ideal free-boundary MHD equations with surface tension. It is worth pointing out that the $1/2$-extra spatial regularity for the flow map $η$ is no longer required in this manuscript thanks to the presence of the surface tension on the boundary.
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。
The Incompressible Limit of the Equations of Compressible Ideal Magneto-Hydrodynamics with perfectly conducting boundary
本文研究半空间中具有完美导电壁边界条件的可压缩理想磁流体力学方程组的不可压缩极限，通过引入考虑边界奇异行为的各向异性Sobolev空间，证明了当马赫数趋于零时解收敛到不可压缩磁流体力学方程组的解。
Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations
本文通过尺度分析、线性化近似和谐函数单调性构造迭代序列，将纳维-斯托克斯技术推广到含外力的MHD方程，证明其恰当弱解奇点的一维抛物型豪斯多夫测度为零，建立了正则性理论。
The global regularity for the incompressible magnetohydrodynamic equations (MHD) in three dimensions is a long standing open problem of fluid dynamics and PDE theory. The Navier-Stokes equations can be viewed as a special case of MHD with a constant magnetic field, whose global regularity problem is known as a Clay Millennium Prize Problem. In this article, we prove the global regularity of axially symmetric solutions to the ideal MHD in three dimensions for a family of non-trivial magnetic fields. The proofs are based on the special structures of MHD and can of course also applied to the resi…
We consider the three-dimensional incompressible magnetohydrodynamics (MHD) equations in a bounded domain with small volume and free moving surface boundary. We establish a priori estimate for solutions with minimal regularity assumptions on the initial data in Lagrangian coordinates. In particular, due to the lack of the Cauchy invariance for MHD equations, the smallness assumption on the fluid domain is required to compensate a loss of control of the flow map. Moreover, we show that the magnetic field has certain regularizing effect which allows us to control the vorticity of the fluid and t…
In this paper, we consider suitable weak solutions of the four dimensional incompressible magneto-hydrodynamic equations. We give two different kind $\varepsilon$-regularity criteria. One only requires the smallness of scaling $L^{p,q}$ norm of $u$, another requires the smallness of scaling space time $L^2$ norm of $\nabla u$ and boundedness of scaling norm of $H$ or $\nabla H$. And as an application of the second kind criteria, we also prove that up to the boundary, the two-dimensional Hausdorff measure of the set of singular points is equal to zero.
This paper is concerned with a Cauchy problem for the three-dimensional (3D) nonhomogeneous incompressible heat conducting magnetohydrodynamic (MHD) equations in the whole space. First of all, we establish a weak Serrin-type blowup criterion for the strong solutions. It is shown that for the Cauchy problem of the 3D nonhomogeneous heat conducting MHD equations, the strong solution exists globally if the velocity satisfies the weak Serrin's condition. In particular, this criterion is independent of the absolute temperature and magnetic field. Then as an immediate application, we prove the globa…
We demonstrate that the three dimensional incompressible magneto-hydrodynamics (MHD) system is ill-posed in the largest scaling invariant space $\dot{B}_{\infty,\infty}^{-1}$. The construction method of initial data used in this paper is different from the one in a previous work [DQS] of the author. Specifically, we construct initial data which has finite energy and is small enough in $H^s\cap \dot{B}_{\infty,\infty}^{-1}$, with $s<1/2$, such that any Leray-Hopf type of weak solution to the MHD system starting from this initial data is discontinuous at time $t=0$ in the metric of $\dot{B}_{\in…
In this paper, we are concerned with the two-dimensional (2D) incompressible magnetohydrodynamic (MHD) equations with velocity dissipation given by $(-Δ)^α$ and magnetic diffusion given by reducing about logarithmic diffusion from standard Laplacian diffusion. More precisely, we establish the global regularity of solutions to the system as long as the power $α$ is a positive constant. In addition, we prove several global \emph{a priori} bounds for the case $α=0$. In particular, our results significantly improve previous works and take us one step closer to a complete resolution of the global r…
This paper studies the regularity problem for the 3D incompress- ible resistive viscous Hall-magneto-hydrodynamic (Hall-MHD) system. The Kolmogorov 41 phenomenological theory of turbulence predicts that there exists a critical wavenumber above which the high frequency part is dominated by the dissipation term in the fluid equation. Inspired by this idea, we apply an approach of splitting the wavenumber combined with an estimate of the energy flux to obtain a new regularity criterion. The regularity condition presented here is weaker than conditions in the existing criteria (Prodi-Serrin type c…
Low regularity solutions for the Cauchy problem of the ideal incompressible Magnetohydrodynamics equations
本文在拉格朗日坐标下研究了理想不可压缩MHD方程的低正则性解的局部适定性，通过引入退化波-椭圆系统和改进的乘积估计，降低了正则性要求。
Global well-posedness for 3D incompressible magneto-micropolar fluids without resistivity and spin viscosity in strip domains
本文研究了三维不可压缩磁流体-微极流体系统在带状域中无电阻率和自转粘性时的经典解的全局存在性与唯一性，通过改进的能量方法和迹估计克服了三个主要障碍，并证明了解向平衡态的代数时间衰减。
Stability of vertically charged steady magnetic field in 3D incompressible magneto-micropolar fluids without magnetic and angular viscosity in a strip domain
研究三维不可压缩磁微极方程在带状区域无磁和角粘性时的正则性与稳定性问题，给出稳态设置，证明小扰动有全局正则稳定解且解随时间趋于无穷以近指数速率收敛到稳态。
Kolmogorov's dissipation number and determining wavenumber for dyadic models
针对不可压缩磁流体力学与 Navier-Stokes 二进模型，本文建立不动点存在性和稳定性，证明确定波数的时间平均受 Kolmogorov 耗散波数控制，并通过数值模拟展示惯性区能谱。
On the Vanishing Viscosity and Magnetic Diffusion Limit for Incompressible Magnetohydrodynamics
研究不可压缩磁流体动力学方程的无粘和无电阻极限，通过推导充分条件、建立准则及分析衰减率，得到MHD系统解的极限及磁普朗特数极限，是该问题数学上的首个严格结果。
We treat hydrodynamic limits of the Vlasov-Maxwell-Boltzmann system for one and two species of particles in a viscous incompressible regime.
Local Well-posedness of the Free-Boundary Incompressible Magnetohydrodynamics with Surface Tension
本文研究带表面张力的三维自由边界不可压缩理想 MHD 方程组，通过人工粘性近似和极限论证证明其局部适定性，并克服速度—磁场强耦合及对称结构丧失带来的困难。
In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by a free surface. The free boundary problem for MHD is an important problem not only for mathematical fluid dynamics but also some application to the field of engineering In fact, when a thermonuclear reaction is caused artificially, a high-temperature plasma is sometimes subjected to a magnetic field and held in the air, and the boundary of the fluid at this time is a free one. In this paper, an electromagnetic field generated by some currents in an external domain keeps an…
This paper concerns the Cauchy problem of the nonhomogeneous incompressible magnetohydrodynamic (MHD) equations on the whole two-dimensional (2D) space with vacuum as far field density. In particular, the initial density can have compact support. We prove that the 2D Cauchy problem of the nonhomogeneous incompressible MHD equations admits a unique local strong solution provided the initial density and the initial magnetic decay not too slow at infinity.
In this paper, we solve a long-standing open problem: nonlinear stability of current-vortex sheet in the ideal incompressible Magneto-Hydrodynamics under the linear stability condition. This result gives a first rigorous confirmation of the stabilizing effect of the magnetic field on Kelvin-Helmholtz instability.
Global current-vortex sheets in the two-dimensional ideal incompressible MHD
本文研究二维理想不可压缩MHD中的电流-涡旋片自由边界问题，通过加权多层能量估计并利用强水平磁场的稳定作用，证明其全局非线性稳定性。
Strong solution of the three-dimensional $(3D)$ incompressible magneto-hydrodynamic $(MHD)$ equationss with a modified damping
该研究针对三维不可压缩MHD方程，通过在对流项引入幂次修正以适配Darcy-Forchheimer定律，在初始数据小性及特定参数条件下证明了全局存在性与唯一性，并辅以对数型扰动函数支持结论。
In the present paper, we show the ill-posedness of the free boundary problem of the incompressible ideal magnetohydrodynamics (MHD) equations in two spatial dimensions for any positive vacuum permeability $μ_0$, in Sobolev spaces. The analysis is uniform for any $μ_0>0$.
In this paper, we prove the local well-posedness of plasma-vacuum interface problem for ideal incompressible magnetohydrodynamics under the stability condition: the magnetic field $\mathbf{h}$ and the vacuum magnetic field $\hat{\mathbf{h}}$ are non-collinear on the interface(i.e., $|\mathbf{h}\times \hat{\mathbf{h}}|>0$), which was introduced by Trakhinin as a stability condition for the compressible plasma-vacuum interface problem.
In this paper, we study the two phase flow problem with surface tension in the ideal incompressible magnetohydrodynamics. We first prove the local well-posedness of the two phase flow problem with surface tension, then demonstrate that as surface tension tends to zero, the solution of the two phase flow problem with surface tension converges to the solution of the two phase flow problem without surface tension.
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.
Exact controllability of incompressible ideal magnetohydrodynamics in $2$D
本研究探讨了平面不可压缩理想磁流体力学的能控性，提出分而治之策略以克服MHD耦合与磁场拓扑障碍，并利用J.-M. Coron的返回方法求解非线性速度控制子问题，扩展了内部与边界能控性结果。
This paper is concerned with the Cauchy problem of the multi-dimensional incompressible magnetohydrodynamic equations with inhomogeneous density and fractional dissipation. It is shown that when $α+β=1+\frac{n}{2}$ satisfying $1\leq β\leq α\leq\min \{\frac{3β}{2},\frac{n}{2},1+\frac{n}{4}\}$ and $\frac{n}{4}<α$ for $n\geq3$ , then the inhomogeneous incompressible MHD equations has a unique global strong solution for the initial data in Sobolev space which do not need a small condition.
Stability for the 2D incompressible MHD equations with only magnetic diffusion
本文针对周期域上仅有磁扩散的二维不可压缩MHD方程，证明了背景磁场附近扰动的全局稳定性，并得出解的显式长时间衰减率，揭示了磁场的稳定作用。
We study the Rayleigh-Taylor problem for two incompressible, immiscible, viscous magnetohydrodynamic (MHD) flows, with zero resistivity, surface tension (or without surface tenstion) and special initial magnetic field, evolving with a free interface in the presence of a uniform gravitational field. First, we reformulate in Lagrangian coordinates MHD equations in a infinite slab as one for the Navier-Stokes equations with a force term induced by the fluid flow map. Then we analyze the linearized problem around the steady state which describes a denser immiscible fluid lying above a light one wi…
Inspired by an approach proposed previously for the incompressible Navier-Stokes (NS) equations, we present a general framework for the a posteriori analysis of the equations of incompressible magnetohydrodynamics (MHD) on a torus of arbitrary dimension d; this setting involves a Sobolev space of infinite order, made of C^infinity vector fields (with vanishing divergence and mean) on the torus. Given any approximate solution of the MHD Cauchy problem, its a posteriori analysis with the method of the present work allows to infer a lower bound on the time of existence of the exact solution, and …
We prove local existence of smooth solutions for large data and global smooth solutions for small data to the incompressible, resitive, viscous or inviscid Hall-MHD model. We also show a Liouville theorem for the stationary solutions.
A global stability result for incompressible magnetohydrodynamics
本文研究了不可压缩磁流体动力学方程在任意维度上的全局稳定性，提出了一种基于高阶Sobolev范数的显式估计方法，并引入了Beltrami型初始数据以证明全局解的稳定性。
This work focuses on the 3D incompressible magnetohydrodynamic (MHD) equations with mixed pressure-velocity-magnetic field in view of Lorentz spaces. Our main result shows the weak solution is regular, provided that $${\frac{π}{\left( e^{-\left\vert x\right\vert ^{2}}+\left\vert u\right\vert +\left\vert b\right\vert \right) ^{θ}}\in L}^{p}(0,T;L^{q,\infty }(\mathbb{R}^{3})),\text{ where }\frac{2}{p}+\frac{3}{% q}=2-θ\text{ and }0\leq θ\leq 1.$$
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