Zhouyu Li, Pan Liu
We prove several Liouville type results for the stationary Hall-MHD and MHD equations. In particular, we show that the velocity and magnetic field, belonging to some Lorentz spaces, must be zero.
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We prove several Liouville type results for the stationary MHD and Hall-MHD equations. In particular, we show that the velocity and magnetic field, belonging to some Lorentz spaces or satisfying a priori decay assumption, must be zero.
In this paper we prove a Liouville type theorem for the stationary MHD and the stationary Hall-MHD systems. Assuming suitable growth condition at infinity for the mean oscillations for the potential functions, we show that the solutions are trivial. These results generalize the previous results obtained by two of the current authors in [6]. To prove our main theorems we use a refined iteration argument.
In this note we investigate Liouville type theorems for the steady three dimensional MHD and Hall-MHD equations, and show that the velocity field $u$ and the magnetic field $B$ are vanishing provided that $B\in L^{6,\infty}(\mathbb{R}^3)$ and $u\in BMO^{-1}(\mathbb{R}^3)$, which state that the velocity field plays an important role. Moreover, the similar result holds in the case of partial viscosity or diffusivity for the three dimensional MHD equations.
This paper is concerned with the Liouville type theorems for the 3D stationary incompressible Hall-MHD equations. We establish that under some sufficient conditions in local Morrey spaces, solutions of the stationary Hall-MHD equations are identically zero. In particular, we also prove Liouville type results for the stationary incompressible MHD equations on $\mathbb{R}^3$. Our theorems extend and generalize the classical results for the stationary incompressible Navier-Stokes equations.
New Liouville type theorems for 3D steady incompressible MHD equations and Hall-MHD equations
本文通过能量方法和迭代论证，研究了三维稳态不可压缩磁流体动力学方程和霍尔磁流体动力学方程的李普希茨型定理，并通过引入对数因子改进了增长条件。
In this paper, we investigate the three dimensional stationary compressible Navier-Stokes equations, and obtain Liouville type theorems if a smooth solution $(ρ, \mathbf{u})$ satisfies some suitable conditions. In particular, our results improve and generalize the corresponding result.
For the two dimensional stationary MHD equations, we proved that Liouville type theorems hold if the velocity is growing at infinity, where the magnetic field is assumed to be bounded under a smallness condition. The key point is to overcome the nonlinear terms, since no maximum principle holds for the MHD case with respect to the Navier-Stokes equations.
This note is devoted to investigating Liouville type properties of the two dimensional stationary incompressible Magnetohydrodynamics equations. More precisely, under smallness conditions only on the magnetic field, we show that there are no non-trivial solutions to MHD equations either the Dirichlet integral or some $L^p$ norm of the velocity-magnetic fields are finite. In particular, these results generalize the corresponding Liouville type properties for the 2D Navier-Stokes equations, such as Gilbarg-Weinberger \cite{GW1978} and Koch-Nadirashvili-Seregin-Sverak \cite{KNSS}, to the MHD sett…
New Liouville type theorems for the stationary MHD equations in $\mathbb{R}^3$
本文在频率空间研究三维稳态 MHD 方程，针对有限 Dirichlet 能量解建立两个新 Liouville 型定理，揭示速度场低频部分的主导作用并改进 Chae-Weng 的结果。
Liouville Type Theorem for the Fractional MHD and Hall-MHD equations in $\mathbb{R}^{3}
本文基于分数磁流体动力学和霍尔磁流体动力学方程，利用Caffarelli-Sivestre延伸方法和Yuan-Xiao方法，解决了非局部算子$(-\triangle)^{s}$的困难，推导了利乌维尔型定理，并作为副产品获得了纳维-斯托克斯方程的结果。
In this paper, we study the Liouville--type theorems for three--dimensional stationary incompressible MHD and Hall--MHD systems in a slab with periodic boundary condition. We show that, under the assumptions that $(u^θ,b^θ)$ or $(u^r,b^r)$ is axisymmetric, or $(ru^r,rb^r)$ is bounded, any smooth bounded solution to the MHD or Hall--MHD system with local Dirichlet integral growing as an arbitrary power function must be constant. This hugely improves the result of \cite[Theorem 1.2]{pan2021Liouville}, where the Dirichlet integral of $\mathbf{u}$ is assumed to be finite. Motivated by \cite[Bang--…
In this paper we prove a Liouville type theorem for the stationary magnetohydrodynamics(MHD) system in $\Bbb R^3$. Let $(v, B, p)$ be a smooth solution to the stationary MHD equations in $\Bbb R^3$. We show that if there exist smooth matrix valued potential functions ${\bf Φ}$, ${\bf Ψ}$ such that $ \nabla \cdot {\bf Φ} =v$ and $\nabla \cdot {\bf Ψ}= B$, whose $L^6$ mean oscillations have certain growth condition near infinity, namely $$-\!\!\!\!\!\int_{B(r)} |\mathbfΦ - \mathbfΦ_{ B(r)} |^6 dx + -\!\!\!\!\!\int_{B(r)} |\mathbfΨ- \mathbfΨ_{ B(r)} |^6 dx\le C r\quad \forall 1< r< +\infty,$$ the…
Liouville Type Theorem for the Steady Fractional Compressible MHD Equations in $\mathbb{R}^{3}$
针对稳态分数阶可压缩磁流体动力学方程，结合 Caffarelli-Silvestre 延拓技术与截断论证，在适当正则性和衰减条件下证明唯一解是平凡的，建立了 Liouville 型定理。
Saint-Venant Estimates and Liouville-Type Theorems for the Stationary MHD Equation in $\mathbb{R}^3$
本文研究定常MHD方程的Liouville型定理，通过Froullani积分细化Saint-Venant型估计，证明在速度与磁场势函数的$L^s$平均振荡增长受控时解平凡，且增长假设弱于已有结果。
Liouville-type theorem for steady helically symmetric MHD system in $\mathbb{R}^{3}$
本文证明R³中稳态螺旋对称MHD系统的有界光滑解必为常向量，将此前Navier-Stokes系统的刘维尔型定理推广至MHD系统，证明基于Saint-Venant型估计刻画Dirichlet积分增长。
Liouville type theorems for the 3D stationary MHD and Hall-MHD equations with non-zero constant vectors at infinity
本文研究了三维定常MHD和Hall-MHD系统在无穷远处特定渐近条件下的Liouville型定理，通过Lizorkin理论和Kato稳定性定理处理Hall项难点，证明了速度场和磁场在特定条件下均为常向量。
Liouville type theorem for double Beltrami solutions of the Hall-MHD system in $\Bbb R^3$
本文针对 $\n\Bbb R^3$ 中稳态 Hall-MHD 方程的光滑双 Beltrami 解，证明在 $q\in[2,3)$ 的 $L^q$ 可积条件下解必为零，并涵盖 $B=0$ 时 Euler 方程的已有结果。
We establish new Liouville-type theorems for the two-dimensional stationary magneto-hydrodynamic incompressible system assuming that the velocity and magnetic field have bounded Dirichlet integral. The key tool in our proof is observing that the stream function associated to the magnetic field satisfies a simple drift-diffusion equation for which a maximum principle is available.
In dimension $n=3$, we prove that the singular set of any stationary solution to the Liouville equation $-Δu=e^u$, which belongs to $W^{1,2}$, has Hausdorff dimension at most 1.
In this paper we consider the entire weak solutions of the equations for stationary flows of shear thickening fluids in the plane and prove Liouville theorem under the global boundedness condition of velocity fields.
In this paper we consider the entire weak solutions $u$ of the equations for stationary flows of shear thickening fluids in the plane and prove Liouville theorems under the conditions on the finiteness of energy and under the integrability condition of the solutions.
It is shown that any smooth solution to the stationary Navier-Stokes system in $R^3$ with the velocity field, belonging globally to $L_6$ and $BM0^{-1}$, must be zero.
Wiener Algebras Methods for Liouville Theorems on the Stationary Navier-Stokes System
本文利用维纳代数方法，为不可压缩流体静止纳维-斯托克斯系统建立了利奥维勒定理的充分条件。
Using an rotation of Yuan, we observe that the gradient graph of any semiconvex function is a Liouville manifold, that is, does not admit bounded harmonic functions. As a corollary, we find that any entire solution of the fourth order Hamiltonian stationary equation with Lagrangian phase angle uniformly larger than the critical angle must be a quadratic.
We consider here the simplified Ericksen-Leslie system on the whole three-dimensional space. This system deals with the incompressible Navier-Stokes equations strongly coupled with a harmonic map flow which models the dynamical behavior for nematic liquid crystals. For both the stationary (time independing) case and the non-stationary (time depending) case, using the fairly general framework of a kind of local Morrey spaces, we obtain some a priori conditions on the unknowns of this coupled system to prove that they vanish identically. This results are known as Liouville-type theorems. As a bi…
The problem for the stationary Navier-Stokes equation in 3D under finite Dirichlet norm is open. In this paper we answer the analogous question on the 3D hyperbolic space. We also address other dimensions and more general manifolds.
The Liouville problem for the stationary Navier-Stokes equations on the whole space is a challenging open problem who has know several recent contributions. We prove here some Liouville type theorems for these equations provided the velocity field belongs to some Lorentz spaces and then in the more general setting of Morrey spaces. Our theorems correspond to a improvement of some recent result on this problem and contain some well-known results as a particular case.
Non-uniqueness of weak solutions to the 3-D stationary MHD equations in Besov space with negative regularity index
通过凸积分方法证明三维环面上分数阶稳态MHD方程存在无穷多个非平凡奇异解，从而在次临界Besov空间中建立弱解的非唯一性。
In this paper we study the Liouville type problem for the stationary Navier-Stokes equations in $\Bbb R^3$. We deduce an asymptotic formula for an integral involving the head pressure, $Q=\frac12 |v|^2 +p$, and its derivative over domains enclosed by level surfaces of $Q$. This formula provides us with new sufficient condition for the triviality of solution to the Navier-Stokes equations.
Uniqueness of Leray solutions of the 3D Navier-Stokes equations is a challenging open problem. In this article we will study this problem for the 3D stationary Navier-Stokes equations and under some additional hypotheses, stated in terms of Lebesgue and Morrey spaces, we will show that the trivial solution U = 0 is the unique solution. This type of results are known as Liouville theorems.
In this paper we study the asymptotic behavior of sequences of stationary weak solutions to the following Liouville-type equation $-Δu=e^u~~~{in }~~~Ω$, where $Ω$ is an open set of $R^3$. By improving the partial regularity estimates obtained by the first author for the above equation, we succeed in performing a blow-up analysis without Morrey-type assumptions on the solutions $u$ and on the nonlinearity $e^u.$
Logarithmic improvement of a Liouville-type theorem for the stationary Navier--Stokes equations
本文改进了 stationary Navier-Stokes 方程的 Liouville 定理，基于速度场在无穷远处的 $L^p$ 增长性质，引入对数因子提升结果。
Liouville type theorems for stationary Navier-Stokes equations with Lebesgue spaces of variable exponent
本文研究稳态三维Navier-Stokes方程的Liouville型定理，通过变指数Lebesgue空间为弱解唯一性提供新的灵活性条件。
We show that any smooth solution $(\mathbf{u},\mathbf{H})$ to the stationary equations of magneto-hydrodynamics (MHD) belonging to both spaces $L^6 (\mathbb{R}^3)$ and $BMO^{-1}(\mathbb{R}^3)$ must be identically zero. This is an extension of previous results, all of which systematically required stronger integrability and the additional assumption $\nabla \mathbf{u}, \nabla \mathbf{H} \in L^2 (\mathbb{R}^3) $, i.e., finite Dirichlet integral.
We consider variational integrals of linear growth satisfying the condition of $μ$-ellipticity for some exponent $μ>1$ and prove that stationary points $u$: $\mathbb{R}^2 \to \mathbb{R}^N$ with the property \[ \limsup_{|x|\to \infty} \frac{|u(x)|}{|x|} < \infty \] must be affine functions. The latter condition can be dropped in the scalar case together with appropriate assumptions on the energy density providing an extension of Bernstein's theorem.
We consider entire solutions $u$ of the equations describing the stationary flow of a generalized Newtonian fluid in 2D concentrating on the question, if a Liouville-type result holds in the sense that the boundedness of $u$ implies its constancy. A positive answer is true for $p$-fluids in the case $p>1$ (including the classical Navier-Stokes system for the choice $p=2$), and recently we established this Liouville property for the Prandtl-Eyring fluid model, for which the dissipative potential has nearly linear growth. Here we finally discuss the case of perfectly plastic fluids whose flow is…
Small-energy regularity for stationary weak solutions of the Liouville equation
该研究在任意维度上证明了Liouville方程平稳弱解的小能量正则性，解决了Da Lio等人提出的开放问题，证明基于对几乎平稳场集中现象的分析。
New Liouville type theorems for the stationary Navier-Stokes equations
本文通过建立 Dirichlet 积分的频率空间新公式，证明 $\mathbb{R}^3$ 中稳态 Navier-Stokes 方程的新 Liouville 型定理，并推广至分数阶情形、解决 $s=5/6$ 的问题。
Stability of a Stationary Solution to a 1-D Model for the MHD
本文研究一维磁流体力学模型稳态解的稳定性，借助De Gregorio方程的相关技术，证明了稳态点附近初值具有全局时间适定性，并得到类似基态指数稳定性的结果。
In this paper we prove Liouville type result for the stationary solutions to the compressible Navier-Stokes-Poisson equations(NSP) and the compressible Navier-Stokes equations(NS) in $\Bbb R^N$, $N\geq 2$. Assuming suitable integrability and the uniform boundedness conditions for the solutions we are led to the conclusion that $v=0$. In the case of (NS) we deduce that the similar integrability conditions imply $v=0$ and $ρ=$constant on $\Bbb R^N$. This shows that if we impose the the non-vacuum boundary condition at spatial infinity for (NS), $v\to 0$ and $ρ\to ρ_\infty >0$, then $v=0$, $ρ=ρ_\…
We study the Liouville type problem for the stationary 3D Navier-Stokes equations on $\Bbb R^3$. Specifically, we prove that if $v$ is a smooth solution to (NS) satisfying $ω={\rm curl}\,v \in L^q (\Bbb R^3) $ for some $\frac32 \leq q< 3$, and $|v(x)|\to 0$ as $|x|\to +\infty$, then either $v=0$ on $\Bbb R^3$, or $\int_{\Bbb R^6} Φ_+ dxdy=\int_{\Bbb R^6} Φ_- dxdy=+\infty$, where $Φ(x,y) :=\frac{1}{4π}\frac{ω(x)\cdot(x-y)\times (v(y)\times ω(y) )}{|x-y|^3} $, and $Φ_\pm:=\max\{ 0, \pm Φ\}$. The proof uses crucially the structure of nonlinear term of the equations.
This work studies the system of $3D$ stationary Navier-Stokes equations. Several Liouville type theorems are established for solutions in mixed-norm Lebesgue spaces and weighted mixed-norm Lebesgue spaces. In particular, we show that, under some sufficient conditions in mixed-norm Lebesgue spaces, solutions of the stationary Navier-Stokes equations are identically zero. This result covers the important case that solutions may decay to zero with different rates in different spatial directions, and some these rates could be significantly slow. In the un-mixed norm case, the result recovers avail…
New Liouville-type theorem for the stationary tropical climate model
本文针对稳态三维热带气候模型的光滑解，提出基于环域 $L^p$ 范数增长条件的新 Liouville 型定理，改进了 2021 年和 2023 年文献中的既有结果。
Liouville-type theorem for the stationary inhomogeneous Navier-Stokes equations
本文针对三维稳态非齐次 Navier-Stokes 方程，通过两次频率局部化获得 Dirichlet 能量局部公式，并据此估计 $u$ 与 $ρ$ 的不同频率部分，证明新的 Liouville 型定理。
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。
Some Liouville-type theorems for the stationary 3D magneto-micropolar fluids
本文研究三维稳态磁微极流体的 Liouville 型问题，分别基于势平均振荡增长、$L^p$ 可积性及无穷远消失条件证明解必平凡。
Refined Liouville-Type Theorems for the Stationary Navier--Stokes Equations
研究三维 stationary Navier-Stokes 方程的光滑解，基于 Cho 等人的工作，引入新的积分性准则，证明当解的 $L^p$ 增长率在 $3/2 < p < 3$ 范围内有界时，解必须为平凡解。
Saint-Venant Estimates and Liouville-Type Theorems for the Stationary Navier-Stokes Equation in $\mathbb{R}^3$
本文通过Saint-Venant型估计刻画非平凡解Dirichlet能量的增长，在关于速度场势函数$L^s$平均振荡增长或无穷远处压头相对衰减的较弱条件下，证明了$\mathbb{R}^3$中定常Navier-Stokes方程的两个Liouville型定理。
Two improved Liouville-type theorems for the 3D stationary tropical climate model without temperature assumptions
该研究针对三维定常热带气候模型，在无温度假设下，通过L²估计等方法建立改进型Liouville型定理，获特定可积性条件下解平凡性，引入对数修正优化定理。
Three dimensional stationary solutions of the Electron MHD equations
本文利用凸积分方法，在低正则性 Sobolev 空间中构造了三维电子磁流体力学方程的非平凡稳态弱解，展示了该方法在经典三维 Navier-Stokes 方程中尚无法实现的可行性。
In this paper we prove a Liouville type theorem for the stationary equations of a non-Newtonian fluid in $\mathbb{R}^3$ with the viscous part of the stress tensor $\mathbf{A}_p(u) = \mathrm{div} ( | \mathbf{D}(u) |^{p-2} \mathbf{D}(u) )$, where $\mathbf{D}(u) = \frac 12 ( \nabla u + ( \nabla u )^{\top})$ and $\frac 95 < p < 3$. We consider a weak solution $u \in W^{1,p}_{loc}(\mathbb{R}^3)$ and its potential function $\mathbf{V} = (V_{ij}) \in W^{2,p}_{loc}(\mathbb{R}^3)$, i.e. $\nabla \cdot \mathbf{V} = u$. We show that there exists a constant $s_0=s_0(p)$ such that if the $L^s$ mean oscillat…
We show that any smooth stationary solution of the 3D incompressible Navier-Stokes equations in the whole space, the half space, or a periodic slab must vanish under the condition that for some $0 \le δ\le 1<L$ and $q=6(3-δ)/(6-δ)$, $$\liminf_{R \to \infty} \frac 1R \|u\|^{3-δ}_{L^{q}(R<|x|<LR)}=0.$$ We also prove sufficient conditions allowing shrinking radii ratio $L= 1+R^{-α}$. Similar results hold on a slab with zero boundary condition by assuming stronger decay rates. We do not assume global bound of the velocity. The key is to estimate the pressure locally in the annuli with radii ratio …
An improved Liouville-type theorem for the stationary tropical climate model
针对稳态三维热带气候模型，本文证明在更弱的正则性条件下光滑解必为平凡解，改进了已有Liouville型定理。
Some new Liouville type theorems for the 3D stationary magneto-micropolar fluid equations
本文通过迭代方法和插值技术，研究了三维静止磁流体-微极流体方程的Liouville型定理，放宽了速度和磁场的增长条件，并对角速度提出了最宽松的限制。
Liouville Theorems for Stationary Navier-Stokes Equations via the Radial Velocity Component
本文研究静止纳维-斯托克斯方程在三维空间中的李普希茨型结果，证明在仅对速度径向分量施加积分条件时，任何$\dot{H}^1(\mathbb{R}^3)$解都是平凡的，并在变指数设定中建立唯一性结果。
Liouville-type theorems for the stationary non-Newtonian fluids in a slab
本文研究了slab中非牛顿流体 stationary 解的Liouville型定理，证明了在特定条件下解必须为平凡解，并引入了Korn不等式常数的估计方法。
Notes on Liouville-type theorems for the 3D stationary Navier-Stokes equations
本文在变量 Lebesgue 空间框架下，扩展了 Chamorro 和 Vergara-Hermosilla 的 Liouville 型定理，发现两个新的非可忽略区域，使定理在某些 p(·) 假设下仍成立。
Asymptotic behavior and Liouville-type theorems for axisymmetric stationary Navier-Stokes equations outside of an infinite cylinder with a periodic boundary condition
本文研究无限长圆柱外无旋、轴向周期的定常 Navier-Stokes 流，在广义有限 Dirichlet 积分条件下建立涡量逐点衰减估计，并证明 Liouville 型定理。
This paper mainly considers three iterations based on charge-conservative finite element approximation in Lipschitz domain for the stationary thermally coupled inductionless MHD equations. Based on the hybrid finite element method, the unknowns of hydrodynamic are discretized by the stable velocity-pressure finite element pair, and the current density along with electric potential are similarly discretized by the comforming finite element pair in $\boldsymbol{H}(\boldsymbol{div}, Ω)\times L^2(Ω)$. And on account of the strong nonlinearity of the equations, we present three coupled iterative me…
A first-order Liouville theorem is obtained for random ensembles of uniformly parabolic systems under the mere qualitative assumptions of stationarity and ergodicity. Furthermore, the paper establishes, almost surely, an intrinsic large-scale $C^{1,α}$-regularity estimate for caloric functions.
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