Ken Abe
We study Liouville theorems for the non-stationary Stokes equations in exterior domains in $ \mathbb{R}^{n}$ under decay conditions for spatial variables. As applications, we prove that the Stokes semigroup is a bounded analytic semigroup on $L^{ \infty}_{ σ}$ of angle $ π/2$ for $n \geq 3$. We also prove large time estimates for $n=2$ with zero net force.
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Liouville Theorem for 2D Navier-Stokes equations in Half Space
本文证明了半空间中二维Navier-斯托克斯方程的弱有界古解的Liouville定理。
A Liouville type theorem is proven for the steady-state Navier-Stokes equations. It follows from the corresponding theorem on the Stokes equations with the drift. The drift is supposed to belong to a certain Morrey space.
In this note, we investigate the 3D steady axially symmetric Navier-Stokes equations, and obtained Liouville type theorems if the velocity or the vorticity satisfies some a priori decay assumptions.
In this paper, we investigate the 3D inhomogeneous Navier-Stokes flows with vacuum, and obtain regularity criteria and Liouville type theorems in the Lorentz space if a smooth solution $(ρ, \mathbf{u})$ satisfies suitable conditions.
Liouville type theorems for the stationary Navier-Stokes equations are proven under certain assumptions. These assumptions are motivated by conditions that appear in Liouvile type theorems for the heat equations with a given divergence free drift.
Our aim is to prove Liouville type theorems for the three dimensional steady-state Navier-Stokes equations provided the velocity field belongs to some Lorentz spaces. The corresponding statement contains several known results as a particular case.
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.
Conditional Liouville theorems for the Navier-Stokes equations
本文通过新方法证明了Navier-Stokes方程的条件Liouville定理，改进了速度反导数的假设条件。
We prove Liouville type theorems for the self-similar solutions to the Navier-Stokes equations. One of our results generalizes the previous ones by Nečas-Ružička-Šverak and Tsai. Using the Liouville type theorem we also remove a scenario of asymtotically self-similar blow-up for the Navier-Stokes equations with the profile belonging to $L^{p, \infty} (\Bbb R^3)$ with $p> \frac{3}{2}$.
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。
Liouville theorems in unbounded domains for the time-dependent Stokes system
本文刻画了具有零边值的不定常Stokes方程组在不同区域（包括半空间）上的有界古解。
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.
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.
Liouville-type theorems for steady solutions of the barotropic Navier--Stokes system
本文研究了静态正压 Navier--Stokes 系统零速度解的唯一性问题，建立了无界物理空间中的 Liouville 型定理。
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.
Liouville type of theorems play a key role in the blow-up approach to study the global regularity of the three-dimensional Navier-Stokes equations. In this paper, we will prove Liouville type of theorems to the 3-D axisymmetric Navier-Stokes equations with swirls under some suitable assumptions on swirl component velocity $u_θ$ which are scaling invariant. It is known that $ru_θ$ satisfies the maximum principle. The assumptions on $u_θ$ will be natural and useful to make further studies on the global regularity to the three-dimensional incompressible axisymmetric Navier-Stokes equations.
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.
A Liouville theorem for the planer Navier-Stokes equations with the no-slip boundary condition and its application to a geometric regularity criterion
本文建立了半平面中Navier-斯托克斯方程在无滑移边界条件下向后整体解的Liouville型结果。没有对涡量和速度场的空间衰减作任何假设。我们用涡量方程代替原来的Navier-斯托克斯方程。作为应用，我们推广了三维半空间中Navier-斯托克斯方程在无滑移边界条件下的几何正则性准则。
We establish a Liouville theorem for bounded mild ancient solutions to the axi-symmetric incompressible Navier-Stokes equations on $(-\infty, 0] \times (\mathbb{R}^2 \times \mathbb{T}^1)$. This is a step forward to completely solve the conjecture on $(-\infty, 0] \times \mathbb{R}^3$ which was made in \cite{KNSS} to describe the potential singularity structures of the Cauchy problem.
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。
We study axially symmetric $D$-solutions of three dimensional steady Navier-Stokes equations. We prove that if the velocity $u$ decays like $|x'|^{-(\frac{2}{3})^+}$ uniformly for $z$, or the vorticity $ω$ decays like $|x'|^{-(\frac{5}{3})^+}$ uniformly for $z$, then $u$ vanishes. Here $|x'|$ denotes the distance to the axis.
On some Liouville Type Theorems for the Compressible Navier-Stokes Equations
本文证明了$d$维可压缩Navier-斯托克斯方程定常解的几个Liouville型结果.特别是，我们表明，当尺寸$d \geqslant 4$，自然要求$ρ\in L^{\infty} (\mathbbm{R}^d)$，$v \in \dot{H}^1 (\mathbbm{R}^d)$足以保证解决方案是平凡的。对于维度$d=2,3$，我们假设额外条件$v \in L^{\frac{3d}{d-1}}(\mathbb R^d)$。这改进了Chae（2012）的最近结果。
We prove that suitable weak solutions of the Navier-Stokes equations exhibit Type I singularities if and only if there exists a non-trivial mild bounded ancient solution satisfying a Type I decay condition. The main novelty is in the reverse direction, which is based on the idea of zooming out on a regular solution to generate a singularity. By similar methods, we prove a Liouville theorem for ancient solutions of the Navier-Stokes equations bounded in $L^3$ along a backward sequence of times.
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。
In the paper \cite{KNSS:1}, the authors make the following conjecture: {\it any bounded ancient mild solution of the 3D axially symmetric Navier-Stokes equations is constant.} And it is proved in the case that the solution is swirl free. Our purpose of this paper is to improve their result by allowing that the solution can grow with a power smaller than 1 with respect to the distance to the origin. Also, we will show that such a power is optimal to prove the Liouville type theorem since we can find counterexamples for the Navier-Stokes equations such that the Liouville theorem fails if the sol…
In this paper we prove Liouville type theorem for the stationary Navier-Stokes equations in $\Bbb R^3$ under the assumptions on the relative decays of velocity, pressure and the head pressure. More precisely, we show that any smooth solution $(u,p)$ of the stationary Navier-Stokes equations satisfying $u(x) \to 0$ as $|x|\to +\infty$ and the condition of finite Dirichlet integral $\int_{\Bbb R^3} | \nabla u|^2 dx <+\infty $ is trivial, if either $|u|/|Q|=O(1)$ or $|p|/|Q| =O(1) $ as $|x|\to \infty$, where $|Q|=\frac12 |u|^2 +p$ is the head pressure.
On the Liouville theorem for the Navier-Stokes and the Euler equations in $\Bbb R^N$
由于（2.20）式中的第二个不等式不允许$R$的指数为负号，因而用这种方法不能得到（2.20）式中所要求的消元，故本文已被作者撤回。
In this paper we prove three different Liouville type theorems for the steady Navier-Stokes equations in $\Bbb R^3$. In the first theorem we improve logarithmically the well-known $L^{\frac92} (\Bbb R^3)$ result. In the second theorem we present a sufficient condition for the trivially of the solution($v=0$) in terms of the head pressure, $Q=\frac12 |v|^2 +p$. The imposed integrability condition here has the same scaling property as the Dirichlet integral. In the last theorem we present Fubini type condition, which guarantee $v=0$.
On the Liouville type theorem for stationary compressible Navier-Stokes-Poisson equations in $\Bbb R^N$
本文证明了可压缩Navier-Stokes-泊松方程（NSP）和可压缩斯托克斯方程（NS）在$\Bbb R^N$，$N\geq 2$中定常解的Liouville型结果.在适当的可积性和解的一致有界性条件下，我们得出结论：$v=0$。在（NS）情形下，我们推导出类似的可积性条件意味着$\Bbb R^N$上的$v=0$和$ρ=$常数.这表明，如果我们对（NS），$v\to 0$和$ρ\to ρ_\infty >0$在空间无穷远处施加非真空边界条件，则$v=0$，$ρ=ρ_\infty$是解.
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。
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…
Liouville type of theorems with weights for the Navier-Stokes equations and the Euler equations
研究了$\Bbb R^N$，$N\geq 2$上Navier-Stokes方程和Euler方程的Liouville型定理.具体地说，我们证明了如果弱解$(v,p)$对某些权函数$w_1(x)$和$w_2 (x)$满足$|v|^2 +|p| \in L^1 (0,T; L^1(\Bbb R^N, w_1(x)dx))$和$\int_{\Bbb R^N} p(x,t)w_2 (x)dx \geq0$，则解是平凡的，即$v=0$几乎处处在$\Bbb R^N \times (0, T)$上.类似的结果也适用于$\Bbb R^N$、$N\geq3$上的MHD方程。
Liouville theorems for the Navier-Stokes equations and applications
研究了Navier-斯托克斯方程的有界古解.这些都是为过去所有时间定义的解决方案。在两个空间维度，我们证明了这样的解决方案要么是常数或时间的函数，这取决于可采纳解的确切定义。一般的三维问题似乎超出了现有的技术，但部分结果可以得到的情况下，轴对称的解决方案。我们将这些结果应用到轴对称解的潜在奇点形成的一些情况。
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.
Liouville type theorems for the fractional Navier-Stokes equations without the integrability condition of velocity in $\mathbb{R}^3$
本文在速度无需可积性条件下，利用Lizorkin乘子定理和Coifman-McIntosh-Meyer型交换子估计，证明了三维分数阶Navier-Stokes方程在无穷远处一致趋于非零常向量的光滑解必为平凡解。
Uniqueness of the trivial solution (the zero solution) for the steady-state Navier-Stokes equations is an interesting problem who has known several recent contributions. These results are also known as the Liouville type problem for the steady-state Navier-Stokes equations. In the setting of the $L^p-$ spaces, when $3\leq p \leq 9/2$ it is known that the trivial solution of these equations is the unique one. In this note, we extend this previous result to other values of the parameter $p$. More precisely, we prove that the velocity field must be zero provided that it belongs to the $L^p -$ spa…
Liouville type of theorems for the Euler and the Navier-Stokes equations
证明了Navier-Stokes方程和Euler方程的弱解的Liouville型定理。特别地，如果压力满足$ p\in L^1 (0,T; L^1 (\Bbb R^N))$与$\int_{\Bbb R^N} p(x,t)dx \geq 0$，则相应的速度应该是微不足道的，即$v=0$对$\Bbb R^N \times (0,T)$。特别地，当$p\in L^1 (0,T; \mathcal{H}^1 (\Bbb R^N))$时就是这种情况，其中$\mathcal{H}^1 (\Bbb R^N)$是Hardy空间。另一方面，如果$p\in L^1 (0,T; L^1 (\Bbb R^N))$与$\int_{\Bbb R^N} p(x,t)dx <0$，我们在每个分量上有能量均分。类似的结果也适用于磁流体动力学方程。
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。
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。
We investigate existence, Liouville type theorems and regularity results for the 3D stationary and incompressible fractional Navier-Stokes equations: in this setting the usual Laplacian is replaced by its fractional power $(-Δ)^{\fracα{2}}$ with $0<α<2$. By applying a fixed point argument, weak solutions can be obtained in the Sobolev space $\dot{H}^{\fracα{2}}(\mathbb{R})$ and if we add an extra integrability condition, stated in terms of Lebesgue spaces, then we can prove for some values of $α$ that the zero function is the unique smooth solution. The additional integrability condition is al…
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。
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。
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 …
Introducing a new notion of generalized suitable weak solutions, we first prove validity of the energy inequality for such a class of weak solutions to the Navier-Stokes equations in the whole space $\mathbb{R}^n$. Although we need certain growth condition on the pressure, we may treat the class even with infinite energy quantity except for the initial velocity. We next handle the equation for vorticity in 2D unbounded domains. Under a certain condition on the asymptotic behavior at infinity, we prove that the vorticity and its gradient of solutions are both globally square integrable. As thei…
In this paper we prove a Liouville type theorem for generalized stationary Navier-Stokes systems in $\Bbb R^3$, which model non-Newtonian fluids, where the Laplacian term $Δu$ is replaced by the corresponding non linear operator $\bA_p( u)=\nabla \cdot ( |\bD(u)|^{p-2} \bD(u))$ with $ \bD(u) = \frac{1}{2} (\nabla u + (\nabla u)^{ \top})$, $3/2<p< 3$. In the case $3/2< p\le 9/5$ we show that a suitable weak solution $u\in W^{1, p}(\Bbb R^3)$ satisfying $ \liminf_{R \rightarrow \infty} |u_{ B(R)}| =0$ is trivial, i.e. $u\equiv 0$. On the other hand, for $9/5<p<3$ we impose the condition for the …
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 forced Euler equations and the Navier-Stokes equations
本文研究了$\Bbb R^N$中定常不可压Euler方程带力解的Liouville型性质。如果我们假设力的“单符号条件”，那么我们可以证明一个$C^1 (\Bbb R^N)$解$(v,p)$与$|v|^2+ |p|\in L^{\frac{q}{2}}(\Bbb R^N)$，$q\in (\frac{3N}{N-1}, \infty)$是平凡的，$v=0$。对于定常Navier-斯托克斯方程的解，满足$v(x)\to 0$作为$|x|\to \infty$，条件$\int_{\Bbb R^3} |Δv|^{\frac65} dx<\infty$比重要的D-条件$\int_{\Bbb R^3} |\nabla v|^2 dx <\infty$更强，但两者具有相同的标度性质，这意味着$v=0$。在附录中我们直接利用自相似欧拉方程重新证明了定理1.1（\cite{cha0}）。
Liouville-type theorems for steady Navier-Stokes system under helical symmetry or Navier boundary conditions
本文证明定常 Navier-Stokes 方程组在螺旋对称及板层 Navier 边界条件下的 Liouville 型定理，关键在于利用 Saint-Venant 型估计刻画非平凡解 Dirichlet 积分的增长。
In this paper, we consider the Liouville property for ancient solutions of the incompressible Navier-Stokes equations. In 2D and the 3D axially symmetric case without swirl, we prove sharp Liouville theorems for smooth ancient mild solutions: velocity fields $v$ are constants if vorticity fields satisfy certain condition and $v$ are sublinear with respect to spatial variables, and we also give counterexamples when $v$ are linear with respect to spatial variables. The condition which vorticity fields need to satisfy is $\lim\limits_{|x|\rightarrow +\infty}|w(x,t)|=0$ and $\lim\limits_{r\rightar…
Liouville-type theorems for Axisymmetric solutions to steady Navier-Stokes system in a layer domain
本文针对层状区域中稳态 Navier–Stokes 系统的轴对称解，在无滑移与 Navier 边界条件下，借助 Saint-Venant 型 Dirichlet 积分增长估计，证明了有界解和 D-解在出口宽度增长受限时的 Liouville 型平凡性结论。
A Liouville Theorem for the Axially-symmetric Navier-Stokes Equations
令$v(x, t)= v^r e_r + v^θe_θ+ v^z e_z$为三维不可压轴对称Navier-Stokes方程斯托克斯方程的解。用$b = v^r e_r + v^z e_z$表示径向-轴向矢量场。在$b$上的一般标度不变条件下，证明了量$Γ= r v^θ$在$r = 0$，$t = 0$上是Hölder连续的.作为应用，我们给出了Koch-Nadirashvili-Seregin-Sverak在\cite{KNSS}和Seregin-Sverak在\cite{SS}中关于Liouville性质的一个猜想的部分证明.作为另一个应用，我们证明了如果$b \in L^\infty([0, T], BMO^{-1})$，则$v$是正则的.这为Koch和Tataru在\cite{KochTataru}中提出的关于轴对称情形下Navier-斯托克斯方程的唯一性和正则性的一个开放性问题提供了答案。
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 of Li and Yu (2014) [8].
Liouville-type theorems for the stationary fractional Navier-Stokes equations in $\mathbb{R}^n$
本文在速度场$u$满足可积性条件和分数阶能量的大尺度Morrey型界条件下，证明了稳态分数阶Navier-Stokes方程的Liouville型定理，并推导出在有限分数阶能量条件下$\frac{n}{3} \le α< \frac{n+2}{3}$范围内的Liouville结果。
In this paper we study Liouville properties of smooth steady axially symmetric solutions of the Navier-Stokes equations. First, we provide another version of the Liouville theorem of \cite{kpr15} in the case of zero swirl, where we replaced the Dirichlet integrability condition by mild decay conditions. Then we prove some Liouville theorems under the assumption $\|\f{u_r}{r}{\bf 1}_{\{u_r< -\f 1r\}}\|_{L^{3/2}(\mbR^3)}< C_{\sharp}$ where $C_{\sharp}$ is a universal constant to be specified. In particular, if $u_r(r,z)\geq -\f1r$ for $\forall (r,z)\in[0,\oo)\times\mbR$, then ${\bf u}\equiv 0$. …
Liouville Theorems Above the Critical $9/2$ Threshold for Stationary Navier-Stokes Equations
本文研究了 stationary Navier-Stokes 方程在临界9/2阈值以上的 Liouville 型定理，证明了在更弱的全局积分性条件下，解的平凡性可以成立，并展示了局部 Liouville 定理的结论。
Liouville-type theorems for the steady incompressible Navier-Stokes system are investigated for solutions in a three-dimensional slab with either no-slip boundary conditions or periodic boundary conditions. When the no-slip boundary conditions are prescribed, we prove that any bounded solution is trivial if it is axisymmetric or $ru^r$ is bounded, and that general three-dimensional solutions must be Poiseuille flows when the velocity is not big in $L^\infty$ space. When the periodic boundary conditions are imposed on the slab boundaries, we prove that the bounded solutions must be constant vec…
Geometric refinements of Liouville-type theorems for the stationary Navier--Stokes equations in $\mathbb{R}^3$
该研究针对三维定常Navier–Stokes方程的光滑解，通过改进两个Osgood型准则，得到仅涉及速度与涡量切向相互作用的Liouville型定理，允许次临界增长。
Liouville theorems for the fractional Navier-Stokes equations with arbitrary asymptotic state at infinity
针对三维稳态分数阶Navier-Stokes方程，通过建立精细的Lp估计和频率局部化方法，证明了在不同参数条件下速度场的Liouville定理。
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.
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