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.
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.
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.
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.
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 $p> \frac{3}{2}$.
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.
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.
We establish a Liouville type result for a backward global solution to the Navier-Stokes equations in the half plane with the no-slip boundary condition. No assumptions on spatial decay for the vorticity nor the velocity field are imposed. We study the vorticity equations instead of the original Navier-Stokes equations. As an application, we extend the geometric regularity criterion for the Navier-Stokes equations in the three-dimensional half space under the no-slip boundary condition.
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.
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.
We prove several Liouville type results for stationary solutions of the $d$-dimensional compressible Navier-Stokes equations. In particular, we show that when the dimension $d \geqslant 4$, the natural requirements $ρ\in L^{\infty} (\mathbbm{R}^d)$, $v \in \dot{H}^1 (\mathbbm{R}^d)$ suffice to guarantee that the solution is trivial. For dimensions $d=2,3$, we assume the extra condition $v \in L^{\frac{3d}{d-1}}(\mathbb R^d)$. This improves a recent result of Chae (2012).
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.
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.
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…
This paper has been withdrawn by the author due to the fact that the negative sign of the exponent of $R$ in (2.20) is not allowed by the second inequality of (2.2), and thus the desired vanishing in (2.20) could not be obtained by this method.
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 $v=0$.
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_\…
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…
We study Liouville type of theorems for the Navier-Stokes and the Euler equations on $\Bbb R^N$, $N\geq 2$. Specifically, we prove that if a weak solution $(v,p)$ satisfies $|v|^2 +|p| \in L^1 (0,T; L^1(\Bbb R^N, w_1(x)dx))$ and $\int_{\Bbb R^N} p(x,t)w_2 (x)dx \geq0$ for some weight functions $w_1(x)$ and $w_2 (x)$, then the solution is trivial, namely $v=0$ almost everywhere on $\Bbb R^N \times (0, T)$. Similar results hold for the MHD Equations on $\Bbb R^N$, $N\geq3$.
We study bounded ancient solutions of the Navier-Stokes equations. These are the solutions which are defined for all past time. In two space dimensions we prove that such solutions are either constant or functions of time only, depending on the exact definition of admissible solutions. The general three dimensional problem seems to be out of reach of existing techniques, but partial results can be obtained in the case of axi-symmetric solutions. We apply these results to some scenarios of potential singularity formation for axi-symmetric solutions.
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…
We prove Liouville type of theorems for weak solutions of the Navier-Stokes and the Euler equations. In particular, if the pressure satisfies $ p\in L^1 (0,T; L^1 (\Bbb R^N))$ with $\int_{\Bbb R^N} p(x,t)dx \geq 0$, then the corresponding velocity should be trivial, namely $v=0$ on $\Bbb R^N \times (0,T)$. In particular, this is the case when $p\in L^1 (0,T; \mathcal{H}^1 (\Bbb R^N))$, where $\mathcal{H}^1 (\Bbb R^N)$ the Hardy space. On the other hand, we have equipartition of energy over each component, if $p\in L^1 (0,T; L^1 (\Bbb R^N))$ with $\int_{\Bbb R^N} p(x,t)dx <0$. Similar results h…
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…
Asymptotic behavior and Liouville-type theorems for axisymmetric stationary Navier-Stokes equations outside of an infinite cylinder with a periodic boundary condition
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 …
In this paper we study the Liouville-type properties for solutions to the steady incompressible Euler equations with forces in $\Bbb R^N$. If we assume "single signedness condition" on the force, then we can show that a $C^1 (\Bbb R^N)$ solution $(v,p)$ with $|v|^2+ |p|\in L^{\frac{q}{2}}(\Bbb R^N)$, $q\in (\frac{3N}{N-1}, \infty)$ is trivial, $v=0$. For the solution of of the steady Navier-Stokes equations, satisfying $v(x)\to 0$ as $|x|\to \infty$, the condition $\int_{\Bbb R^3} |Δv|^{\frac65} dx<\infty$, which is stronger than the important D-condition, $\int_{\Bbb R^3} |\nabla v|^2 dx <\in…
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…
Let $v(x, t)= v^r e_r + v^θe_θ+ v^z e_z$ be a solution to the three-dimensional incompressible axially-symmetric Navier-Stokes equations. Denote by $b = v^r e_r + v^z e_z$ the radial-axial vector field. Under a general scaling invariant condition on $b$, we prove that the quantity $Γ= r v^θ$ is Hölder continuous at $r = 0$, $t = 0$. As an application, we give a partial proof of a conjecture on Liouville property by Koch-Nadirashvili-Seregin-Sverak in \cite{KNSS} and Seregin-Sverak in \cite{SS}. As another application, we prove that if $b \in L^\infty([0, T], BMO^{-1})$, then $v$ is regular. Th…
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].
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-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…
Liouville theorems for the fractional Navier-Stokes equations with arbitrary asymptotic state at infinity
Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education)（计算与随机数学教育部重点实验室）；School of Mathematics and Statistics, Hunan Normal University（湖南师范大学数学与统计学院）
In this paper we study the Liouville type properties for solutions to the steady incompressible Navier-Stoks equations in $\mathbf{R}^{3}$. It is shown that any solution to the steady Navier-Stokes equations in $\mathbf{R}^{3}$ with finite Dirichlet integral and vanishing velocity field at far fields must be trivial. This solves an open problem. The key ingredients of the proof include a Hodge decomposition of the energy-flux and the observation that the square of the deformation matrix lies in the local Hardy space. As a by-product, we also obtain a Liouville type theorem for the steady densi…
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.
