相关论文: Liouville type of theorems with weights for the Na…
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 $\rho \in L^{\infty}…
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…
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…
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 $\mathbf{A}_p(u) = \mathrm{div} ( | \mathbf{D}(u) |^{p-2} \mathbf{D}(u) )$,…
In the paper, a Liouville theorem for mild bounded ancient solutions to the 2D Navier-Stokes equations in half space has been proven.
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.
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…
In this paper, the Liouville-type theorems for the steady Navier-Stokes system are investigated. First, we prove that any bounded smooth helically symmetric solution in $\mathbb{R}^3$ must be a constant vector. Second, for steady…
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 $\omega={\rm curl}\,v \in L^q (\Bbb R^3) $ for some $\frac32 \leq q<…
We examine the weighted elliptic system \begin{equation*} \begin{cases} -\Delta u=(1+|x|^2)^{\frac{\alpha}{2}} v,\\ -\Delta v=(1+|x|^2)^{\frac{\alpha}{2}} u^p, \end{cases} \quad \mbox{in}\;\ \mathbb{R}^N, \end{equation*}where $N \ge 5$,…
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…
We study regularity criteria for the $d$-dimensional incompressible Navier-Stokes equations. We prove in this paper that if $u\in L_\infty^tL_{d}^x((0,T)\times {\mathbb R}^d)$ is a Leray-Hopf weak solution, then $u$ is smooth and unique in…
We establish Liouville type theorems for elliptic systems with various classes of non-linearities on $\mathbb{R}^N$. We show among other things, that a system has no semi-stable solution in any dimension, whenever the infimum of the…
Motivated by the classification of solutions of harmonic functions, we investigate Liouville type theorems for the fractional Navier-Stokes equations in $\mathbb{R}^3$ under some conditions on the boundedness of fractional derivatives. We…
We establish various results concerning the uniqueness of zero velocity solutions for the static barotropic Navier--Stokes system. Some of them can be seen as Liouville-type theorems for problems in unbounded physical space.
In this paper, we establish Liouville-type theorems for the steady compressible Navier-Stokes system. Assuming a smooth solution \(u \in L^p(\mathbb{R}^3)\), \(3 \le p \le \frac{9}{2}\), with bounded density, one obtains \(u \equiv0\). This…
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 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.…
In this paper, we prove Liouville type theorems for stable solutions to the weighted fractional Lane-Emden system \begin{align*} (-\Delta)^s u = h(x)v^p,\quad (-\Delta)^s v= h(x)u^q, \quad u,v>0\quad \mbox{in }\;\mathbb{R}^N, \end{align*}…
Let $r,s \in [2,\infty]$ and consider the Navier-Stokes equations on $\mathbb{R}^3$. We study the following two questions for suitable $s$-homogeneous Banach spaces $X \subset \mathcal{S}'$: does every $u_0 \in L^2_\sigma$ have a weak…