Related papers: Liouville type theorems for stationary Navier-Stok…
We examine the conditional regularity of the solutions of Navier-Stokes equations in the entire three-dimensional space under the assumption that the data are axially symmetric. We show that if positive part of the radial component of…
In this paper, we consider the conditional regularity of weak solution to the 3D Navier--Stokes equations. More precisely, we prove that if one directional derivative of velocity, say $\partial_3 u,$ satisfies $\partial_3 u \in…
In this paper, we investigate the vanishing viscosity limit for solutions to the Navier-Stokes equations with a Navier slip boundary condition on general compact and smooth domains in $\mathbf{R}^3$. We first obtain the higher order…
An old problem since Leray \cite{Le:1} asks whether homogeneous D solutions of the 3 dimensional Navier-Stokes equation in $\mathbb{R}^3$ or some noncompact domains are 0. In this paper, we give a positive solution to the problem in two…
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…
In this paper, we investigate some priori estimates to provide the critical regularity criteria for incompressible Navier-Stokes equations on $\mathbb{R}^3$ and super critical surface quasi-geostrophic equations on $\mathbb{R}^2$.…
We present a novel approach to the Liouville problem for the stationary Navier-Stokes equations. As an application of our method, we prove conditional Liouville theorems with assumptions on the antiderivative of the velocity that represent…
Smooth solutions of the stationary Navier-Stokes equations in an infinitely long pipe, equipped with the Navier-slip or Navier-Hodge-Lions boundary condition, are considered in this paper. Three main results are presented. First, when…
We consider the three-dimensional incompressible Navier-Stokes equation on the whole space. We observe that this system admits a $L^\infty$ family of global spatial plane wave solutions, which are connected with the two-dimensional…
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 $(u^\theta,b^\theta)$ or…
In the present paper, we prove a sufficient condition of local regularity for suitable weak solutions to the Navier-Stokes equations having axial symmetry. Our condition is an axially symmetric analog of the so-called $L_{3,\infty}$-case in…
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…
Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $\R^3$ with non-trivial swirl. Let $z$ denote the axis of symmetry and $r$ measure the distance to the z-axis. Suppose the solution satisfies either $|v…
In this article we study some Liouville-type theorems for the stationary 3D Navier-Stokes equations. These results are related to the uniqueness of weak solutions for this system under some additional information over the velocity field,…
This paper investigates the existence and regularity of strong solutions to the incompressible Navier-Stokes equations within a bounded domain $\Omega \subset \mathbb{R}^3$, subject to the boundary condition $(u\cdot \vec{n})|_{\partial…
In this paper, we consider the regularity criterion for 3D incompressible Navier-Stokes equations in terms of one directional derivative of the velocity in anisotropic Lebesgue spaces. More precisely, it is proved that u becomes a regular…
We consider the stationary Navier-Stokes equations on the whole plane $\mathbb{R}^2$. We show that for a given small and smooth external force around a radial flow, there exists a classical solution decaying like $|x|^{-1}$. In our result,…
In this paper, we give a brief survey of recent results on axially symmetric Navier-Stokes equations (ASNS) in the following categories: regularity criterion, Liouville property for ancient solutions, decay and vanishing of stationary…
We prove that if $u$ is a suitable weak solution to the three dimensional Navier-Stokes equations from the space $L_{\infty}(0,T;\dot{B}_{\infty,\infty}^{-1})$, then all scaled energy quantities of $u$ are bounded. As a consequence, it is…
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…