Related papers: Liouville Type Theorem for Stationary Navier-Stoke…
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}…
This paper studies the stability of a stationary solution of the Navier-Stokes system with a constant velocity at infinity in an exterior domain. More precisely, this paper considers the stability of the Navier-Stokes system governing the…
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…
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 this paper, we investigate Liouville-type theorems for stationary solutions to the shear thickening fluid equations in a slab. We show that the axisymmetric solution must be trivial if its local $L^\infty$-norm grows mildly as the radius…
In this paper, we establish two major classes of Liouville type results for the three-dimensional stationary tropical climate model. The first class is obtained under the assumptions imposed on $u,v,\theta$ whereas the second one relies on…
In this paper, we study the stationary magnetohydrodynamics system in $\mathbb{R}^2\times\mathbb{T}$. We prove trivialness of D-solutions (the velocity field $u$ and the magnetic field $h$) when they are swirl-free. Meanwhile, this…
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…
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.
We consider the stationary (time-independent) Navier-Stokes equations in the whole threedimensional space, under the action of a source term and with the fractional Laplacian operator (--$\Delta$) $\alpha$/2 in the diffusion term. In the…
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…
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…
We study the stationary Navier--Stokes equations in the region between two rotating concentric cylinders. We first prove that, under the small Reynolds number, if the fluid is axisymmetric and if its velocity is sufficiently small in the…
A sufficient condition of regularity for solutions to the Navier-Stokes equations is proved. It generalizes the so-called $L_{3,\infty}$-case.
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 establish the existence of a uniformly bounded $ C^\infty $ solution of the Navier-Stokes equations on $\mathbb{R}^3 x\ [0, \infty) $ without external forces or boundaries for a divergence free initial condition $ u_o \in \cap_m H^m $…
It is known that the three dimensional Navier-Stokes system for an incompressible fluid in the whole space has a one parameter family of explicit stationary solutions, which are axisymmetric and homogeneous of degree -1. We show that these…
In this paper, we investigate the global regularity to 3-D inhomogeneous incompressible Navier-Stokes system with axisymmetric initial data which does not have swirl component for the initial velocity. We first prove that the $L^\infty$…
In this paper, we investigate the Liouville-type theorems for axisymmetric solutions to steady Navier-Stokes system in a layer domain. The both cases for the flows supplemented with no-slip boundary and Navier boundary conditions are…
We provide an example of a smooth, divergence-free $\nabla \cdot \vec{u}(\vec{x})=0$ velocity vector field $\vec{u}(\vec{x})$ for incompressible fluid occupying all of $R^{3}$ space, and smooth vector field $\vec{f}(\vec{x}, t)$ for which…