The existence of global regular axially symmetric solutions to the Navier-Stokes equations in a bounded cylinder is proved. The main step in the proof is a proof of the Holder continuity of the swirl. This gives a possibility to prove global a priori estimate for regular solutions. Then a local solution can be extended step by step in time.
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.
Axially symmetric solutions to the Navier-Stokes equations in a bounded cylinder are considered. On the boundary the normal component of the velocity and he angular components of the velocity and vorticity are assumed to vanish. If the norm of the initial swirl is sufficiently small, then the regularity of axially symmetric, weak solutions is shown. The key tool is a new estimate for the stream function in certain weighted Sobolev spaces.
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.
A domain in $\mathbb{R}^3$ that touches the $x_3$ axis at one point is found with the following property. For any initial value in a $C^2$ class, the axially symmetric Navier Stokes equations with Navier slip boundary condition has a finite energy solution that stays bounded for any given time, i.e. no finite time blow up of the fluid velocity occurs. The result seems to be the first case where the Navier-Stokes regularity problem is solved beyond dimension 2.
Two main results will be presented in our paper. First, we will prove the regularity of solutions to axially symmetric Navier-Stokes equations under a $log$ supercritical assumption on the horizontally radial component $u^r$ and vertical component $u^z$, accompanied by a $log$ subcritical assumption on the horizontally angular component $u^θ$ of the velocity. Second, the precise Green function for the operator $-(Δ-\frac{1}{r^2})$ under the axially symmetric situation, where $r$ is the distance to the symmetric axis, and some weighted $L^p$ estimates of it will be given. This will serve as a t…
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…
We investigate the decay properties of smooth axially symmetric D-solutions to the steady Navier-Stokes equations. The achievements of this paper are two folds. One is improved decay rates of $u_þ$ and $\na {\bf u}$, especially we show that $|u_þ(r,z)|\leq c\left(\f{\log r}{r}\right)^{\f 12}$ for any smooth axially symmetric D-solutions to the Navier-Stokes equations. These improvement are based on improved weighted estimates of $\om_þ$, integral representations of ${\bf u}$ in terms of $\bm{\om}=\textit{curl }{\bf u}$ and $A_p$ weight for singular integral operators, which yields good decay e…
Smooth solutions to the axially symmetric Navier-Stokes equations obey the following maximum principle:$\|ru_θ(r,z,t)\|_{L^\infty}\leq\|ru_θ(r,z,0)\|_{L^\infty}.$ We first prove the global regularity of solutions if $\|ru_θ(r,z,0)\|_{L^\infty}$ or $ \|ru_θ(r,z,t)\|_{L^\infty(r\leq r_0)}$ is small compared with certain dimensionless quantity of the initial data. This result improves the one in Zhen Lei and Qi S. Zhang \cite{1}. As a corollary, we also prove the global regularity under the assumption that $|ru_θ(r,z,t)|\leq\ |\ln r|^{-3/2},\ \ \forall\ 0<r\leqδ_0\in(0,1/2).$
Let $v$ be the velocity of Leray-Hopf solutions to the axially symmetric three-dimensional Navier-Stokes equations. Under suitable conditions for initial values, we prove the following a priori bound \[ |v(x, t)| \le \frac{C}{r^2} |\ln r|^{1/2}, \]where $r \in (0, 1/2)$ is the distance from $x$ to the z axis, and $C$ is a constant depending only on the initial value. This provides a pointwise upper bound (worst case scenario) for possible singularities while the recent papers \cite{CSTY2} and \cite{KNSS} gave a lower bound. The gap is polynomial order 1 modulo a half log term.
In this paper, we derive asymptotic properties of both the velocity and the vorticity fields to the 3-dimensional axially symmetric Navier-Stokes equations at infinity under the generalized D-solution assumption $\int_{\mathbb{R}^3}|\nabla u|^qdx<\infty$ for $2<q<\infty$. We do not impose any zero or nonzero constant vector asymptotic assumption on the solution at infinity. Our results generalize those in \cite{CJ:2009JMFM,Ws:2018JMFM,CPZ2018} where the authors focused on the case $q=2$ and the velocity field approaches zero at infinity. Meanwhile, when $q\to 2_+$ and the velocity field approa…
We study axially symmetric D-solutions of the 3 dimensional Navier-Stokes equations. The first result is an a priori decay estimate of the velocity for general domains. The second is an a priori decay estimate of the vorticity in $\bR^3$, which improves the corresponding results in the literature. In addition, we prove a similar decay of full 3d solutions except for a small set of angles. Next we turn to D-solutions which are periodic in the third variable and prove vanishing result under a reasonable condition. As a corollary we prove that axially symmetric D-solutions in the slab $\bR^2 \tim…
An old problem asks whether bounded mild ancient solutions of the 3 dimensional Navier-Stokes equations are constants. While the full 3 dimensional problem seems out of reach, in the works \cite{KNSS, SS09}, the authors expressed their belief that the following conjecture should be true. For incompressible axially-symmetric Navier-Stokes equations (ASNS) in three dimensions: \textit{bounded mild ancient solutions are constant}. Understanding of such solutions could play useful roles in the study of global regularity of solutions to the ASNS. In this article, we essentially prove this conjectur…
Let $v$ be the velocity of Leray-Hopf solutions to the axially symmetric three-dimensional Navier-Stokes equations. It is shown that $v$ is regular if the angular velocity $v_θ$ satisfies an integral condition which is critical under the standard scaling. This condition allows functions satisfying \[ |v_θ(x, t)| \le \frac{C}{r |\ln r|^{2+ε}}, \quad r<1/2, \] where $r$ is the distance from $x$ to the axis, $C$ and $ε$ are any positive constants. Comparing with the critical a priori bound \[ |v_θ(x, t)| \le \frac{C}{r}, \qquad 0< r \le 1/2, \]our condition is off by the log factor $|\ln r|^{2+ε}…
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 $\lim\limits_{r\rightar…
We show that the Navier-Stokes as well as a random perturbation of this equation can be derived from a stochastic variational principle where the pressure is introduced as a Lagrange multiplier. Moreover we describe how to obtain corresponding constants of the motion.
In this paper, we rigorously derive the compressible one-fluid Navier-Stokes equation from the scaled compressible two-fluid Navier-Stokes-Maxwell equations locally in time under the assumption that the initial data are well prepared. We justify the singular limit by proving the uniform decay of the error system, which is obtained by elaborate energy estimates.
The three-dimensional Navier-Stokes-$α$ model for fast rotating geophysical fluids is considered. The Navier-Stokes-$α$ model is a nonlinear dispersive regularization of the exact Navier-Stokes equations obtained by Lagrangian averaging and tend to the Navier-Stokes equations as $α\rightarrow 0^+$. We estimate upper bounds for the dimensions of global attractors and study the dependence of the dimensions on the parameter $α$. All the estimates are uniform in $α$, and our estimate of attractor dimensions remain finite when $α\rightarrow 0^+$.
We study Caccioppoli's inequalities of the non-stationary Stokes equations and Navier-Stokes equations. Our analysis is local near boundary and we prove that, in contrast to the interior case, the Caccioppoli's inequalities of the Stokes equations and the Navier-Stokes equations, in general, fail near boundary.
An algebraic upper bound for the decay rate of solutions to the Navier-Stokes and Navier-Stokes-Coriolis equations in the critical space $\dot{H} ^{\frac{1}{2}} (\mathbb{R} ^3)$ is derived using the Fourier Splitting Method. Estimates are framed in terms of the decay character of initial data, leading to solutions with algebraic decay and showing in detail the roles played by the linear and nonlinear parts.
In this paper, we first prove the global well-posedness of a scaled anisotropic Navier-Stokes system and the hydrostatic Navier-Stokes system in a 2-D striped domain with small analytic data in the tangential variable. Then we justify the limit from the anisotropic Navier-Stokes system to the hydrostatic Navier-Stokes system with analytic data.
In this paper, we shall establish the global well-posedness, the space-time analyticity of the Navier-Stokes equations for a class of large periodic data $u_0 \in BMO^{-1}(\mathbb{R}^3)$. This improves the classical result of Koch \& Tataru \cite{koch-tataru}, for the global well-posedness with small initial data $u_0 \in BMO^{-1}(\mathbb{R}^n)$.
We study the stationary Stokes and Navier-Stokes equations with non-homogeneous Navier boundary condition in a bounded domain $Ω\subset\mathbb{R}^{3}$ of class $\mathcal{C}^{1,1}$. We prove existence, uniqueness of weak and strong solutions in $\mathbf{W}^{1,p}(Ω)$ and $\mathbf{W}^{2,p}(Ω)$ for all $1<p<\infty$ considering minimal regularity on the friction coefficient $α$. Moreover, we deduce uniform estimates on the solution with respect to $α$ which enables us to analyze the behavior of the solution when $α\rightarrow \infty$.
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 ${\bf u}\equiv 0$. …
We introduce a framework to justify hydrodynamic limits of the Vlasov-Navier-Stokes system. We specifically study high friction regimes, which take into account the fact that particles of the dispersed phase are light (resp. small) compared to the fluid part, and lead to the derivation of Transport-Navier-Stokes (resp. Inhomogeneous Navier-Stokes) systems.
In the note, a local regularity condition for axisymmetric solutions to the non-stationary 3D Navier-Stokes equations is proven. It reads that axially symmetric energy solutions to the Navier-Stokes equations have no Type I blowups.
Fundamental solutions to the time-periodic Stokes and Oseen linearizations of the Navier-Stokes equations in dimension $n\geq 2$ are investigated. Integrability properties and pointwise estimates are established.
In this paper, we justify the limit from the Navier-Stokes system in a thin domain to the hydrostatic Navier-Stokes/Prandtl system for the convex initial data with Gevrey 9/8 regualrity in $x$.
These notes are dedicated to the analysis of the one-dimensional free-congested Navier-Stokes equations. After a brief synthesis of the results obtained in [4] related to the existence and the asymptotic stability of partially congested profiles associated to the soft congestion Navier-Stokes system, we present a first local well-posedness result for the one-dimensional free-congested Navier-Stokes equations.
An enstrophy cascade is exhibited for the Navier-Stokes equations in physical scales independently of boundary conditions under physically reasonable assumptions on the flow.
In this paper, we study the upper bound of the time decay rate of solutions to the Navier-Stokes equations and generalized Navier-Stokes equations with damping term $|u|^{β-1}u$ ($β>1$) in $\mathbb{R}^3$.
We study 2D Navier-Stokes equations with a constraint on $L^2$ energy of the solution. We prove the existence and uniqueness of a global solution for the constrained Navier-Stokes equation on $\R^2$ and $\T$, by a fixed point argument. We also show that the solution of constrained Navier-Stokes converges to the solution of Euler equation as viscosity $ν$ vanishes.
We prove in this paper a decay estimate for scaling invariant local energy solutions for some toy-models related to the incompressible Navier-Stokes system.
