In this paper, the existence and uniqueness of strong axisymmetric solutions with large flux for the steady Navier-Stokes system in a pipe are established even when the external force is also suitably large in $L^2$. Furthermore, the exponential convergence rate at far fields for the arbitrary steady solutions with finite $H^2$ distance to the Hagen-Poiseuille flows is established as long as the external forces converge exponentially at far fields. The key point to get the existence of these large solutions is the refined estimate for the derivatives in the axial direction of the stream function and the swirl velocity, which exploits the good effect of the convection term. An important observation for the asymptotic behavior of general solutions is that the solutions are actually small at far fields when they have finite $H^2$ distance to the Hagen-Poiseuille flows. This makes the estimate for the linearized problem play a crucial role in studying the convergence of general solutions at far fields.
In this paper we prove a global well-posedness result for tridimensional Navier-Stokes-Boussinesq system with axisymmetric initial data. This system couples Navier-Stokes equations with a transport equation governing the density.
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.
In the note, a new regularity condition for axisymmetric solutions to the non-stationary 3D Navier-Stokes equations is proven. It is slightly supercritical.
We study in this paper the axisymmetric 3-D Navier-Stokes system where the horizontal viscosity is zero. We prove the existence of a unique global solution to the system with initial data of Yudovitch type
In the paper, a new {\it slightly supercritical} condition, providing {\it local} regularity of axially symmetric solutions to the non-stationary 3D Navier-Stokes equations, is discussed. It generalises almost all known results in the local regularity theory of weak axisymmetric solutions.
In this paper we prove the global well-posedness for a three-dimensional Boussinesq system with axisymmetric initial data. This system couples the Navier-Stokes equation with a transport-diffusion equation governing the temperature. Our result holds uniformly with respect to the heat conductivity coefficient $κ\geq 0$ which may vanish.
We consider a family of 3D models for the axi-symmetric incompressible Navier-Stokes equations. The models are derived by changing the strength of the convection terms in the axisymmetric Navier-Stokes equations written using a set of transformed variables. We prove the global regularity of the family of models in the case that the strength of convection is slightly stronger than that of the original Navier-Stokes equations, which demonstrates the potential stabilizing effect of convection.
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.
We study the asymptotic behavior of axisymmetric solutions with no swirl to the steady Navier-Stokes equations in the outside of the cylinder. We prove an a priori decay estimate of the vorticity under the assumption that the velocity has generalized finite Dirichlet integral. As an application, we obtain a Liouville-type theorem.
A note on potential Type II blowups of axisymmetric solutions to the Navier-Stokes equations
University of Oxford（牛津大学）；St Petersburg Department of Steklov Mathematical Institute, RAS（俄罗斯科学院圣彼得堡斯捷克洛夫数学研究所）；MPI for Mathematics in the Sciences（莱比锡马克斯·普朗克科学促进会数学研究所）
In this note, a new local regularity criteria for the axisymmetric solutions to the 3D Navier--Stokes equations is investigated. It is slightly supercritical and implies an upper bound for the oscillation of $Γ=r u^θ$: for any $0< τ<1$, there exists a constant $c>0$, $$ |Γ(r,x_{3},t)|\leq N e^{-c\, |\ln r|^τ},\ 0<r\leq \frac{1}{4}. $$
We point out some criteria that imply regularity of axisymmetric solutions to Navier-Stokes equations. We show that boundedness of $\|{v_{r}}/{\sqrt{r^3}}\|_{L_2({\rm R}^3\times (0,T))}$ as well as boundedness of $\|{ω_φ}/{\sqrt{r}}\|_{L_2({\rm R}^3\times (0,T))},$ where $v_r$ is the radial component of velocity and $ω_φ$ is the angular component of vorticity, imply regularity of weak solutions.
Global Existence and Incompressible Limit for the Three-Dimensional Axisymmetric Compressible Navier-Stokes Equations with Large Bulk Viscosity and Large Initial Data
In this paper, we study the three-dimensional axisymmetric Navier-Stokes system with nonzero swirl. By establishing a new key inequality for the pair $(\frac{ω^{r}}{r},\frac{ω^θ}{r})$, we get several Prodi-Serrin type regularity criteria based on the angular velocity, $u^θ$. Moreover, we obtain the global well-posedness result if the initial angular velocity $u_{0}^θ$ is appropriate small in the critical space $L^{3}(\R^{3})$. Furthermore, we also get several Prodi-Serrin type regularity criteria based on one component of the solutions, say $ω^3$ or $u^3$.
We consider a family of weights which permit to generalize the Leray procedure to obtain weak suitable solutions of the 3D incom-pressible Navier-Stokes equations with initial data in weighted L 2 spaces. Our principal result concerns the existence of regular global solutions when the initial velocity is an axisymmetric vector field without swirl such that both the initial velocity and its vorticity belong to L 2 ((1 + r 2) -- $γ$ 2 dx), with r = x 2 1 + x 2 2 and $γ$ $\in$ (0, 2).
We classify all (-1)-homogeneous axisymmetric no-swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus the south and north poles, parameterizing them as a four dimensional surface with boundary in appropriate function spaces. Then we establish smoothness properties of the solution surface in the four parameters. The smoothness properties will be used in a subsequent paper where we study the existence of (-1)-homogeneous axisymmetric solutions with non-zero swirl on $\mathbb{S}^2\setminus\{S,N\}$, emanating from the fou…
By applying the delicate \textit{a priori} estimates for the equations of $(Φ,Γ)$, which is introduced in the previous work, we obtain some multi-scale regularity criteria of the swirl component $u^θ$ for the 3D axisymmetric Navier-Stokes equations. In particularly, the solution $\mathbf{u}$ can be continued beyond the time $T$, provided that $u^θ$ satiesfies $$ u^θ \in L^{p}_{T}L^{q_{v}}_{v}L^{q_{h},w}_{h},~~\frac{2}{p}+\frac{1}{q_{v}}+\frac{2}{q_{h}}\leq 1, ~2<q_{h}\leq\infty,~\frac{1}{q_{v}}+\frac{2}{q_{h}}<1. $$
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 $L^\infty$ norm to the quotient of the inhomogeneity by $r,$ namely $a/r\eqdefa\bigl(1/\r-1\bigr)\bigl/r,$ controls the regularity of the solutions. Then we prove the global regularity of such solutions provided that the $L^\infty$ norm of $a_0/r$ is sufficiently small. Finally, with additional assumption that the initial velocity belongs to $L^p$ for some $p\in [1,2),$ w…
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 (x,t)| \le C_*{|t|^{-1/2}} $ or, for some $\e > 0$, $|v (x,t)| \le C_* r^{-1+ε} |t|^{-ε/2}$ for $-T_0\le t < 0$ and $0<C_*<\infty$ allowed to be large. We prove that $v$ is regular at time zero.
This paper is devoted to the study of the Cauchy problem for the stratified Navier-Stokes system in space dimension three. In the first part of the paper, we prove the existence of a unique global solution $(v_ν,ρ_ν)$ for this system with axisymmetric initial data belonging to the Sobolev spaces $H^{s}\times H^{s-2}$ with $s>5/2.$ The bounds of the solution are uniform with respect to the viscosity. In the second part, we analyse the inviscid limit problem. We prove the strong convergence in the space $L^{\infty}_{\text{loc}}(\RR_+; H^{s}\times H^{s-2})$ of the viscous solutions $(v_ν,ρ_ν)_{ν>…
Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $\R^3$ with non-trivial swirl. Such solutions are not known to be globally defined, but it is shown in \cite{MR673830} that they could only blow up on the axis of symmetry. Let $z$ denote the axis of symmetry and $r$ measure the distance to the z-axis. Suppose the solution satisfies the pointwise scale invariant bound $|v (x,t)| \le C_*{(r^2 -t)^{-1/2}} $ for $-T_0\le t < 0$ and $0<C_*<\infty$ allowed to be large, we then prove that $v$ is regular at time zero.
In this paper, we investigate the global well-posedness for the 3-D inhomogeneous incompressible Navier-Stokes system with the axisymmetric initial data. We prove the global well-posedness provided that $$\|\frac{a_{0}}{r}\|_{\infty} \textrm{ and } \|u_{0}^θ\|_{3} \textrm{ are sufficiently small}. $$ Furthermore, if $\mathbf{u}_0\in L^1$ and $ru^θ_0\in L^1\cap L^2$, we have \begin{equation*} \|u^θ(t)\|_{2}^{2}+\langle t\rangle \|\nabla (u^θ\mathbf{e}_θ)(t)\|_{2}^{2}+t\langle t\rangle(\|u_{t}^θ(t)\|_{2}^{2}+\|Δ(u^θ\mathbf{e}_θ)(t)\|_{2}^{2}) \leq C \langle t\rangle^{-\frac{5}{2}},\ \forall\ t>0…
In this paper, we investigate the decay properties of an axisymmetric D-solutions to stationary incompressible Navier-Stokes systems in $\mathbb{R}^3$. We obtain the optimal decay rate $|{\bf u}(x)|\leq \frac{C}{|x|+1}$ for axisymmetric flows without swirl. Furthermore, we find a dichotomy for the decay rates of the swirl component $u_θ$, that is, either $O(\frac{1}{r+1})\leq |u_θ(r,z)|\leq \frac{C\log(r+1)}{(r+1)^{1/2}}$ or $|u_θ(r,z)|\leq \frac{C r}{(ρ+1)^3}$, where $ρ=\sqrt{r^2+z^2}$. In the latter case, we can further deduce that the other two components of the velocity field also attain t…
Motivated by applications to vortex rings, we study the Cauchy problem for the three-dimensional axisymmetric Navier-Stokes equations without swirl, using scale invariant function spaces. If the axisymmetric vorticity is integrable with respect to the two-dimensional measure dr dz, where (r,θ,z) denote the cylindrical coordinates in R^3, we show the existence of a unique global solution, which converges to zero in L^1 norm as time goes to infinity. The proof of local well-posedness follows exactly the same lines as in the two-dimensional case, and our approach emphasizes the similarity between…
$(-1)$-homogeneous axisymmetric no-swirl solutions of three dimensional incompressible stationary Navier-Stokes equations which are smooth on the unit sphere minus the north and south poles have been classified, %as a four parameter family for each viscosity. In this paper we study the vanishing viscosity limit of sequences of these solutions. As the viscosity tends to zero, some sequences of solutions $C^m_{loc}$ converge to solutions of Euler equations on the sphere minus the poles, while for other sequences of solutions, transition layer behaviors occur. For every latitude circle, there are…
Asymptotic behavior and Liouville-type theorems for axisymmetric stationary Navier-Stokes equations outside of an infinite cylinder with a periodic boundary condition
Global axisymmetric solutions and incompressible limit for the 3D isentropic compressible Navier-Stokes equations in annular cylinders with swirl and large initial data
School of Mathematics and Statistics, Southwest University（西南大学数学与统计学院）；School of Mathematics and Statistics, Xiamen University of Technology（厦门理工学院数学与统计学院）
Classification of local $(-1)$-homogeneous axisymmetric solutions of the $3$D stationary Navier-Stokes equations with logarithmic singular-ray behavior
In this paper, we establish several new anisotropic Hardy-Sobolev inequalities in mixed Lebesgue spaces and mixed Lorentz spaces, which covers many known corresponding results. As an application, this type of inequalities allows us to generalize some regularity criteria of the 3D axisymmetric Navier-Stokes equations.
All $(-1)$-homogeneous axisymmetric no-swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus north and south poles have been classified in our earlier work as a four dimensional surface with boundary. In this paper, we establish near the no-swirl solution surface existence, non-existence and uniqueness results on $(-1)$-homogeneous axisymmetric solutions with nonzero swirl which are smooth on the unit sphere minus north and south poles.
We classify all $(-1)-$homogeneous axisymmetric no swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus the south pole, parameterize them as a two dimensional surface with boundary, and analyze their pressure profiles near the north pole. Then we prove that there is a curve of $(-1)-$homogeneous axisymmetric solutions with nonzero swirl, having the same smoothness property, emanating from every point of the interior and one part of the boundary of the solution surface. Moreover we prove that there is no such curve of …
A regularity criterion for the angular component of velocity in the norm $L_\infty(0,T;L_p(Ω)),\;\frac 3 p <1$ in axisymmetric Navier Stokes equations in a cylinder
Landau solutions are special solutions to the stationary incompressible Navier-Stokes equations in the three dimensional space excluding the origin. They are self-similar and axisymmetric with no swirl. In fact, any self-similar smooth solution must be a Landau solution. In the effort of extending this result to the one for the solution class with the pointwise scale-invariant bound $|u(x)|\leq C_0|x|^{-1}$ for some $C_0>0$, we consider axisymmetric discretely self-similar solutions, and investigate the existence of such solution curve emanating from some Landau solution. We prove that the inc…
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 solutions are asymptotically stable under any $L^2$-perturbation.
In this paper, we consider the 3D Navier-Stokes equations in the whole space. We investigate some new inequalities and \textit{a priori} estimates to provide the critical regularity criteria in terms of one directional derivative of the velocity field, namely $\partial_3 \mathbf{u} \in L^p((0,T); L^q(\mathbb{R}^3)), ~\frac{2}{p} + \frac{3}{q} = 2, ~\frac{3}{2}<q\leq 6$. Moreover, we extend the range of $q$ while the solution is axisymmetric, i.e. the axisymmetric solution $\mathbf{m}{u}$ is regular in $(0,T]$, if $ \partial_3 u^3 \in L^p((0,T); L^q(\mathbb{R}^3)), ~\frac{2}{p} + \frac{3}{q} = …
We study an initial-boundary value problem of the three-dimensional Navier-Stokes equations in the exterior of a cylinder $Π=\{x=(x_{h}, x_3)\ |\ |x_{h} |>1\}$, subject to the slip boundary condition. We construct unique global solutions for axisymmetric initial data $u_0\in L^{3}\cap L^{2}(Π)$ satisfying the decay condition of the swirl component $ru^θ_{0}\in L^{\infty}(Π)$.
We derive refined estimates of the Green tensor of the stationary Stokes system in the half space. We then investigate the spatial asymptotics of stationary solutions of the incompressible Navier-Stokes equations in the half space. We also discuss the asymptotics of fast decaying flows in the whole space and exterior domains. In the Appendix we consider axisymmetric self-similar solutions.
In this paper, we study the Vlasov-Navier-Stokes system in a 2D pipe with partially absorbing boundary conditions. We show the existence of stationary states for this system near small Poiseuille flows for the fluid phase, for which the kinetic phase is not trivial. We prove the asymptotic stability of these states with respect to appropriately compactly supported perturbations. The analysis relies on geometric control conditions which help to avoid any concentration phenomenon for the kinetic phase.
We study the long-time behaviour of axisymmetric solutions without swirl for the threedimensional Navier-Stokes equations in the whole space. Assuming that the initial vorticity is sufficiently localised, we compute explicitly the leading terms in the asymptotic expansion of the solution, both for the vorticity and the velocity field. In particular, we identify optimal temporal decay rates depending on the spatial localisation of the initial data. Our approach relies on accurate $L^p$-$L^q$ estimates for the linearised evolution equation and its Taylor expansion in self-similar variables.
In this paper, the global strong axisymmetric solutions for the inhomogeneous incompressible Navier-Stokes system are established in the exterior of a cylinder subject to the Dirichlet boundary conditions. Moreover, the vacuum is allowed in these solutions. One of the key ingredients of the analysis is to obtain the ${L^{2}(s,T;L^{\infty}(Ω))}$ bound for the velocity field, where the axisymmetry of the solutions plays an important role.
In this paper, uniqueness and uniform structural stability of Poiseuille flows in an infinitely long pipe with Navier boundary conditions are established for axisymmetric solutions of steady Navier-Stokes system. The crucial point is that the estimate is uniform with respect both the flux of flows and slip coefficient which appeared in Navier boundary conditions. With the aid of special structure of Navier-Stokes system and the refined estimate for some quantities such as radial velocity, the uniqueness and existence of steady solutions of Navier-Stokes system can be obtained even when the ext…
In this paper, we prove the uniqueness and structural stability of Poiseuille flows for axisymmetric solutions of steady Navier-Stokes system supplemented with Navier boundary conditions in a periodic pipe. Moreover, the stability is uniform with respect to both the flux and the slip coefficient of Navier boundary conditions. It is also showed that the non-zero frequency part of the velocity is bounded by a power function of the flux with negative power as long as the flux is suitably large. One of the key ingredients of the analysis is to prove the uniform linear structural stability, where t…
