对于 $R^3$ 中具有低粘性 $ν>0$ 的不可压缩 Navier-Stokes 方程，我们考虑如下 Cauchy 问题：其初始涡度 $ω_0$ 表示支撑在一个圆上、强度为任意给定值 $Γ$ 的无限细涡丝。解的涡度场 $ω(x,t)$ 在任意正时刻都是光滑的，对应于厚度为 $\sqrt{νt}$ 的涡环，该涡环由于自感应而沿其对称轴平移，这一效应由 Helmholtz 于 1858 年预见，并由 Kelvin 于 1867 年量化。对于小粘性，我们证明在一个大时间区间内，$ω(x,t)$ 能被 $ω_{lin}(x-a(t),t)$ 很好地近似，其中 $ω_{lin}(\cdot,t)=\exp(νtΔ)ω_0$ 是以 $ω_0$ 为初始数据的热方程的解，而 $\dot a(t)$ 是由 Kelvin 公式给出的瞬时速度。这为弱粘性流体中圆形涡丝的双法向运动提供了严格的证明。证明依赖于构造一个精确的近似解，使用了在自相似变量中的微扰展开。为了验证该近似的稳定性，需要排除来自线性化算子中极大平流项的潜在不稳定性。这是通过将 V. I. Arnold 在无粘情况 $ν=0$ 下发展的几何稳定性方法，调整适用于微粘性情况来完成的。结果表明，尽管 Arnold 方法背后的几何结构不再被 $ν> 0$ 时的方程所保持，但相关的二次型在比 Arnold 理论最初使用的更大的子空间上表现良好，并与粘性项产生有利的相互作用。
英文摘要：
For the incompressible Navier-Stokes equations in $R^3$ with low viscosity $ν>0$, we consider the Cauchy problem with initial vorticity $ω_0$ that represents an infinitely thin vortex filament of arbitrary given strength $Γ$ supported on a circle. The vorticity field $ω(x,t)$ of the solution is smooth at any positive time and corresponds to a vortex ring of thickness $\sqrt{νt}$ that is translated along its symmetry axis due to self-induction, an effect anticipated by Helmholtz in 1858 and quantified by Kelvin in 1867. For small viscosities, we show that $ω(x,t)$ is well-approximated on a large time interval by $ω_{lin}(x-a(t),t)$, where $ω_{lin}(\cdot,t)=\exp(νtΔ)ω_0$ is the solution of the heat equation with initial data $ω_0$, and $\dot a(t)$ is the instantaneous velocity given by Kelvin's formula. This gives a rigorous justification of the binormal motion for circular vortex filaments in weakly viscous fluids. The proof relies on the construction of a precise approximate solution, using a perturbative expansion in self-similar variables. To verify the stability of this approximation, one needs to rule out potential instabilities coming from very large advection terms in the linearized operator. This is done by adapting V. I. Arnold's geometric stability methods developed in the inviscid case $ν=0$ to the slightly viscous situation. It turns out that although the geometric structures behind Arnold's approach are no longer preserved by the equation for $ν> 0$, the relevant quadratic forms behave well on larger subspaces than those originally used in Arnold's theory and interact favorably with the viscous terms.
We study the vanishing viscosity limit for the three-dimensional incompressible Navier-Stokes equations in terms of the relative vorticity in the setting of axisymmetric velocity fields without swirl. We show that the weak convergence of relative vorticity to a renormalized solution of the Euler equations, established by Nobili and Seis, can be upgraded to strong convergence.
The current paper establishes the global well-posedness issue for the full viscous MHD equations in the axisymmetric setting. Global solutions are obtained in critical Besov spaces uniformly to the viscosity when the resistivity is fixed in the spirit of \cite{AKH,Hassainia,hz0}. Furthermore, strong convergence in the resolution spaces with a rate of convergence is also studied.
Following the recent papers [9] and [10] by T. Gallay and V. uSverák, in the line of work initiated by H. Feng and V. uSverák in their paper [3], we prove the uniqueness of a solution of the axisymmetric Navier-Stokes equations without swirl when the initial data is a positive linear combination of Dirac masses.
Global Existence and Incompressible Limit for the Three-Dimensional Axisymmetric Compressible Navier-Stokes Equations with Large Bulk Viscosity and Large Initial Data
$(-1)$-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…
In this paper, we are concerned with the global well-posedness of a tri-dimensional MHD system with only vertical viscosity in velocity equation for the large axisymmetric initial data. By making good use of the axisymmetric structure of flow and the maximal smoothing effect of vertical diffusion, we show that $\displaystyle\sup_{2\leq p<\infty}\int_0^t\frac{\|\partial_{z}u(τ)\|_{L^p}^{2}}{p^{3/4}}\,\mathrm{d}τ<\infty$. With this regularity for the vertical first derivative of velocity vector field, we further establish losing estimates for the anisotropy tri-dimensional MHD system to get the …
We consider a model of axisymmetric flows for a free boundary vortex embedded in a statically stable fluid at rest. We identify the boundary of the vortex by solving a variational problem. Then, we reduce the analysis of the dynamics of the vortex to the study of a class of continuity equations for which we construct a solution.
The incompressible Navier-Stokes equations in R^3 are shown to admit a unique axisymmetric solution without swirl if the initial vorticity is a circular vortex filament with arbitrarily large circulation Reynolds number. The emphasis is on uniqueness, as existence has already been established in [10]. The main difficulty which has to be overcome is that the nonlinear regime for such flows is outside of applicability of standard perturbation theory, even for short times. The solutions we consider are archetypal examples of viscous vortex rings, and can be thought of as axisymmetric analogues of…
In this paper, we study nonlinear desingularization of steady vortex rings of three-dimensional incompressible Euler flows. We construct a family of steady vortex rings (with and without swirl) which constitutes a desingularization of the classical circular vortex filament in $\mathbb{R}^3$. The construction is based on a study of solutions to the similinear elliptic problem \begin{equation*} -\frac{1}{r}\frac{\partial}{\partial r}\Big(\frac{1}{r}\frac{\partialψ^\varepsilon}{\partial r}\Big)-\frac{1}{r^2}\frac{\partial^2ψ^\varepsilon}{\partial z^2}=\frac{1}{\varepsilon^2}\left(g(ψ^\varepsilon)…
In this paper, we present strong numerical evidences that the incompressible axisymmetric Euler equations with degenerate viscosity coefficients and smooth initial data of finite energy develop a potential finite-time locally self-similar singularity at the origin. An important feature of this potential singularity is that the solution develops a two-scale traveling wave that travels towards the origin. The two-scale feature is characterized by the scaling property that the center of the traveling wave is located at a ring of radius $O((T-t)^{1/2})$ surrounding the symmetry axis while the thic…
We consider axisymmetric incompressible inviscid flows without swirl in $\mathbb{R}^3$, under the assumption that the axial vorticity is non-positive in the upper half space and odd in the last coordinate, which corresponds to the flow setup for head-on collision of anti-parallel vortex rings. For any such data, we establish monotonicity and infinite growth of the vorticity impulse on the upper half-space. As an application, we achieve infinite growth of Sobolev norms for certain classical/smooth and compactly supported vorticity solutions in $\mathbb{R}^{3}$.
We construct global weak solutions of the Euler equations in an infinite cylinder $Π=\{x\in \mathbb{R}^{3}\ |\ x_h=(x_1,x_2),\ r=|x_h|<1\}$ for axisymmetric initial data without swirl when initial vorticity $ω_{0}=ω^θ_{0}e_θ$ satisfies $ω^θ_{0}/r\in L^{q}$ for $q\in [3/2,3)$. The solutions constructed are Hölder continuous for spatial variables in $\overlineΠ$ if in addition that $ω^θ_{0}/r\in L^{s}$ for $s\in (3,\infty)$ and unique if $s=\infty$. The proof is by a vanishing viscosity method. We show that the Navier-Stokes equations subject to the Neumann boundary condition is globally well-po…
We present a study on the nonlinear dynamics of small long-wave disturbances to the laminar state in non-rotating axisymmetric Poiseuille pipe flows. At high Reynolds numbers, the associated Navier-Stokes equations can be reduced to a set of coupled Korteweg-de Vries-type (KdV) equations that support inviscid and smooth travelling waves numerically computed using the Petviashvili method. In physical space they correspond to localized toroidal vortices concentrated near the pipe boundaries (wall vortexons) or that wrap around the pipe axis (centre vortexons), in agreement with the analytical so…
The dynamics along the particle trajectories for the 3D axisymmetric Euler equations are considered. It is shown that if the inflow is rapidly increasing (pushy) in time, the corresponding laminar profile of the incompressible Euler flow is not (in some sense) stable provided that the swirling component is not zero. It is also shown that if the vorticity on the axis is not zero (with some extra assumptions), then there is no steady flow. We can rephrase these instability to an instantaneous blow-up. In the proof, Frenet-Serret formulas and orthonormal moving frame are essentially used.
In this paper, we consider the small viscosity limit problem for the isentropic compressible Navier-Stokes equations in a 2D exterior domain with impermeable boundary conditions , and the corresponding Euler equations have vortex sheet solutions.We obtain that away from the boundary and the contact discontinuous the isentropic compressible viscous flow can be approximated by the corresponding inviscid flow, near the boundary (the contact discontinuous) there is a boundary layer (vortex layer)for the angular velocity in the leading order expansion of solution, while the radial velocity and the …
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
There are a few examples of solutions to the incompressible Euler equations which are piecewise smooth with a discontinuity of the tangential velocity across a hypersurface evolving in time: the so-called vortex sheets. An important open problem is to determine whether or not these solutions can be obtained as zero viscosity limits of the incompressible Navier-Stokes solutions in the energy space. In this paper we establish a couple of sufficient conditions similar to the one obtained by Kato in [T.~Kato. Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary. Semi…
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）
We construct traveling wave solutions to the 3d Euler equations by axisymmetric Beltrami fields with a non-constant proportionality factor. They form a vortex ring with nested invariant tori consisting of level sets of the proportionality factor.
Spiral structure is one of the most common structures in the nature flows. A general steady spiral solution of incompressible inviscid axisymmetric flow was obtained analytically by applying separation of variables to the 3D Euler equations. The solution, depending on 3 parameters, describes the spiral path of the fluid material element on the Bernoulli surface, whereas some new exact solutions were obtained to be bounded within the whole region. The first one is a continued typhoon-like vortex solution, where there are two intrinsic length scales. One is the radius of maximum circular velocit…
We study the time evolution of a viscous incompressible fluid with axial symmetry without swirl, when the initial vorticity is very concentrated in $N$ disjoint rings. We show that in a suitable joint limit, in which both the thickness of the rings and the viscosity tend to zero, the vorticity remains concentrated in $N$ disjointed rings, each one of them performing a simple translation along the symmetry axis with constant speed.
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…
We study vanishing viscosity solutions to the axisymmetric Euler equations with (relative) vorticity in $L^p$ with $p>1$. We show that these solutions satisfy the corresponding vorticity equations in the sense of renormalized solutions. Moreover, we show that the kinetic energy is preserved provided that $p>3/2$ and the vorticity is nonnegative and has finite second moments.
We consider Navier-Stokes equations for compressible viscous fluids in the one-dimensional case with general viscosity coefficients. We prove the existence of global weak solution when the initial momentum $ρ_0 u_0$ belongs to the set of the finite measure ${\cal M}(\mathbb{R})$ and when the initial density $ρ_0$ is in the set of bounded variation functions $BV(\mathbb{R})$. In particular it allows to deal with initial momentum which are Dirac masses and initial density which admit shocks. We can observe in particular that this type of initial data have infinite energy. Furthermore we show tha…
In this paper, we prove the global well-posedness for the three-dimensional magnetohydrodynamics (MHD) equations with zero viscosity and axisymmetric initial data. First, we analyze the problem corresponding to the Sobolev regularities $ H^s\times H^{s-2}$, with $ s>5/2$. Second, we address the same problem but for the Besov critical regularities $ B_{p,1}^{3/p+1}\times B_{1,p}^{3/p-1}$, $2\leq p\leq \infty$. This case turns out to be more subtle as the Beale-Kato-Majda criterion is not known to be valid for rough regularities.
In this paper, we are concerned with the vanishing viscosity problem for the three-dimensional Navier-Stokes equations with helical symmetry, in the whole space. We choose viscosity-dependent initial $\bu_0^ν$ with helical swirl, an analogue of the swirl component of axisymmetric flow, of magnitude $\mathcal{O}(ν)$ in the $L^2$ norm; we assume $\bu_0^ν\to \bu_0$ in $H^1$. The new ingredient in our analysis is a decomposition of helical vector fields, through which we obtain the required estimates.
We study desingularization of steady vortex rings in three-dimensional axisymmetric incompressible Euler fluids with swirl. Using the variational method, we construct a two-parameter family of steady vortex rings, which constitute a desingularization of the classical circular vortex filament, in several kinds of domains. The precise localization of the asymptotic singular vortex filament is shown to depend on the circulation and the velocity at far fields of the vortex ring and the geometry of the domains. We also discuss other qualitative and asymptotic properties of these vortices.
In this paper, we are concerned with the tridimensional anisotropic Boussinesq equations which can be described by {equation*} {{array}{ll} (\partial_{t}+u\cdot\nabla)u-κΔ_{h} u+\nabla Π=ρe_{3},\quad(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{3}, (\partial_{t}+u\cdot\nabla)ρ=0, \text{div}u=0. {array}. {equation*} Under the assumption that the support of the axisymmetric initial data $ρ_{0}(r,z)$ does not intersect the axis $(Oz)$, we prove the global well-posedness for this system with axisymmetric initial data. We first show the growth of the quantity $\fracρr$ for large time by taking advantage …
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>…
For the axisymmetric incompressible Euler equations, we prove linear in time filamentation near Hill's vortex: there exists an arbitrary small outward perturbation growing linearly for all times. This is based on combining the recent nonlinear orbital stability obtained in [13] with a dynamical bootstrapping scheme for particle trajectories. These results rigorously confirm numerical simulations by Pozrikidis [45] in 1986.
Whether, in the presence of a boundary, solutions of the Navier-Stokes equations converge to a solution to the Euler equations in the vanishing viscosity limit is unknown. In a seminal 1983 paper, Tosio Kato showed that the vanishing viscosity limit is equivalent to having sufficient control of the gradient of the Navier-Stokes velocity in a boundary layer of width proportional to the viscosity. In a 2008 paper, the present author showed that the vanishing viscosity limit is equivalent to the formation of a vortex sheet on the boundary. We present here several observations that follow on from …
Vortex Stretching in the Navier-Stokes Equations and Information Dissipation in Diffusion Models: A Reformulation from a Partial Differential Equation Viewpoint
Graduate School of Economics, Hitotsubashi University（早稻田大学经济学研究科）
We study the three-dimensional Navier-Stokes equations in the presence of the axisymmetric linear strain, where the strain rate depends on time in a specific manner. It is known that the system admits solutions which blow up in finite time and whose profiles are in a backward self-similar form of the familiar Burgers vortices. In this paper it is shown that the existing stability theory of the Burgers vortex leads to the stability of these blow-up solutions as well. The secondary blow-up is also observed when the strain rate is relatively weak.
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Department of Mathematical Sciences and RIM, Seoul National University（首尔大学数学科学系及RIM）；School of Mathematics, Korea Institute for Advanced Study（韩国高等研究院数学学院）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
The two-dimensional Navier-Stokes equations are rewritten as a system of coupled nonlinear ordinary differential equations. These equations describe the evolution of the moments of an expansion of the vorticity with respect to Hermite functions and of the centers of vorticity concentrations. We prove the convergence of this expansion and show that in the zero viscosity and zero core size limit we formally recover the Helmholtz-Kirchhoff model for the evolution of point-vortices. The present expansion systematically incorporates the effects of both viscosity and finite vortex core size. We also…
We establish the inviscid limit of the incompressible Navier-Stokes equations on the whole plane $\mathbb{R}^2$ for initial data having vorticity as a superposition of point vortices and a regular component. In particular, this rigorously justifies the vortex-wave system from the physical Navier-Stokes flows in the vanishing viscosity limit, a model that was introduced by Marchioro and Pulvirenti in the early 90s to describe the dynamics of point vortices in a regular ambient vorticity background. The proof rests on the previous analysis of Gallay in his derivation of the vortex-point system.
We say that the vanishing viscosity limit holds in the classical sense if the velocity for a solution to the Navier-Stokes equations converges in the energy norm uniformly in time to the velocity for a solution to the Euler equations. We prove, for a bounded domain in dimension 2 or higher, that the vanishing viscosity limit holds in the classical sense if and only if a vortex sheet forms on the boundary.
