On the global existence for the axisymmetric Euler equations

关于四维及更高维度欧拉方程轴对称、无旋解的正则性

On the regularity of axisymmetric, swirl-free solutions of the Euler equation in four and higher dimensions

研究四维及更高维度欧拉方程轴对称无旋解的正则性，证明在四维及以上维度下，此类解可能在有限时间内形成奇点，且随着维度增加，引发奇点的条件变弱。

On slowly rotating axisymmetric solutions of the Euler-Poisson equations

We construct stationary axisymmetric solutions of the Euler-Poisson equations, which govern the internal structure of polytropic gaseous stars, with small constant angular velocity when the adiabatic exponent $γ$ belongs to $(\frac65,\frac32]$. The problem is formulated as a nonlinear integral equation, and is solved by iteration technique. By this method, not only we get the existence, but also we clarify properties of the solutions such as the physical vacuum condition and oblateness of the star surface.

On slowly rotating axisymmetric solutions of the Einstein-Euler equations

In recent works we have constructed axisymmetric solutions to the Euler-Poisson equations which give mathematical models of slowly uniformly rotating gaseous stars. We try to extend this result to the study of solutions of the Einstein-Euler equations in the framework of the general theory of relativity. Although many interesting studies have been done about axisymmetric metric in the general theory of relativity, they are restricted to the region of the vacuum. Mathematically rigorous existence theorem of the axisymmetric interior solutions of the stationary metric corresponding to the energy…

3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler-Poisson system

We address the structural stability of 3-D axisymmetric subsonic flows with nonzero swirl for the steady compressible Euler-Poisson system in a cylinder supplemented with non small boundary data. A special Helmholtz decomposition of the velocity field is introduced for 3-D axisymmetric flow with a nonzero swirl(=angular momentum density) component. With the newly introduced decomposition, a quasilinear elliptic system of second order is derived from the elliptic modes in Euler-Poisson system for subsonic flows. Due to the nonzero swirl, the main difficulties lie in the solvability of a singula…

Global Axisymmetric Euler Flows with Rotation

We construct a class of global, dynamical solutions to the 3d Euler equations near the stationary state given by uniform "rigid body" rotation. These solutions are axisymmetric, of Sobolev regularity, have non-vanishing swirl and scatter linearly, thanks to the dispersive effect induced by the rotation. To establish this, we introduce a framework that builds on the symmetries of the problem and precisely captures the anisotropic, dispersive mechanism due to rotation. This enables a fine analysis of the geometry of nonlinear interactions and allows us to propagate sharp decay bounds, which is c…

Euler 和 Navier-Stokes 方程具有旋流的轴对称流动

Axisymmetric flows with swirl for Euler and Navier-Stokes equations

本文研究轴对称 Euler 和 Navier-Stokes 方程的定常自相似解，证明无滑移间断解的不存在性，并证明 Navier-Stokes 解在零粘性极限下收敛到 Euler 解。

On Vorticity Gradient Growth for the Axisymmetric 3D Euler Equations Without Swirl

We consider the 3D axisymmetric Euler equations without swirl on some bounded axial symmetric domains. In this setting, well-posedness is well known due to the essentially 2D geometry. The quantity $ω^θ/r$ plays the role of vorticity in 2D. First, we prove that the gradient of $ω^θ/r$ can grow at most double exponentially with improving a priori bound close to the axis of symmetry. Next, on the unit ball, we show that at the boundary, one can achieve double exponential growth of the gradient of $ω^θ/r$.

On the blow-up problem for the axisymmetric 3D Euler equations

In this paper we study the finite time blow-up problem for the axisymmetric 3D incompressible Euler equations with swirl. The evolution equations for the deformation tensor and the vorticity are reduced considerably in this case. Under the assumption of local minima for the pressure on the axis of symmetry with respect to the radial variations we show that the solution blows-up in finite time. If we further assume that the second radial derivative vanishes on the axis, then system reduces to the form of Constantin-Lax-Majda equations, and can be integrated explicitly.

On the Finite-Time Blowup of a 1D Model for the 3D Axisymmetric Euler Equations

In connection with the recent proposal for possible singularity formation at the boundary for solutions of 3d axi-symmetric incompressible Euler's equations (Luo and Hou, 2013), we study models for the dynamics at the boundary and show that they exhibit a finite-time blow-up from smooth data.

On the Global Existence for the Axisymmetric Euler-Boussinesq System in Critical Besov Spaces

This paper is devoted to the global existence and uniqueness results for the three-dimensional Boussinesq system with axisymmetric initial data $v^{0}{\in}B_{2,1}^{5/2}(\RR^3)$ and$ $ρ^{0}{\in}B_{2,1}^{1/2}(\RR^3)\cap L^{p}(\RR^3)$ with $p>6.$ This system couples the incompressible Euler equations with a transport-diffusion equation governing the density. In this case the Beale-Kato-Majda criterion is not known to be valid and to circumvent this difficulty we use in a crucial way some geometric properties of the vorticity.

轴对称欧拉流中的源驱动转变

Source-Driven Transitions in Axisymmetric Euler Flow

本文研究轴对称欧拉流中有限时间的源驱动转变，通过精确恒等式和区间算术证明固定标签收缩与水平集建立，并列出未解决的全局步骤。

Lagrangian aspects of the axisymmetric Euler equation

In this paper we are interested in geometric aspects of blowup in the axisymmetric 3D Euler equations with swirl on a cylinder. Writing the equations in Lagrangian form for the flow derivative along either the axis or the boundary and imposing oddness on the vertical component of the flow, we extend some blowup criteria due to Chae, Constantin, and Wu related to assumptions on the sign of the pressure Hessian. In addition we give a geometric interpretation of the results, both in terms of the local geometry along trajectories and in terms of the Riemannian geometry of the volume-preserving dif…

On Finite Time Singularity and Global Regularity of an Axisymmetric Model for the 3D Euler Equations

We investigate the large time behavior of an axisymmetric model for the 3D Euler equations. In \cite{HL09}, Hou and Lei proposed a 3D model for the axisymmetric incompressible Euler and Navier-Stokes equations with swirl. This model shares many properties of the 3D incompressible Euler and Navier-Stokes equations. The main difference between the 3D model of Hou and Lei and the reformulated 3D Euler and Navier-Stokes equations is that the convection term is neglected in the 3D model. In \cite{HSW09}, the authors proved that the 3D inviscid model can develop a finite time singularity starting fr…

Global existence of weak solutions to the 3D incompressible axisymmetric Euler equations without swirl

In this paper, we mainly investigate the tridimensional incompressible axisymmetric Euler equations without swirl in the whole space. Specifically, we prove the global existence of weak solutions if the swirl component of initial vorticity $w_0^θ$ satisfies that $\frac{w_0^θ}r\in L^1\cap L^p({\Bbb R}^3)$ for some $p>1$. To achieve this goal, we establish the $L_{\rm loc}^{2+α}({\Bbb R}^3)$ estimate of velocity fields for some $α>0$, which is innovative to the best of our knowledge. Our result extends previous work in the literature.

四维空间中轴对称、无旋欧拉方程解的全局正则性

Global regularity for axisymmetric, swirl-free solutions of the Euler equation in four dimensions

本文证明了四维空间中所有光滑轴对称无旋欧拉方程解的全局正则性，改进了对涡旋拉伸项的界，仅需涡旋初始条件满足L^{2,1}条件，从而克服了之前对L^∞条件的限制。

On Rotating Axisymmetric Solutions of the Euler-Poisson Equations

We consider stationary axisymmetric solutions of the Euler-Poisson equations, which govern the internal structure of barotropic gaseous stars. We take the general form of the equation of states which cover polytropic gaseous stars indexed by $6/5<γ<2$ and also white dwarfs. A generic condition of the existence of stationary solutions with differential rotation is given, and the existence of slowly rotating configurations near spherically symmetric equilibria is shown. The problem is formulated as a nonlinear integral equation, and is solved by an application of the infinite dimensional implici…

关于轴对称障碍物后三维亚音速欧拉流的全球存在性

Global three-dimensional subsonic Euler flows past an axisymmetric obstacle with large vorticity

本文研究了轴对称障碍物后三维亚音速欧拉流的存在性，通过强最大原理和连续性论证证明了亚音速解的唯一性，并考虑了大涡量流的结构条件。

扰动圆柱中带外力的三维轴对称欧拉流的跨音速激波

Transonic shocks for three-dimensional axisymmetric Euler flows with an external force in a perturbed cylinder

研究在入流超音速流、出口压力、外力和喷嘴壁扰动下，三维轴对称欧拉系统在圆柱中跨音速激波解的存在性与稳定性，通过修正拉格朗日变换和双曲-椭圆模态分解克服奇异性，并利用代数方程唯一确定激波位置。

不可压缩欧拉方程轴对称爆破解的不稳定性

Instability for Axisymmetric Blow-up Solutions to Incompressible Euler Equations

本文证明不可压缩欧拉方程若存在有限时间爆破，其解在轴对称情形下同样会在爆破时刻附近线性不稳定，并建立了仅依赖涡量环向分量的爆破准则。

Removing Type II singularities off the axis for the 3D axisymmetric Euler equations

We prove local blow-up criterion for smooth axisymmetric solutions to the 3D incompressible Euler equation. If the vorticity satisfies $ \intl_{0}^{t_*} (t_*-t) \| ω(t)\|_{ L^\infty(B(x_{ \ast}, R_0))} dt <+\infty$ for a ball $B(x_{ \ast}, R_0)$ away from the axis of symmetry, then there exists no singularity at $t=t_*$ in the torus $T(x_*, R)$ generated by rotation of the ball $B(x_{ \ast}, R_0)$ around the axis. This implies that possible singularity at $t=t_*$ in the torus $T(x_*, R)$ is excluded if the vorticity satisfies the blow-up rate $ \|ø(t)\|_{L^\infty (T(x_*, R))}= O\left(\frac{1}{…

On Admissible Positions of Transonic Shocks for Steady Euler Flows in a 3-D Axisymmetric Cylindrical Nozzle

This paper concerns with the existence of transonic shocks for steady Euler flows in a 3-D axisymmetric cylindrical nozzle, which are governed by the Euler equations with the slip boundary condition on the wall of the nozzle and a receiver pressure at the exit. Mathematically, it can be formulated as a free boundary problem with the shock front being the free boundary to be determined. In dealing with the free boundary problem, one of the key points is determining the position of the shock front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solut…

On the steady axisymmetric vortex rings for 3-D incompressible Euler flows

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)…

Axisymmetric Solutions of the Euler Equations for Sub-Square Polytropic Gases

We establish rigorously the existence of a three-parameter family of self-similar,globally bounded, and continuous weak solutions in two space dimensions to the compressible Euler equations with axisymmetry for gamma-law polytropic gases with gamma between 1 and 2, including 1. The initial data of these solutions have constant densities and outward-swirling velocities. We use the axisymmetry and self-similarity assumptions to reduce the equations to a system of three ordinary differential equations, from which we obtain detailed structures of solutions besides their existence. These solutions …

3-D axisymmetric transonic shock solutions of the full Euler system in divergent nozzles

We establish the stability of 3-D axisymmetric transonic shock solutions of the steady full Euler system in divergent nozzles under small perturbations of an incoming radial supersonic flow and a constant pressure at the exit of the nozzles. To study 3-D axisymmetric transonic shock solutions of the full Euler system, we use a stream function formulation of the full Euler system for a 3-D axisymmetric flow. We resolve the singularity issue arising in stream function formulations of the full Euler system for a 3-D axisymmetric flow. We develop a new scheme to determine a shock location of a tra…

具有接触间断的轴对称亚音速欧拉流的两相自由边界问题

A Two-Phase Free Boundary Problem for Axisymmetric Subsonic Euler Flows with Contact Discontinuities

研究三维无限长圆柱中稳态可压缩欧拉方程的自由边界问题，通过亥姆霍兹分解和耦合迭代框架，证明存在接触间断分隔两个旋转亚音速欧拉流。

定常Euler-Poisson方程组柱形超音速解的结构稳定性

Structural stability of cylindrical supersonic solutions to the steady Euler-Poisson system

本文研究喷管内柱对称超音速Euler-Poisson流的结构稳定性，分别证明三维势流解和轴对称带涡旋流解的存在唯一性，通过双曲-椭圆系统分析与双层迭代建立非线性稳定性。

An instability mechanism of pulsatile flow along particle trajectories for the axisymmetric Euler equations

The dynamics along the particle trajectories for the 3D axisymmetric Euler equations in an infinite cylinder are considered. It is shown that if the inflow-outflow is rapidly increasing in time, the corresponding laminar profile of the Euler flow is not (in some sense) stable provided that the swirling component is not small. This exhibits an instability mechanism of pulsatile flow. In the proof, Frenet-Serret formulas and orthonormal moving frame are essentially used.

Low Mach number limit for the isentropic Euler system with axisymmetric initial data

This paper is devoted to the study of the low Mach number limit for the isentropic Euler system with axisymmetric initial data without swirl. In the first part of the paper we analyze the problem corresponding to the subcritical regularities, that is $H^s$ {with $s>\frac52$.} Taking advantage of the Strichartz estimates and using the special structure of the vorticity we show that the {lifespan $T_ε$} of the solutions is bounded below by $\log\log\log\frac1ε$, where $ε$ denotes the Mach number. Moreover, we prove that the incompressible parts converge to the solution of the incompressible Eule…

粗糙轴对称欧拉动力学：全局能量紧性与跨越正则性破缺的动力学通道

Rough Axisymmetric Euler Dynamics: Global Energy Compactness and Kinetic Passage Across Regularity Breakdown

该研究建立了粗糙轴对称欧拉流的全局能量紧性，并利用它构造了碰撞动力学方程的流体动力学极限，实现了跨越欧拉正则性破缺的能量守恒延拓。

Stability of Blow Up for a 1D model of Axisymmetric 3D Euler Equation

The question of the global regularity vs finite time blow up in solutions of the 3D incompressible Euler equation is a major open problem of modern applied analysis. In this paper, we study a class of one-dimensional models of the axisymmetric hyperbolic boundary blow up scenario for the 3D Euler equation proposed by Hou and Luo based on extensive numerical simulations. These models generalize the 1D Hou-Luo model suggested in Hou & Luo's paper, for which finite time blow up has been established in the paper by Kyudong Choi, Thomas Y. Hou, Alexander Kiselev, Guo Luo, Vladimir Sverak, Yao Yao. …

扩张喷管中三维轴对称流动的跨音速激波

Transonic shocks for three-dimensional axisymmetric flows in divergent nozzles

本文通过 Helmholtz 分解推导自由边界问题，并利用 Schauder 不动点定理与隐函数定理构造迭代格式，证明了扩张喷管中带跨音速激波的三维轴对称稳态 Euler 系统解在出口压力和上游超音速解扰动下的稳定性。

轴对称欧拉流中的边界双曲包

Boundary Hyperbolic Packets in Axisymmetric Euler Flow

针对轴对称欧拉方程，推导双曲侧壁包的五维格林核展开及估计，得到条件放大准则，表明在特定假设下比较振幅有限时间爆破，但未证明轨迹假设的持续性。

带有外力的定常 Euler 方程的光滑轴对称跨声速无旋流

Smooth axisymmetric transonic irrotational flows to the steady Euler equations with an external force

本文证明了一维定常 Euler 系统带外力光滑跨声速流的存在唯一性，并通过分析 Keldysh 型混合方程建立了轴对称扰动下光滑跨声速无旋流的首个存在唯一性结果。

Global regularity for some axisymmetric Euler flows in $\mathbb{R}^{d}$

We consider axisymmetric Euler flows without swirl in $\mathbb{R}^{d}$ with $d\geq 4$, for which the global regularity of smooth solutions is an open problem. When $d = 4$, we obtain global regularity under the assumption that the initial vorticity satisfies some decay at infinity and is vanishing at the axis. Assuming further that the initial vorticity is of one sign guarantees global regularity for $d\leq 7$.

Axisymmetric Euler-$α$ Equations without Swirl: Existence, Uniqueness, and Radon Measure Valued Solutions

The global existence of weak solutions for the three-dimensional axisymmetric Euler-$α$ (also known as Lagrangian-averaged Euler-$α$) equations, without swirl, is established, whenever the initial unfiltered velocity $v_0$ satisfies $\frac{\nabla \times v_0}{r}$ is a finite Randon measure with compact support. Furthermore, the global existence and uniqueness, is also established in this case provided $\frac{\nabla \times v_0}{r} \in L^p_c(\mathbb{R}^3)$ with $p>{3/2}$. It is worth mention that no such results are known to be available, so far, for the three-dimensional Euler equations of ideal…

A Remark on the Matter-Vacuum Matching Problem for Axisymmetric Metrics Governed by the Einstein-Euler Equations

Axially symmetric stationary metrics governed by the Einstein-Euler equations for slowly rotating perfect fluids have been constructed in an arbitrarily large bounded domain containing the support of the mass density. However the problem of global prolongation of the metric is still open. On the other hand the so called matter-vacuum matching problem, particularly as the source problem for the Kerr metric, has been discussed by several authors. This can be regarded as the approach to the same open problem in the opposite direction. We give a remark on this open problem.

Finite time blowup explicit solution for 3D incompressible Euler equations

This paper construct a family of explicit self-similar blowup axisymmetric solutions for the 3D incompressible Euler equations in R^3. Those singular solutions admit infinite energy.

Mathematical analysis of pulsatile flow, vortex breakdown and instantaneous blow-up for the axisymmetric Euler equations

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.

无旋轴对称欧拉方程的增长估计

Growth estimates for axisymmetric Euler equations without swirl

本文研究了无旋轴对称欧拉方程中解的增长行为，证明了径向矩和涡度范数的幂律增长，并首次建立了光滑初始涡度的$L^p$-范数增长结果。

Axi-symmetric solutions for active vector models generalizing 3D Euler and electron--MHD equations

We study systems interpolating between the 3D incompressible Euler and electron--MHD equations, given by \begin{equation*} \partial_t B + V \cdot \nabla B = B\cdot \nabla V, \qquad V = -\nabla\times (-Δ)^{-a} B, \qquad \nabla\cdot B = 0, \end{equation*} where $B$ is a time-dependent vector field in $\mathbb{R}^3$. Under the assumption that the initial data is axi-symmetric without swirl, we prove local well-posedness of Lipschitz continuous solutions and existence of traveling waves in the range $1/2<a<1$. These generalize the corresponding results for the 3D axisymmetric Euler equations and s…

On the vanishing viscosity limit for 3D axisymmetric flows without swirl

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.

Global existence and uniqueness for anisotropic axisymmetric Navier-Stokes system

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

轴对称Navier-Stokes方程具有Euler长度的正则性

Regularity for axisymmetric Navier-Stokes with an Euler length

本文证明三维轴对称Navier-Stokes方程在Euler长度尺度下的Type II逐点界蕴含局部正则性，通过环量和位势涡度论证改编至古老极限，并以Hölder界替代二阶导数假设。

Vanishing viscosity limit for homogeneous axisymmetric no-swirl solutions of stationary Navier-Stokes equations

$(-1)$-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…

关于Navier-Stokes方程轴对称解潜在II型爆破的注记

A note on potential Type II blowups of axisymmetric solutions to the Navier-Stokes equations

该注记研究Navier-Stokes方程轴对称解的潜在II型爆破情景，以Euler尺度变换为主要工具展开分析。

关于无旋轴对称欧拉流的涡旋拉伸最优速率

On the optimal rate of vortex stretching for axisymmetric Euler flows without swirl

本研究针对无旋紧支集初始涡量的轴对称欧拉流，证明了涡量最大值增长的$t^{4/3}$上界（验证Childress猜想），核心方法是利用动能和含涡量守恒量估计速度最大值。

度数为$-\alpha\notin [-2,0]$的齐次Euler流的存在性

Existence of homogeneous Euler flows of degree $-α\notin [-2,0]$

本研究证明了在互补范围内轴对称齐次Euler流的存在性，首次给出了无显式形式的齐次解，并提供了新的Beltrami/Euler流例子。

亚音速激波解在有限圆柱喷嘴中稳态三维轴对称全欧拉流中大涡旋速度的情况

Transonic shock solutions for steady 3-D axisymmetric full Euler flows with large swirl velocity in a finite cylindrical nozzle

本文研究了有限圆柱喷嘴中稳态三维轴对称全欧拉流中大涡旋速度下的亚音速激波解的存在性和位置，通过新分解技术与特殊解构造解决了复合系统和非平凡解的难题。

具有非零涡量的 Chaplygin 气体二维可压缩 Euler 方程的全局光滑轴对称解

Global smooth axisymmetric solutions to 2D compressible Euler equations of Chaplygin gases with non-zero vorticity

本文研究 Chaplygin 气体二维可压缩等熵 Euler 方程的轴对称小扰动问题，利用光锥附近的零条件及远离光锥处非线性 ODE 对衰减与非衰减部分的分解，建立加权全局能量估计并证明全局光滑解存在。

具有大范围 $C^α$ 初始涡量的轴对称欧拉方程的潜在奇异性

Potential Singularity of the Axisymmetric Euler Equations with $C^α$ Initial Vorticity for A Large Range of $α$

本文通过自适应网格数值方法研究无旋流轴对称欧拉方程在 $C^α$ 初始涡量下的有限时间奇异性，发现临界 Hölder 指数 $α^*$ 可能为 $1/3$，并推广到 $n$ 维情形及提出一维近似模型。

带大涡量的三维轴对称可压缩亚声速流的射流问题

The jet problem for three-dimensional axisymmetric compressible subsonic flows with large vorticity

该研究针对带大涡量的三维轴对称可压缩亚声速射流问题，采用变分法证明了定常Euler系统下此类射流的存在性，明确了上游轴向速度给定时，足够大的质量通量对应唯一外压可使光滑射流存在并具特定远场特性。

Vanishing viscosity limits for axisymmetric flows with boundary

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…

Potential singularity formation of incompressible axisymmetric Euler equations with degenerate viscosity coefficients

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…

具有大体积粘性和大初值的三维轴对称可压缩Navier-Stokes方程的整体存在性与不可压缩极限

Global Existence and Incompressible Limit for the Three-Dimensional Axisymmetric Compressible Navier-Stokes Equations with Large Bulk Viscosity and Large Initial Data

针对不含轴圆柱域内带滑移边界条件的三维轴对称可压缩Navier-Stokes方程，在大体积粘性系数假设下，证明大初值含真空的各类解整体存在、指数衰减，以及大体积粘性极限下收敛到非齐次不可压缩方程的解。