Thomas C. Sideris, Luis Vega
The motion of incompressible and ideal fluids is studied in the plane. The stability in $L^1$ of circular vortex patches is established among the class of all bounded vortex patches of equal strength without any restriction on the size of the initial perturbation.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper, we prove nonlinear stability of planar vortex patches concentrated near an isolated minimum point of the Robin function in a general bounded domain. These vortex patches are stationary solutions of the two-dimensinal incompressible Euler equations. The result is obtained by showing that these concentrated vortex patches are in fact isolated maximizers of the kinetic energy among isovortical patches.
In this paper we prove the existence of steady multiple vortex patch solutions to the vortex-wave system in a planar bounded domain. The construction is performed by solving a certain variational problem for the vorticity and studying its asymptotic behavior as the vorticity strength goes to infinity.
Cusp Formation in Vortex Patches
针对涡旋斑块的尖点形成问题，证明了任何具锐角的初始涡旋斑块会瞬时形成尖点，验证了数值文献中的推测。
In a companion paper, we gave a detailed account of the well-posedness theory for singular vortex patches. Here, we discuss the long-time dynamics of some of the classes of vortex patches we showed to be globally well-posed in the companion work. In particular, we give examples of time-periodic behavior, cusp formation in infinite time at an exponential rate, and spiral formation in infinite time.
In this paper, we prove nonlinear orbital stability for steady vortex patches that maximize the kinetic energy among isovortical rearrangements in a planar bounded domain. As a result, nonlinear stability for an isolated vortex patch is proved. The proof is based on conservation of energy and vorticity, which is an analogue of the classical Liapunov function method.
In this paper we consider steady vortex flows for the incompressible Euler equations in a planar bounded domain. By solving a variational problem for the vorticity, we construct steady double vortex patches with opposite signs concentrating at a strict local minimum point of the Kirchhoff-Routh function with $k=2$. Moreover, we show that such steady solutions are in fact local maximizers of the kinetic energy among isovortical patches, which correlates stability to uniqueness.
In this paper, we study the vortex patch problem in an ideal fluid in a planar bounded domain. By solving a certain minimization problem and studying the limiting behavior of the minimizer, we prove that for any harmonic function $q$ corresponding to a nontrivial irrotational flow, there exists a family of steady vortex patches approaching the set of extremum points of $q$ on the boundary of the domain. Furthermore, we show that each finite collection of strict extreme points of $q$ corresponds to a family of steady multiple vortex patches approaching it.
We construct a family of rotating vortex patches with fixed angular velocity for the two-dimensional Euler equations in a disk. As the vorticity strength goes to infinity, the limit of these rotating vortex patches is a rotating point vortex whose motion is described by the Kirchhoff-Routh equation. The construction is performed by solving a variational problem for the vorticity which is based on an adaption of Arnold's variational principle. We also prove nonlinear orbital stability of the set of maximizers in the variational problem under $L^p$ perturbation when $p\in[{3}/{2},+\infty)$.
In this note, we consider the radial symmetry property of rotating vortex patches for the 2D incompressible Euler equations in the unit disc. By choosing a suitable vector field to deform the patch, we show that each simply-connected rotating vortex patch $D$ with angular velocity $Ω$, $Ω\geq \max\{{1}/{2},({2 l^2})/{(1-l^2)^2}\}$ or $Ω\leq -({2 l^2})/{(1-l^2)^2}$, where $l=\sup_{x\in D}|x|$, must be a disc. The main idea of the proof, which has a variational flavor, comes from a very recent paper of Gómez-Serrano--Park--Shi--Yao, arXiv:1908.01722, where radial symmetry of rotating vortex patc…
This article concerns the equations of motion of perfect incompressible fluids in a 3-D, smooth, bounded, simply connected domain. We suppose that the curl of the initial velocity field is a vortex patch, and examine the classical problems of the existence of a solution, either locally or globally in time, and of the persistence of the initial regularity.
The main result is that given a generic self-similarly expanding configuration of 3 point vortices that start sufficiently far out, we can instead take compactly supported vorticity functions, and the resulting solution to 2D incompressible Euler will evolve like a nearby point vortex configuration for all time, with the size of the patches growing at most as $t^{1/4+ε}$ and the distance between them growing as $\sqrt{t}$.
Boundary regularity of uniformly rotating vortex patches and an unstable elliptic free boundary problem
本文研究了与二维不可压缩欧拉方程均匀旋转涡旋片解相关的变号自由边界问题，证明了涡旋片边界在奇异边界点附近局部形成90度角，若涡旋片为Lipschitz连续。
Stability of the Rankine Vortex and Perimeter Growth in Vortex Patches
本文证明兰金涡旋相关涡旋的L¹偏差可由伪能量偏差等界定（m重对称时无需角动量），并证实其单连通扰动周长随时间线性增长，支持了先前模拟结果。
In this paper, we derive some quantitative estimates for uniformly-rotating vortex patches. We prove that if a non-radial simply-connected patch $D$ is uniformly-rotating with small angular velocity $0 < Ω\ll 1$, then the outmost point of the patch must be far from the center of rotation, with distance at least of order $Ω^{-1/2}$. For $m$-fold symmetric simply-connected rotating patches, we show that their angular velocity must be close to $\frac{1}{2}$ for $m\gg 1$ with the difference at most $O(1/m)$, and also obtain estimates on $L^{\infty}$ norm of the polar graph which parametrizes the b…
Existence of stationary vortex patches for the gSQG in bounded domains
研究有界区域内 gSQG 方程的时间周期涡斑，针对 $\gamma\in(1,2)$，结合轮廓动力学线性化、隐函数定理与合适函数空间证明其存在性。
This paper deals with the existence of $N$ vortex patches located at the vertex of a regular polygon with $N$ sides that rotate around the center of the polygon at a constant angular velocity. That is done for Euler and (SQG)$_β$ equations, with $β\in(0,1)$, but may be also extended to more general models. The idea is the desingularization of the Thomsom polygon for the $N$ point vortex system, that is, $N$ point vortices located at the vertex of a regular polygon with $N$ sides. The proof is based on the study of the contour dynamics equation combined with the application of the infinite dime…
Survey on periodic vortex patches
该综述聚焦二维欧拉方程和涡斑解，重新审视不可压缩流体动力学中周期解存在性的经典结果，通过结合原著思想与最新进展，对经典构造进行统一简化阐述。
We investiage the (slightly) super-critical 2-D Euler equations. The paper consists of two parts. In the first part we prove well-posedness in $C^s$ spaces for all $s>0.$ We also give growth estimates for the $C^s$ norms of the vorticity for $0< s \leq 1.$ In the second part we prove global regularity for the vortex patch problem in the super-critical regime.This paper extends the results of Chae, Constantin, and Wu where they prove well-posedness for the so-called LogLog-Euler equation. We also extend the classical results of Chemin and Bertozzi-Constantin on the vortex patch problem to the s…
Illposedness of $C^{2}$ vortex patches
该研究针对涡斑适定性问题，在边界曲率$L^p$可积的$W^{2,p}$空间框架下，证明了$1<p<\rm{\text{∞}}$时$W^{2,p}$正则性的保持性，同时构造出$C^2$初始涡斑数据验证$C^2$涡斑的不适定性，核心分析工具为曲率演化方程的线性色散主导项。
By applying implicit function theorem on contour dynamics, we prove the existence of co-rotating and travelling patch solutions for both Euler and the generalized surface quasi-geostrophic equation. The solutions obtained constitute a desingularization of points vortices when the size of patch support vanishes. In particular, solutions constructed in this paper consist of doubly connected components, which is essentially different from all known results.
We rigorously construct continuous curves of rotating vortex patch solutions to the two-dimensional Euler equations. The curves are large in that, as the parameter tends to infinity, the minimum along the interface of the angular fluid velocity in the rotating frame becomes arbitrarily small. This is consistent with the conjectured existence [WOZ84,Ove86] of singular limiting patches with 90$^\circ$ corners at which the relative fluid velocity vanishes. For solutions close to the disk, we prove that there are "Cat's eyes"-type structures in the flow, and provide numerical evidence that these s…
We develop a new structure of the Green's function of a second-order elliptic operator in divergence form in a 2D bounded domain. Based on this structure and the theory of rearrangement of functions, we construct concentrated traveling-rotating helical vortex patches to 3D incompressible Euler equations in an infinite pipe. By solving an equation for vorticity \begin{equation*} w=\frac{1}{\varepsilon^2}f_\varepsilon\left(\mathcal{G}_{K_H}w-\fracα{2}|x|^2|\ln\varepsilon|\right) \ \ \text{in}\ Ω\end{equation*} for small $ \varepsilon>0 $ and considering a certain maximization problem for the vor…
Existence of nested polygonal vortex patches for the generalized SQG equation
本文通过隐函数论证，在非退化条件下构造了广义SQG方程在奇异区域中的共旋转嵌套多边形涡补丁解，精确描述了边界几何，深化了对活性标量方程中相干旋转结构的理解。
Steady vortex patches on flat torus with a constant background vorticity
本文在平坦环面上利用轮廓动力学方程构造点涡平衡附近的稳态涡斑，通过背景涡量附加项与位置中心化条件解决紧致性和平移不变性带来的问题，并证明边界的光滑性与凸性。
The regularity of the boundary of vortex patches for the quasi-geostrophic shallow-water equations
研究准地转浅水方程中涡旋斑块边界光滑性的保持，并证明当Rossby半径参数趋于0时解在little Hölder空间中局部收敛到Euler方程解。
Degenerate bifurcations of two-fold doubly-connected uniformly rotating vortex patches
本文通过详细分析非线性泛函并扰动截断多项式定义的实代数簇，克服了强退化性困难，获得了二维不可压Euler方程的二重双连通均匀旋转涡块族的分岔曲线，部分回答了Hmidi和Mateu提出的问题。
In this paper, we consider the inviscid limit of the incompressible Navier-Stokes equations in a smooth, bounded and simply connected domain $Ω\subset \mathbb{R}^d, d=2,3$. We prove that for a vortex patch initial data the weak Leray solutions of the incompressible Navier-Stokes equations with Navier boundary conditions will converge (locally in time for $d=3$ and globally in time for $d=2$) to a vortex patch solution of the incompressible Euler equation as the viscosity vanishes. In view of the results obtained in [1] and [19] which dealt with the case of the whole space, we derive an almost …
Desingularization of nondegenerate rotating vortex patches
本文研究了非退化旋转涡旋片解的去奇异化问题，通过分析二维不可压缩欧拉方程的稳态旋转解空间，证明了涡旋片状态是具有无限光滑性、紧致涡量支持和二面体对称性的旋转欧拉解的极限。
In "Global regularity for vortex patches" (Commun. Math. Phys. 1993), Bertozzi and Constantin formulate the vortex patch problem in the level-set framework and prove a priori estimates for this active scalar equation. By extending the tools used to prove these estimates, we construct solutions and show propagation of higher Hölder regularity. This constitutes a proof of the regularity of vortex patches, carried out solely in the level-set framework.
Rotating vortex patches: rigidity, bifurcation, and unified structures
该专著系统阐述旋转涡片理论，通过柯西积分与共形映射统一分析，并基于分岔理论构建统一框架，涵盖欧拉及多种主动标量模型。
In 1993, two proofs of the persistence of regularity of the boundary of a classical vortex patch for the 2D Euler equations were published, one by Chemin (announced in 1991) the other by Bertozzi and Constantin. Chemin, in fact, proved a more general result, extending it further in 1995 showing, roughly, that vorticity initially having discontinuities only in directions normal to a family of vector fields that together foliate the plane continue to be so characterized by the time-evolved vector fields. A different, four-page "elementary" proof of Chemin's 1993 result was published in 1994 by P…
In this paper, we study steady vortex patch solutions to the incompressible Euler equations in a planar bounded domain $D$. Let $ψ_0$ be the solution of the elliptic problem $-Δψ_{0} =1$ in $D$; $ψ_0=0$ on $\partial D$. We prove that for any finite collection of isolated maximum points of $ψ_0$, say $\{x_1,\cdot\cdot\cdot,x_k\},$ and any $k$-tuple $\vecκ=(κ_1,\cdot,\cdot,\cdot,κ_k)$ with $κ_i>0$ and $|\vecκ|:=\sum_{i=1}^kκ_i<<1,$ there exists a steady solution of the Euler equations such that the vorticity has the form $ω^{\vecκ}=1-I_{\cup_{i=1}^k A^{\vecκ}_i}$, where $I$ denotes the character…
Uniformly Rotating Vortex Patches with Arbitrarily Many Genuine Holes
本文针对平面欧拉方程，构造了带有任意N≥2个真实零涡量孔洞的均匀旋转单位涡量涡块，通过移除收缩共旋转多边形构型实现，确定了边界与角速度的一阶修正。
Sharp local well-posedness of $C^1$ vortex patches
针对涡斑边界在 $C^1$ 中适定性不明的问题，本文在连续模定义的 $C^{1,φ}$ 空间建立局部适定性，并通过分析曲线参数化与 Hilbert 变换系统获得临界估计。
Uniformly Rotating Vortex Patches with 90-Degree Corners
针对二维不可压缩欧拉方程，证明了对所有m≥12存在m重对称的带90度角的均匀旋转涡斑，刻画了相关流的性质，采用新型不动点方法完成构造。
In this paper we construct a family of steady symmetric vortex patches for the incompressible Euler equations in an open disk. The result is obtained by studying a variational problem in which the kinetic energy of the fluid is maximized subject to some appropriate constraints for the vorticity. Moreover, we show that these vortex patches shrink to a given minimum point of the corresponding Kirchhoff-Routh function as the vorticity strength parameter goes to infinity.
The regularity of the boundary of vortex patches for some non-linear transport equations
本文研究一类由密度与齐次奇核卷积定义速度场的非线性输运方程，证明涡斑边界光滑性可持续保持，并涵盖准地转方程与平面 Cauchy 输运方程。
In this paper, we shall implement KAM theory in order to construct a large class of time quasi-periodic solutions for an active scalar model arising in fluid dynamics. More precisely, the construction of invariant tori is performed for quasi-geostrophic shallow-water equations when the {\it Rossby deformation length} belongs to a massive Cantor set. As a consequence, we construct pulsating vortex patches whose boundary is localized in a thin annulus for any time.
The global regularity of vortex patches revisited
本文重新研究涡旋斑块的全局正则性，证明在速度场由涡量与齐次$-1$阶、原点外$C^2$的奇核卷积给出时，一大类平面输运方程中涡旋斑块边界的正则性得以持久保持。
Stability of Vortex Patches in Channels
研究二维不可压缩欧拉方程在满足弱有限体积条件的域和任意宽度带状域中涡斑的轨道稳定性，通过惩罚动能泛函的极小化建立椭圆方程，并证明极小元集在欧拉动力学下轨道稳定。
We study vortex patches for the 2D incompressible Euler equations. Prior works on this problem take the support of the vorticity (i.e., the vortex patch) to be a bounded region. We instead consider the horizontally periodic setting. This includes both the case of a periodic array of bounded vortex patches and the case of vertically bounded vortex layers. We develop the contour dynamics equation for the boundary of the patch in this horizontally periodic setting, and demonstrate global $C^{1,ε}$ regularity of this patch boundary. In the process of formulating the problem, we consider different …
We consider the two-dimensional incompressible Euler equations. We construct vortex patches with smooth boundary on $T^2$ and $R^2$ whose perimeter grows with time. More precisely, for any constant $M > 0$, we construct a vortex patch in $T^2$ whose smooth boundary has length of order 1 at the initial time such that the perimeter grows up to the given constant $M$ within finite time. The construction is done by cutting a thin stick out of an almost square patch. A similar result holds for an almost round patch with a thin handle in $R^2$.
Lyapunov Unstable Motion Bifurcating from a Circular Vortex Filament
本文研究了由局部感应方程支配的三维闭合涡旋线动力学，证明了环形涡旋线存在一种称为轴向螺旋运动的闭合解，这些解从环形涡旋线分岔出来，虽然保持与环形涡旋线轨道的均匀接近，但因轴向速度不同而逐渐远离，揭示了轨道稳定性和Lyapunov稳定性之间的差距。
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.
Time quasi-periodic vortex patches of Euler equation in the plane
本文将平面 Euler 涡斑问题改写为径向位移的拟线性 Hamiltonian 方程，通过角动量的微扰辛约化克服零法向频率退化，证明接近 Kirchhoff 椭圆涡的时间拟周期解存在。
We show that the boundary of a rotating vortex patch (or V-state, in the terminology of Deem and Zabusky) is of class C^infinity provided the patch is close enough to the bifurcation circle in the Lipschitz norm. The rotating patch is convex if it is close enough to the bifurcation circle in the C^2 norm. Our proof is based on Burbea's approach to V-states. Thus conformal mapping plays a relevant role as well as estimating, on Hölder spaces, certain non-convolution singular integral operators of Calderón-Zygmund type.
Long-time behavior of an Arc-shaped Vortex Filament and its Application to the Stability of a Circular Vortex Filament
本文研究了局部感应方程下弧形涡旋丝的稳定性，证明其在非轴向上李雅普诺夫稳定且估计最优，并通过对称性假设将结论推广至圆形涡旋丝，揭示其不稳定性。
The purpose of this work is to discuss the well-posedness theory of singular vortex patches. Our main results are of two types: well-posedness and ill-posedness. On the well-posedness side, we show that globally $m-$fold symmetric vortex patches with corners emanating from the origin are globally well-posed in natural regularity classes as long as $m\geq 3$. In this case, all of the angles involved solve a \emph{closed} ODE system which dictates the global-in-time dynamics of the corners and only depends on the initial locations and sizes of the corners. {Along the way we obtain a global well-…
$L^p$ Stability of Vortex Patches in Two Dimensional Domains
该研究拓展惩罚能量变分框架，在三类典型二维区域上建立了无需先验界的涡斑统一$L^p$稳定性理论，证明极小元满足对应椭圆方程且具有轨道稳定性，显著拓展了可证明稳定的涡结构范围。
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…
We construct time quasi-periodic vortex patch solutions with one hole for the planar Euler equations. These structures are captured close to any annulus provided that its modulus belongs to a massive Borel set. The proof is based on Nash-Moser scheme and KAM theory applied with a Hamiltonian system governing the radial deformations of the patch. Compared to the scalar case, some technical issues emerge due to the interaction between the interfaces. One of them is related to a new small divisor problem in the second order Melnikov non-resonances condition coming from the transport equations adv…
Quadratic lifespan for the sublinear $α$-SQG sharp front problem
研究次线性 $\alpha$-SQG 方程中接近涡盘的涡补丁稳定性，证明其解在 $O(\varepsilon^{-2})$ 时间内保持 $\varepsilon$-接近，方法为仿微分 Birkhoff 正规形约化。
We propose and analyze a system of nonlinear partial differential equations describing the motion of a pair of vortex filaments. Furthermore, for a pair of coaxial circular vortex filaments, we derive a condition for leapfrogging to occur and prove that the condition is necessary and sufficient for the occurrence of leapfrogging.
Stability of the Shape for Circular Vortex Filaments under Non-Symmetric Perturbations
研究圆环涡旋线在非对称扰动下的非线性轨道稳定性，通过几何稳定性引理和等周不等式证明其形状在模去平移和旋转后保持稳定。
This paper aims to study the existence of asymmetric solutions for the two-dimensional generalized surface quasi-geostrophic (gSQG) equations of simply connected patches for $α\in[1,2)$ in the whole plane, where $α=1$ corresponds to the surface quasi-geostrophic equations (SQG). More precisely, we construct non-trivial simply connected co-rotating and traveling patches with unequal vorticity magnitudes. The proof is carried out by means of a combination of a desingularization argument with the implicit function theorem on the linearization of contour dynamics equation. Our results extend recen…
Existence and stability of smooth traveling circular pairs for the generalized surface quasi-geostrophic equation
本文针对广义表面准地转方程，通过能量最大化方法构造了光滑行进的反向旋转圆形涡对，证明了极大元的唯一性与序列紧性，并进一步确立了这些涡对的轨道稳定性。
Dissipative Euler flows originating from circular vortex filaments
针对初始涡量集中于圆上的三维 Euler 方程，论文利用时间加权空间中的凸积分构造弱解，实现有限递减能量与移动增厚环带，且无需磨光初值或重标时间。
We consider the problem of asymptotic stability and linear inviscid damping for perturbations of a point vortex and similar degenerate circular flows. Here, key challenges include the lack of strict monotonicity and the necessity of working in weighted Sobolev spaces whose weights degenerate as the radius tends to zero or infinity. Prototypical examples are given by circular flows with power law singularities or zeros as $r\downarrow 0$ or $r \uparrow \infty$.
This paper aims to provide a robust model for the well-known phenomenon of Kármán Vortex Street arising in Fluid Mechanics. The first theoretical attempt to model this pattern was given by von Kármán [22, 23]. He considered two parallel staggered rows of point vortices, with opposite strength, that translate at the same speed. Following the ideas of Saffman and Schatzman [36], we propose to study this phenomenon in the Euler equations by considering two infinite arrows of vortex patches. The key idea is to desingularize the point vortex model proposed by von Kármán. Our construction is flexibl…
We consider the head-on collision of two coaxial vortex rings described as the motion of two circular vortex filaments under the localized induction approximation. We prove the existence of solutions to a system of nonlinear partial differential equations proposed by the author which exhibit head--on collision. We also give a necessary and sufficient condition for the initial configuration and parameters of the filaments for head-on collision to occur. Our results suggest that there exists a critical value $ γ_{\ast}>1 $ for the ratio $ γ$ of the absolute value of the circulations such that wh…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。