Zineb Hassainia, Taoufik Hmidi
In this paper, we study the existence of co-rotating and counter-rotating unequal-sized pairs of simply connected patches for Euler equations. In particular, we prove the existence of curves of steadily co-rotating and counter-rotating asymmetric vortex pairs passing through a point vortex pairs with unequal circulations.
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In this paper, we study the existence of corotating and counter-rotating pairs of simply connected patches for Euler equations and the $(\hbox{SQG})_α$ equations with $α\in (0,1).$ From the numerical experiments implemented for Euler equations in \cite{DZ, humbert, S-Z} it is conjectured the existence of a curve of steady vortex pairs passing through the point vortex pairs. There are some analytical proofs based on variational principle \cite{keady, Tur}, however they do not give enough information about the pairs such as the uniqueness or the topological structure of each single vortex. We in…
Global bifurcation for corotating and counter-rotating vortex pairs
本文基于Buffoni和Toland的解析全局分岔定理，通过研究非线性问题的拓扑性质，构造了同向旋转与反向旋转涡对局部曲线的全局延拓，并刻画了全局解曲线的角向流体速度消失或涡斑自交现象。
We prove well-posedness of vortex sheets with surface tension in the 3D incompressible Euler equations with vorticity.
We construct co-rotating and traveling vortex sheets for 2D incompressible Euler equation, which are supported on several small closed curves. These examples represent a new type of vortex sheet solutions other than two known classes. The construction is based on Birkhoff-Rott operator, and accomplished by using implicit function theorem at point vortex solutions with suitably chosen function spaces.
In this article we consider the physical justification of the Vortex-Wave equation introduced by Marchioro and Pulvirenti in the case of a single point vortex moving in an ambient vorticity. We consider a sequence of solutions for the Euler equation in the plane corresponding to initial data consisting of an ambient vorticity in $L^1\cap L^\infty$ and a sequence of concentrated blobs which approach the Dirac distribution. We introduce a notion of a weak solution of the Vortex-Wave equation in terms of velocity (or primitive variables) and then show, for a subsequence of the blobs, the solution…
We study weak solutions of the two-dimensional (2D) filtered Euler equations whose vorticity is a finite Radon measure and velocity has locally finite kinetic energy, which is called the vortex sheet solution. The filtered Euler equations are a regularized model based on a spatial filtering to the Euler equations. The 2D filtered Euler equations have a unique global weak solution for measure valued initial vorticity, while the 2D Euler equations require initial vorticity to be in the vortex sheet class with a distinguished sign for the existence of global solutions. In this paper, we prove tha…
In this paper, we study desingularization of vortices for the two-dimensional incompressible Euler equations in the full plane. We construct a family of steady vortex pairs for the Euler equations with a general vorticity function, which constitutes a desingularization of a pair of point vortices with equal magnitude and opposite signs. The results are obtained by using an improved vorticity method.
Cluster of vortex helices in the incompressible $3$d Euler equations
该研究针对三维无粘不可压缩 Euler 方程，证明了多个螺旋丝状涡旋结构可相互坍缩，形成涡旋螺旋簇。
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.
It is well-known that the dynamics of vortices in an ideal incompressible two-dimensional fluid contained in a bounded not necessarily simply connected smooth domain is described by the Kirchhoff--Routh point vortex system. In this paper, we revisit the classical problem of how well solutions to the Euler equations approximate these vortex dynamics and extend previous rigorous results to the case where the vorticity field is unbounded. More precisely, we establish estimates for the $2$-Wasserstein distance between the vorticity and the empirical measure associated with the point vortex dynamic…
Uniqueness and stability of steady vortex rings for 3D incompressible Euler equation
本文通过变分原理和集中紧致性方法，证明了三维欧拉方程中特殊涡环的唯一性和非线性稳定性。
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.
We consider the vortex patch problem for both the 2-D and 3-D incompressible Euler equations. In 2-D, we prove that for vortex patches with $H^{k-0.5}$ Sobolev-class contour regularity, $k \ge 4$, the velocity field on both sides of the vortex patch boundary has $H^k$ regularity for all time. In 3-D, we establish existence of solutions to the vortex patch problem on a finite-time interval $[0,T]$, and we simultaneously establish the $H^{k-0.5}$ regularity of the two-dimensional vortex patch boundary, as well as the $H^k$ regularity of the velocity fields on both sides of vortex patch boundary,…
Self-similar algebraic spiral vortex sheets of 2-D incompressible Euler equations
本文首次严格构造了二维不可压缩欧拉方程的自相似代数螺旋涡面解，并通过处理超出经典奇异积分理论的柯西积分，刻画了涡面曲率奇异性形成后的典型卷起模式。
In this paper, we are concerned with nonlinear desingularization of steady vortex rings of three-dimensional incompressible Euler fluids. We focus on the case when the vorticity function has a simple discontinuity, which corresponding to a jump in vorticity at the boundary of the cross-section of the vortex ring. Using the vorticity method, we construct a 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 proved to depend on the circulati…
We present an alpha-regularization of the Birkhoff-Rott equation, induced by the two-dimensional Euler-alpha equations, for the vortex sheet dynamics. We show that initially smooth self-avoiding vortex sheet remains smooth for all times under the alpha-regularized dynamics, provided the initial density of vorticity is an integrable function over the curve with respect to the arc-length measure.
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 $p\in[{3}/{2},+\infty)$.
Nearly parallel helical vortex filaments in the three dimensional Euler equations
本研究严格证明了Klein等人提出的近平行涡丝渐近运动定律适用于两种中心构型（N条螺旋涡丝正多边形及N+1条含中心直涡丝的构型），验证了该模型在三维欧拉方程中的有效性。
We construct infinitely many admissible weak solutions to the 2D incompressible Euler equations for vortex sheet initial data. Our initial datum has vorticity concentrated on a simple closed curve in a suitable Hölder space and the vorticity may not have a distinguished sign. Our solutions are obtained by means of convex integration; they are smooth outside a "turbulence" zone which grows linearly in time around the vortex sheet. As a byproduct, this approach shows how the growth of the turbulence zone is controlled by the local energy inequality and measures the maximal initial dissipation ra…
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…
We present an alpha-regularization of the Birkhoff-Rott equation, induced by the two-dimensional Euler-alpha equations, for the vortex sheet dynamics. We show the convergence of the solutions of Euler-alpha equations to a weak solution of the Euler equations for initial vorticity being a finite Radon measure of fixed sign, which includes the vortex sheets case. We also show that, provided the initial density of vorticity is an integrable function over the curve with respect to the arc-length measure, (i) an initially Lipschitz chord arc vortex sheet (curve), evolving under the BR-alpha equatio…
We discuss the Lagrangian property and the conservation of the kinetic energy for solutions of the 2D incompressible Euler equations. Existence of Lagrangian solutions is known when the initial vorticity is in $L^p$ with $1\leq p\leq \infty$. Moreover, if $p\geq 3/2$ all weak solutions are conservative. In this work we prove that solutions obtained via the vortex method are Lagrangian, and that they are conservative if $p>1$.
Leapfrogging vortex rings for the 3-dimensional incompressible Euler equations
针对不可压缩三维欧拉方程，构造光滑解严格证明了亥姆霍兹观察到的同轴涡环反复穿越的跳跃式运动现象。
Weakened vortex stretching effect in three scale hierarchy for the 3D Euler equations
本文研究了三维不可压缩 Euler 方程在三尺度层级下的涡旋拉伸效应，证明中尺度流的拉伸效应会被大尺度减弱，且被拉伸涡旋的应力张量也会相应减弱。
In this article we consider the evolution of vortex sheets in the plane both as a weak solution of the two dimensional incompressible Euler equations and as a (weak) solution of the Birkhoff-Rott equations. We begin by discussing the classical Birkhoff-Rott equations with respect to arbitrary parametrizations of the sheet. We introduce a notion of weak solution to the Birkhoff-Rott system and we prove consistency of this notion with the classical formulation of the equations. Our main purpose in this paper is to present a sharp criterion for the equivalence of the weak Euler and weak Birkhoff-…
Helical vortex filaments with compactly supported cross-sectional vorticity for the incompressible Euler equations in $\mathbb{R}^3$
针对三维无粘不可压缩欧拉流的涡丝猜想，利用粘合论证首次构造$\mathbb{R}^3$中截面涡量紧支的光滑螺旋涡丝，推广至多涡解并改进了相关变分结果。
A classical problem for the two-dimensional Euler flow for an incompressible fluid confined to a smooth domain. is that of finding regular solutions with highly concentrated vorticities around $N$ moving {\em vortices}. The formal dynamic law for such objects was first derived in the 19th century by Kirkhoff and Routh. In this paper we devise a {\em gluing approach} for the construction of smooth $N$-vortex solutions. We capture in high precision the core of each vortex as a scaled finite mass solution of Liouville's equation plus small, more regular terms. Gluing methods have been a powerful …
In this paper, we consider the time evolution of an ideal fluid in a planar bounded domain. We prove that if the initial vorticity is supported in a sufficiently small region with diameter $\varepsilon$, then the time evolved vorticity is also supported in a small region with diameter $d$, $d\leq C\varepsilon^α$ for any $α<\frac{1}{3}$, and the center of the vorticity tends to the point vortex, the motion of which is described by the Kirchhoff-Routh equation.
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)…
Growth rates for anti-parallel vortex tube Euler flows in three and higher dimensions
研究三维及更高维度中无旋对称欧拉方程解的增长下界，改进了Choi和Jeong在三维中的结果。
Nonlinear Scale-Local Geometric Deformations of Vortex Rings in Smooth Euler Flows via Bayesian Optimization and Adjoint Methods
本文通过贝叶斯优化与伴随方法，研究了涡环在光滑欧拉流中非线性尺度局部变形的机制，提出了一种新的波方程来描述 swirling particles 的轴线运动。
We show the short-time existence and nonlinear stability of vortex sheets for the nonisentropic compressible Euler equations in two spatial dimensions, based on the weakly linear stability result of Morando--Trebeschi (2008) [20]. The missing normal derivatives are compensated through the equations of the linearized vorticity and entropy when deriving higher-order energy estimates. The proof of the resolution for this nonlinear problem follows from certain \emph{a priori} tame estimates on the effective linear problem {in the usual Sobolev spaces} and a suitable Nash--Moser iteration scheme.
Clustered vortex helices with compactly supported cross-sectional vorticity in the 3D Euler equations
本文基于三维不可压缩欧拉方程，通过粘合技术构造了首个光滑多涡旋解，展示了 collapsing 的螺旋纤维簇，横截面涡量始终在 $\mathbb{R}^2$ 中保持紧凑支撑。
The point vortex dynamics in background fields on surfaces is justified as an Euler-Arnold flow in the sense of de Rham currents. We formulate a current-valued solution of the Euler-Arnold equation with a regular-singular decomposition. For the solution, we first prove that, if the singular part of the vorticity is given by a linear combination of delta functions centered at $q_n(t)$ for $n=1,\ldots,N$, $q_n(t)$ is a solution of the point vortex equation. Conversely, we next prove that, if $q_n(t)$ is a solution of the point vortex equation for $n=1,\ldots,N$, there exists a current-valued sol…
In this article we study the limit when $α\to 0$ of solutions to the $α$-Euler system in the half-plane, with no-slip boundary conditions, to weak solutions of the 2D incompressible Euler equations with non-negative initial vorticity in the space of bounded Radon measures in $H^{-1}$. This result extends the analysis done in arXiv:1611.05300 and arXiv:1403.5682. It requires a substantially distinct approach, analogous to that used for Delort's Theorem, and a new detailed investigation of the relation between (no-slip) filtered velocity and potential vorticity in the half-plane.
An Expanding Self-Similar Vortex Configuration for the 2D Euler Equations
本文针对二维不可压缩欧拉方程，构造了涡量集中在点涡附近并收敛于狄拉克质量之和的解，证明这些点涡沿扩张自相似螺旋轨迹运动，并与 Kirchhoff-Routh 点涡系统相关联。
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…
Time-periodic leapfrogging vortex rings in the 3D Euler equations
研究证明了三维不可压缩欧拉方程中时间周期性跳跃环 vortex 环的存在，首次严谨地实现了海姆霍兹在1858年提出的现象。通过构造两个相互作用的 vortex 纤维的去奇异化，得到近似同心环的哈密顿描述，克服了线性传输动力学中的奇异小分母问题，构造了非平凡的时间周期性解。
We investigate a steady planar flow of an ideal fluid in a (bounded or unbounded) domain $Ω\subset \mathbb{R}^2$. Let $κ_i\not=0$, $i=1,\ldots, m$, be $m$ arbitrary fixed constants. For any given non-degenerate critical point $\mathbf{x}_0=(x_{0,1},\ldots,x_{0,m})$ of the Kirchhoff-Routh function defined on $Ω^m$ corresponding to $(κ_1,\ldots, κ_m)$, we construct a family of stationary planar flows with vortex sheets that have large vorticity amplitude and are perturbations of small circles centered near $x_i$, $i=1,\ldots,m$. The proof is accomplished via the implicit function theorem with su…
We prove the existence of critical points of the $N$-vortex Hamiltonian $H_Ω(x_1,\ldots, x_N) =\sum\limits^N_{i=1}Γ^2_i h(x_i) + \sum\limits_{i,j=1\atop j\not= k}^N Γ_iΓ_jG(x_i,x_j)+2\sum\limits_{i=1}^NΓ_iψ_0(x_i)$ in a bounded domain $Ω\subset{\mathbb R}^2$ which may be simply or multiply connected. Here $G$ denotes the Green function for the Dirichlet Laplace operator in $Ω$, more generally a hydrodynamic Green function, and $h$ the Robin function. Moreover $ψ_0\in C^1(\overlineΩ)$ is a harmonic function on $Ω$. We obtain new critical points $x=(x_1,\dots,x_N)$ for $N=3$ or $N=4$ under condi…
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 Euler equations in ${\mathbb R}^3$ expressed in vorticity form. A classical question that goes back to Helmholtz is to describe the evolution of solutions with a high concentration around a curve. The work of Da Rios in 1906 states that such a curve must evolve by the so-called binormal curvature flow. Existence of true solutions concentrated near a given curve that evolves by this law is a long-standing open question that has only been answered for the special case of a circle travelling with constant speed along its axis, the thin vortex-rings. We provide what appears to be t…
Conservation and Breakdown of Modulus of Continuity in the Propagation of Loglog Vortex for 2D Euler equation
本文研究二维 Euler 方程中 loglog 涡在连续扰动下的奇异结构持续性，证明 $0<α<1$ 时连续模可保持传播，并构造光滑扰动说明 $α>1$ 时范数瞬间爆破，从而确立阈值的尖锐性。
A proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation
本文给出强迫二维欧拉方程在涡度类L¹∩L^p中Vishik非唯一性定理的更简单证明，通过构造稳定的涡旋来简化证明过程。
We study the evolution of solutions to the 2D Euler equations whose vorticity is sharply concentrated in the Wasserstein sense around a finite number of points. Under the assumption that the vorticity is merely $L^p$ integrable for some $p>2$, we show that the evolving vortex regions remain concentrated around points, and these points are close to solutions to the Helmholtz--Kirchhoff point vortex system.
It is well known that the Euler vortex patch in $\mathbb{R}^{2}$ will remain regular if it is regular enough initially. In bounded domains, the regularity theory for patch solutions is less complete. We study here the Euler vortex patch in a disk. We prove global in time regularity by providing the upper bound of the growth of curvature of the patch boundary. For a special symmetric scenario, we construct an example of double exponential curvature growth, showing that such upper bound is qualitatively sharp.
Finite-time blowup for the inviscid vortex stretching equation
针对忽略涡量平流的 3D Euler 模型方程，本文证明无粘涡拉伸方程即便在轴对称、无旋流情形下也存在光滑解的有限时间爆破，从而揭示平流对抑制非线性涡拉伸的关键作用。
Strong Beltrami fields have long played a key role in fluid mechanics and magnetohydrodynamics. In particular, they are the kind of stationary solutions of the Euler equations where one has been able to show the existence of vortex structures (vortex tubes and vortex lines) of arbitrarily complicated topology. On the contrary, there are very few results about the existence of generalized Beltrami fields, that is, divergence-free fields whose curl is the field itself times a non-constant function. In fact, generalized Beltrami fields (which are also stationary solutions to the Euler equations) …
Desingularization of vortex sheets for the 2D Euler equations
本文通过构造光滑紧支撑涡量族，将涡片初始数据正则化，并证明对应欧拉方程解收敛到Birkhoff-Rott系统定义的测度，利用Kelvin-Helmholtz不稳定性的各向异性效应实现去奇异化。
We established the existence, uniqueness and stability of subsonic flows past an airfoil with a vortex line at the trailing edge. Such a flow pattern is governed by the two dimensional steady compressible Euler equations. The vortex line attached to the trailing edge is a contact discontinuity for the Euler system and is treated as a free boundary. The problem is formulated and solved by using the implicit function theorem. The main difficulties are due to the fitting of the vortex line with the profile at the trailing edge and the possible subtle instability of the vortex line at the far fiel…
Steady Contiguous Vortex-Patch Dipole Solutions of the 2D Incompressible Euler Equation
本文通过新颖的不动点方法严格构造了二维不可压缩欧拉方程的首个连续涡斑偶极子稳态行波解，并证明了其边界光滑性等性质。
It is well known that the Euler vortex patch in $\mathbb{R}^{2}$ will remain regular if it is regular enough initially. In bounded domains, the regularity theory for patch solutions is less complete. In this paper, we study Euler vortex patches in a general smooth bounded domain. We prove global in time regularity by providing an upper bound on the growth of curvature of the patch boundary. For a special symmetric scenario, we construct an example of double exponential curvature growth, showing that our upper bound is qualitatively sharp.
We consider self-similar solutions of the 2d incompressible Euler equations. We construct a class of solutions with vorticity forming algebraic spirals near the origin, in analogy to vortex sheets rolling up into algebraic spirals.
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.
We prove a definitive theorem on the asymptotic stability of point vortex solutions to the full Euler equation in 2 dimensions. More precisely, we show that a small, Gevrey smooth, and compactly supported perturbation of a point vortex leads to a global solution of the Euler equation in 2D, which converges weakly as $t\to\infty$ to a radial profile with respect to the vortex. The position of the point vortex, which is time dependent, stabilizes rapidly and becomes the center of the final, radial profile. The mechanism that leads to stabilization is mixing and inviscid damping.
In this article, we construct traveling-rotating helical vortices with small cross-section to the 3D incompressible Euler equations in an infinite pipe, which tend asymptotically to singular helical vortex filament evolved by the binormal curvature flow. The construction is based on studying a general semilinear elliptic problem in divergence form \begin{equation*} \begin{cases} -\varepsilon^2\text{div}(K(x)\nabla u)= (u-q|\ln\varepsilon|)^{p}_+,\ \ &x\in Ω,\\ u=0,\ \ &x\in\partial Ω, \end{cases} \end{equation*} for small values of $ \varepsilon. $ Helical vortex solutions concentrating near s…
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 consider vortex patch solutions of the incompressible Euler equations in the plane. It is shown that the winding number around the origin for most particles in the patch grows linearly in time when the initial patch is close to a disk enough.
We consider the 3D incompressible Euler equations under the following situation: small-scale vortex blob being stretched by a prescribed large-scale stationary flow. More precisely, we clarify what kind of large-scale stationary flows really stretch small-scale vortex blobs in alignment with the straining direction. The key idea is constructing a Lagrangian coordinate so that the Lie bracket is identically zero (c.f. the Frobenius theorem), and investigate the locality of the pressure term by using it.
Non-uniqueness of admissible solutions for the 2D Euler equation with $L^p$ vortex data
本文针对二维Euler方程构造具有$L^p$涡量初值的无穷多个有界容许解，利用自相似子解与凸积分揭示唯一性界限及Onsager临界指数。
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