In-Jee Jeong, Tsuyoshi Yoneda
We consider the 3D incompressible Navier-Stokes equations under the following $2+\frac{1}{2}$-dimensional situation: vertical vortex blob (quasi-streamwise vortices) being stretched by two-dimensional shear flow. We prove enhanced dissipation induced by such quasi-streamwise vortices.
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In the analysis of the long-time behaviour of two-dimensional incompressible viscous fluids, Oseen vortices play a major role as attractors of any homogeneous solution with integrable initial vorticity. As a first step in the study of the density-dependent case, the present paper establishes the asymptotic stability of Oseen vortices for slighty inhomogeneous fluids with respect to localized regular perturbations.
On concentrated vortices of 3D incompressible Euler equations under helical symmetry: with swirl
本文通过变分方法构造了带旋度的三维不可压缩欧拉方程在螺旋对称下的集中螺旋涡旋，并证明其渐近收敛于涡丝。
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…
Desingularization of vortices for the incompressible Euler equation on a sphere
本文构造了标准二维球面上不可压缩 Euler 方程的一族奇对称旋转整体解，通过最大化能量-冲量泛函实现涡旋去奇异化，并证明了其关于奇对称扰动的稳定性。
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.
In this paper, we construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure is carried out by constructing solutions to the following elliptic problem [ -\ep^2 Δu=(u-q-\fracκ{2π}\ln\frac{1}{\ep})_+^p, \quad & x\inΩ, u=0, \quad & x\in\partialΩ, ] where $p>1$, $Ω\subset\mathbb{R}^2$ is a bounded domain, $q$ is a harmonic function. We showed that if $Ω$ is simply-connected smooth domain, then for any given non-degenerate critical point of Kirchhoff-Routh function $\mathcal{W}(x_1,...,x_m)$ with the sa…
Clustered helical vortices for 3D incompressible Euler equation in infinite cylinders
该研究先证明小参数下发散型半线性椭圆问题存在含任意多泡、坍缩至指定极大值点的簇状解，再以此构造无限圆柱中三维不可压缩Euler方程的簇状螺旋涡解，涡支撑为集中在单螺旋附近的多螺旋管。
In this paper, we continue to construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure now is carried out by constructing solutions to the following elliptic problem \[ {cases} -\ep^2 Δu=(u-q-\fracκ{2π}\ln\frac{1}{\ep})_+^p-(q-\fracκ{2π}\ln\frac{1}{\ep}-u)_+^p, \quad & x\inΩ, u=0, \quad & x\in\partialΩ, {cases} \] where $p>1$, $Ω\subset\mathbb{R}^2$ is a bounded domain, $q$ is a harmonic function. We showed that if $Ω$ is a simply-connected smooth domain, then for any given non-degenerate critical…
Dynamics and leapfrogging phenomena of multiple helical vortices for 3D incompressible Euler equations
本文研究了3D不可压缩Euler方程组在一般初始假设下多重螺旋涡旋的时间演化，证明其运动一致收敛到特定动力系统，并首次为数值观察到的“Kelvin波的蛙跳”现象提供了数学证明。
Steady vortices for the three-dimensional Euler equation for inviscid incompressible flows and for the shallow water equation are constructed and showed to tend asymptotically to singular vortex filaments. The construction is based on the asymptotic study of solutions to a semilinear elliptic problem.
By DNS of Navier-Stokes turbulence, Goto-Saito-Kawahara (2017) showed that turbulence consists of a self-similar hierarchy of anti-parallel pairs of vortex tubes, in particular, stretching in larger-scale strain fields creates smaller-scale vortices. Inspired by their numerical result, we examine the Goto-Saito-Kawahara type of vortex-tubes behavior using the 3D incompressible Euler equations, and show that such behavior induces energy cascade in the absence of nonlinear scale-interaction. From this energy cascade, we prove a modified version of the zeroth-law. In the Appendix, we derive Kolmo…
We study the existence of stationary classical solutions of the incompressible Euler equation in the plane that approximate singular stationnary solutions of this equation. The construction is performed by studying the asymptotics of equation $-\eps^2 Δu^\eps=(u^\eps-q-\fracκ{2π} \log \frac{1}{\eps})_+^p$ with Dirichlet boundary conditions and $q$ a given function. We also study the desingularization of pairs of vortices by minimal energy nodal solutions and the desingularization of rotating vortices.
We establish the regularity of weak solutions for the vorticity equation associated to a family of desingularised models for vortex filament dynamics in 3D incompressible viscous flows. These include and generalise the classical model "of an allowance for the thickness of the vortices" due to Louis Rosenhead in 1930. Our approach is based on an interplay between the geometry of vorticity and analytic inequalities in Sobolev spaces.
Co-rotating nearly parallel helical vortices with small cross-section in 3D incompressible Euler equations
本文利用格林函数和有限维约化方法构造半线性椭圆方程的聚簇解，并由此证明三维不可压缩欧拉方程中存在集中于近平行共旋螺旋涡丝附近的小截面螺旋涡量场，验证并推广了已有结果。
We study dynamics of vortices in solutions of the Gross-Pitaevskii equation $i \partial_t u = Δu + \varepsilon^{-2} u (1 - |u|^2)$ on $\mathbb{R}^2$ with nonzero degree at infinity. We prove that vortices move according to the classical Kirchhoff-Onsager ODE for a small but finite coupling parameter $\varepsilon$. By carefully tracking errors we allow for asymptotically large numbers of vortices, and this lets us connect the Gross-Pitaevskii equation on the plane to two dimensional incompressible Euler equations through the work of Schochet [21].
We consider the inviscid limit for the two-dimensional incompressible Navier-Stokes equation in the particular case where the initial flow is a finite collection of point vortices. We suppose that the initial positions and the circulations of the vortices do not depend on the viscosity parameter ν, and we choose a time T > 0 such that the Helmholtz-Kirchhoff point vortex system is well-posed on the interval [0,T]. Under these assumptions, we prove that the solution of the Navier-Stokes equation converges, as ν-> 0, to a superposition of Lamb-Oseen vortices whose centers evolve according to a v…
The global asymptotic dynamics of point vortices for the lake equations is rigorously derived. Vorticity that is initially sharply concentrated around $N$ distinct vortex centers is proven to remain concentrated for all times. Specifically, we prove weak concentration of the vorticity and in addition strong concentration in the direction of the steepest ascent of the depth function. As a consequence, we obtain the motion law of point vortices following at leading order the level lines of the depth function. The lack of strong localization in the second direction is linked to the vortex filamen…
The aim of these notes is to present in a comprehensive and relatively self-contained way some recent developments in the mathematical analysis of two-dimensional viscous flows. We consider the incompressible Navier-Stokes equations in the whole plane, and assume that the initial vorticity is a finite measure. This general setting includes vortex patches, vortex sheets, and point vortices. We first prove the existence of a unique global solution, for any value of the viscosity parameter, and we investigate its long-time behavior. We next consider the particular case where the initial flow is a…
In this paper, we study two-dimensional steady incompressible Euler flows in which the vorticity is sharply concentrated in a finite number of regions of small diameter in a bounded domain. Mathematical analysis of such flows is an interesting and physically important research topic in fluid mechanics. The main purpose of this paper is to prove that in such flows the locations of these concentrated blobs of vorticity must be in the vicinity of some critical point of the Kirchhoff-Routh function, which is determined by the geometry of the domain. The vorticity is assumed to be only in $L^{4/3},…
Arnold's variational principle and its application to the stability of planar vortices
该研究围绕二维不可压缩Euler方程稳态解的Arnold稳定性判据相关变分原理展开，明确了对应二次型可用于稳定性分析的条件，给出Oseen涡旋的新几何稳定性证明，并分析了无黏情形及高斯涡旋的黏性影响。
$C^1$ type regularization for point vortices on $\mathbb S^2$
本文在球面上利用切映射与 Lyapunov--Schmidt 约化构造行进点涡系统的 $C^1$ 型正则化，证明涡旋位于 Kirchhoff--Routh 函数非退化临界点附近，并给出定性与定量估计。
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.
Dynamics of helical vortex filaments in non viscous incompressible flows
本文研究三维欧拉方程螺旋对称无旋涡情形下的集中涡量解，证明初始集中在不同半径螺旋线附近的解保持丝状集中，并利用Biot-Savart核奇异部分的新显式公式刻画丝状物的平移与旋转动态。
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