C. H. Arthur Cheng, Daniel Coutand, Steve Shkoller
We prove well-posedness of vortex sheets with surface tension in the 3D incompressible Euler equations with vorticity.
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In this paper, we study the inviscid limit of the free surface incompressible Navier-Stokes equations with or without surface tension. By delicate estimates, we prove the weak boundary layer of the velocity of the free surface Navier-Stokes equations and the existence of strong or weak vorticity layer for different conditions. When the limit of the difference between the initial Navier-Stokes vorticity and the initial Euler vorticity is nonzero, or the tangential projection on the free surface of the Euler strain tensor multiplying by normal vector is nonzero, there exists a strong vorticity l…
We study modulational stability and instability in the Whitham equation, combining the dispersion relation of water waves and a nonlinearity of the shallow water equations, and modified to permit the effects of surface tension and constant vorticity. When the surface tension coefficient is large, we show that a periodic traveling wave of sufficiently small amplitude is unstable to long wavelength perturbations if the wave number is greater than a critical value, and stable otherwise, similarly to the Benjamin-Feir instability of gravity waves. In the case of weak surface tension, we find inter…
In this paper, we show that the only solution of the vortex sheet equation, either stationary or uniformly rotating with negative angular velocity $Ω$, such that it has positive vorticity and is concentrated in a finite disjoint union of smooth curves with finite length is the trivial one: constant vorticity amplitude supported on a union of nested, concentric circles. The proof follows a desingularization argument and a calculus of variations flavor.
Steady vortex sheets in presence of surface tension
本文研究了存在表面张力时两种相同密度无旋流体模型中圆形分布附近的均匀旋转/静止非平凡涡旋面，通过旋转速度、表面张力系数或平均涡度作为分岔参数证明了分岔结果。
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 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 investigate the convergence rates of inviscid limits for the free-boundary problems of the incompressible magnetohydrodynamics (MHD) with or without surface tension in $\mathbb{R}^3$, where the magnetic field is identically constant on the surface and outside of the domain. First, we establish the vorticity, the normal derivatives and the regularity structure of the solutions, and develop a priori co-norm estimates including time derivatives by the vorticity system. Second, we obtain two independent sufficient conditions for the existence of strong vorticity layers: (I) the l…
Compressible Gravity-Capillary Water Waves with Vorticity: Local Well-Posedness, Incompressible and Zero-Surface-Tension Limits
本文研究了具有涡旋的可压缩重力-表面张力水波，通过精心设计的近似系统和双曲方法证明了局部well-posed性，同时获得不可压缩和零表面张力极限。
On the Incompressible Limit of Current-Vortex Sheets with or without Surface Tension
本文研究理想可压缩MHD中电流-涡旋片的不可压缩极限与零表面张力极限，在稳定性条件下利用拟微分分析、压力波动方程和加权各向异性结构证明了一致估计。
In this paper, we construct new, uniformly-rotating solutions of the vortex sheet equation bifurcating from circles with constant vorticity amplitude. The proof is accomplished via a Lyapunov-Schmidt reduction and a second order expansion of the reduced system.
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.
We consider a sequence of Leray-Hopf weak solutions of the 2D Navier-Stokes equations on a bounded domain, in the vanishing viscosity limit. We provide sufficient conditions on the associated vorticity measures, away from the boundary, which ensure that as the viscosity vanishes the sequence converges to a weak solution of the Euler equations. These assumptions are consistent with vortex sheet solutions of the Euler equations.
In this paper, we study solitary waves propagating along the surface of an infinitely deep body of water in two or three dimensions. The waves are acted upon by gravity and capillary effects are allowed --- but not required --- on the interface. We assume that the vorticity is localized in the sense that it satisfies certain moment conditions, and we permit there to be finitely many point vortices in the bulk of the fluid in two dimensions. We also consider a two-fluid model with a vortex sheet. Under mild decay assumptions, we obtain precise asymptotics for the velocity field and free surface…
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…
This article is concerned with the incompressible, infinite depth water wave equation in two space dimensions, with gravity and constant vorticity but with no surface tension. We consider this problem expressed in position-velocity potential holomorphic coordinates, and prove local well-posedness for large data, as well as cubic lifespan bounds for small data solutions.
Local Well-posedness of the Incompressible Current-Vortex Sheet Problems
本文证明不可压缩电流-涡流片问题在表面张力或Syrovatskij条件下的局部适定性，并建立表面张力消失极限，表明毛细力和大切向磁场可稳定其运动，且无需界面为图形假设。
Global nonlinear stability of vortex sheets for the Navier-Stokes equations with large data
本文研究小马赫数下大振幅、大涡量Navier-Stokes方程涡片的全局非线性稳定性，通过辅助流重构、频率分解及反导数技术，得到马赫数一致的强解全局估计并验证不可压缩极限。
We prove the first bifurcation result of time quasi-periodic traveling waves solutions for space periodic water waves with vorticity. In particular we prove existence of small amplitude time quasi-periodic solutions of the gravity-capillary water waves equations with constant vorticity, for a bidimensional fluid over a flat bottom delimited by a space-periodic free interface. These quasi-periodic solutions exist for all the values of depth, gravity and vorticity, and restricting the surface tension to a Borel set of asymptotically full Lebesgue measure.
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.
For a nonlinear system of coupled PDEs, that describes evolution of a viscous thin liquid sheet and takes account of surface tension at the free surface, we show exponential $(H^1,\,L^2)$ asymptotic decay to the flat profile of its solutions considered with general initial data. Additionally, by transforming the system to Lagrangian coordinates we show that the minimal thickness of the sheet stays positive for all times. This result proves the conjecture formally accepted in the physical literature [1], that a viscous sheet can not rupture in finite time in absence of external forcing. Moreove…
The question of energy concentration in approximate solution sequences $u^ε$, as $ε\to 0$, of the two-dimensional incompressible Euler equations with vortex-sheet initial data is revisited. Building on a novel identity for the structure function in terms of vorticity, the vorticity maximal function is proposed as a quantitative tool to detect concentration effects in approximate solution sequences. This tool is applied to numerical experiments based on the vortex-blob method, where vortex sheet initial data without distinguished sign are considered, as introduced in \emph{[R.~Krasny, J. Fluid …
We consider the dynamics of a vortex sheet that evolves by the Birkhoff-Rott equations. The fluid evolution is understood as a weak solution of the incompressible Euler equations where the vorticity is given by a delta function on a curve multiplied by an amplitude. The solutions we study are with finite energy, which implies zero mean amplitude. In this context we choose a parametrization for the motion of the vortex sheet for which the equation is well-posed for analytic initial data. For the equation of the amplitude we show ill-posedness for non-analytic initial data.
We prove the existence of non-negative measure- and $H^{-1}$-valued vorticity solutions to the stochastic 2D Euler equations with transport vorticity noise, starting from any non-negative vortex sheet. This extends the result by Delort (J.Amer.Math.Soc. 1991) to the stochastic case.
Well-posedness and Incompressible Limit of Current-Vortex Sheets with Surface Tension in Compressible Ideal MHD
本文作为系列论文首篇，证明了带表面张力的可压缩理想MHD电流涡旋片的局部适定性与不可压缩极限，揭示洛伦兹力隐藏结构以建立一致估计，提出不依赖Nash-Moser迭代的迭代逼近框架，并为无表面张力情形的证明奠定基础。
Small scale creation in 2D gravity-capillary water waves with vorticity
研究二维不可压缩欧拉方程在有限深度无界域中的自由表面运动，构造初始数据使涡度梯度的L^∞范数在解寿命内具有双指数增长，扩展Zlatos和Hu-Luo-Yao的结果。
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.
We establish the existence and uniqueness of smooth solutions with large vorticity and weak solutions with vortex sheets/entropy waves for the steady Euler equations for both compressible and incompressible fluids in arbitrary infinitely long nozzles. We first develop a new approach to establish the existence of smooth solutions without assumptions on the sign of the second derivatives of the horizontal velocity, or the Bernoulli and entropy functions, at the inlet for the smooth case. Then the existence for the smooth case can be applied to construct approximate solutions to establish the exi…
Small scale creation for 2D free boundary Euler equations with surface tension
针对带表面张力、周期且有限深的二维自由边界不可压 Euler 方程，本文构造任意小速度初值，证明涡量梯度在解生命期内至少双指数增长，并通过自由边界一致控制与近似 Biot-Savart 估计完成推广。
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…
This paper is concerned with two-dimensional, steady, periodic water waves propagating at the free surface of water in a flow of constant vorticity over an impermeable flat bed. The motion of these waves is assumed to be governed both by surface tension and gravitational forces. A new reformulation of this problem is given, which is valid without restriction on the geometry or amplitude of the wave profiles and possesses a structure amenable to global bifurcation. The existence of global curves and continua of solutions bifurcating at certain parameter values from flat, laminar flows is then e…
We find conditions under which measures belong to . Next we show that measures generated by Prandtl, Kaden as well as Pullin spirals, objects considered by physicists as incompressible flows generating vorticity, satisfy assumptions of our theorem, thus they are (locally) elements of . Moreover, as a by-product, we prove an embedding of the space of Morrey type measures in .
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.
Bifurcation analysis for axisymmetric capillary water waves with vorticity and swirl
该研究针对带一般涡量、旋度及表面张力的定常轴对称水波，通过变量替换克服坐标奇异性，将问题转化为适用于Rabinowitz全局分岔定理的形式，构造出从平坦层流分岔的局部与全局解曲线。
Steady Ring-Shaped Vortex Sheets
本文构造两相欧拉方程中带涡面的薄环形行波解，利用隐函数定理保证局部唯一性，并推广开尔文-希克斯公式以包含表面张力。
Absence of anomalous dissipation for vortex sheets
本文证明二维环面上具有一般初始涡量和平方可积速度的Navier-Stokes流动在小粘性极限下不存在反常耗散，并给出显式耗散估计，基于对Nash不等式的新改进。
We consider the interface problem between two incompressible and inviscid fluids in the presence of surface tension. Following the geometric approach of [Shatah,J.;Zeng,C. A priori estimates for Fluid Interface Problems. CPAM, vol.16, no.6, June 2008] we show that solutions to this problem converge to solutions of the free-boundary Euler equations in vacuum as one of the densities goes to zero.
On the motion of a rigid body in a two-dimensional ideal flow with vortex sheet initial data
本文研究二维不可压缩理想流体中刚体在涡片初值下的运动问题，在初涡量为定号弥散有界Radon测度的特定情形下证明了à la Delort弱解的存在性，且解满足能量不等式、刚体加速度有界。
Axisymmetric capillary water waves with vorticity and swirl connecting to static unduloid configurations
本文研究受表面张力影响的具有一般涡量和旋流的稳态轴对称水波，证明任何静态unduloid构型都通过全局隐函数定理连接到一个非静态解的全局连续统，关键在于涉及完全椭圆积分的函数的严格单调性。
Solitary solutions to the steady Euler equations with piecewise constant vorticity in a channel
本文在无限通道中构造了具有分段常涡量的两层稳态欧拉方程的局部化孤立解，严格给出精确解曲线及渐近首阶项，并定性描述了含驻点、临界层及边界相交的流线粒子路径。
In this paper we prove the local well-posedness (LWP) for the 3D compressible Euler equations describing the motion of a liquid in an unbounded initial domain with moving boundary. The liquid is under the influence of gravity but without surface tension, and it is not assumed to be irrotational. We apply the tangential smoothing method introduced in [10,11] to construct the approximation system with energy estimates uniform in the smooth parameter. It should be emphasized that, when doing the nonlinear a priori estimates, we need neither the higher order wave equation of the pressure and delic…
In this article we examine the interaction of incompressible 2D flows with compact material boundaries. Our focus is the dynamic behavior of the circulation of velocity around boundary components and the possible exchange between flow vorticity and boundary circulation in flows with vortex sheet initial data We begin by showing that the velocity can be uniquely reconstructed from the vorticity and boundary component circulations, which allows to recast 2D Euler evolution using vorticity and the circulations as dynamic variables. The weak form of this vortex dynamics formulation of the equation…
Local-in-time existence of free-surface 3D Euler flow with $H^{2+δ}$ initial vorticity in a neighborhood of the free boundary
本文研究无表面张力自由边界三维 Euler 方程，在自由边界附近初始涡量具有 H^{2+δ} 正则性的条件下，利用拉格朗日方法导出先验估计，从而在 Rayleigh-Taylor 稳定条件下证明了解的局部时间存在性与唯一性。
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.
We consider the motion of the interface separating two domains of the same fluid that moves with different velocity along the tangential direction of the interface. We assume that the fluids occupying the two domains are of constant densities that are equal, are inviscid, incompressible and irrotational, and that the surface tension is zero. We discuss results on the existence and uniqueness of solutions for given data, the regularity of solutions, singularity formation and the nature of solutions after the singularity formation time.
We prove an almost global in time existence result of small amplitude space periodic solutions of the 1D gravity-capillary water waves equations with constant vorticity. The result holds for any value of gravity, vorticity and depth and any surface tension belonging to a full measure set. The proof demands a Hamiltonian paradifferential Birkhoff normal form reduction for quasi-linear PDEs in presence of resonant wave interactions: the normal form may be not integrable but it preserves the Sobolev norms thanks to its Hamiltonian nature. A major difficulty is that usual paradifferential calculus…
Vortex-sheet desingularization for three-dimensional ideal fluids
研究三维不可压缩欧拉方程解析涡旋片去奇异化定理，从伯克霍夫 - 罗特系统解析解出发，构造特定涡度，证明其在\(\varepsilon \to 0\)时收敛到规定涡旋片，得出解析三维涡旋片运动是精确欧拉流极限且寿命有下限的结论。
We present an existence and stability theory for gravity-capillary solitary waves with constant vorticity on the surface of a body of water of finite depth. Exploiting a rotational version of the classical variational principle, we prove the existence of a minimiser of the wave energy $\mathcal H$ subject to the constraint $\mathcal I=2μ$, where $\mathcal I$ is the wave momentum and $0< μ\ll 1$. Since $\mathcal H$ and $\mathcal I$ are both conserved quantities a standard argument asserts the stability of the set $D_μ$ of minimisers: solutions starting near $D_μ$ remain close to $D_μ$ in a suit…
Consider the dynamics of a layer of viscous incompressible fluid under the influence of gravity. The upper boundary is a free boundary with the effect of surface tension taken into account, and the lower boundary is a fixed boundary on which the Navier-slip condition is imposed. It is proved that there is a uniform time interval on which the estimates independent of both viscosity and surface tension coefficients of the solution can be established. This then allows one to justify the vanishing viscosity and surface tension limits by the strong compactness argument. In the presence of surface t…
Studies on singular flows in which either the velocity fields or the vorticity fields change dramatically on small regions are of considerable interests in both the mathematical theory and applications. Important examples of such flows include supersonic shock waves, boundary layers, and motions of vortex sheets, whose studies pose many outstanding challenges in both theoretical and numerical analysis. The aim of this talk is to discuss some of the key issues in studying such flows and to present some recent progress. First we deal with a supersonic flow past a perturbed cone, and prove the gl…
Controllability for 2D water waves: effects of bottom topography and constant vorticity
本文研究重力与表面张力作用下二维水波的可控性，针对一般底部地形（开稠密集合）的无旋波和平底情形的恒定涡量波，分别证明了可控性结果，并克服了特征值非显式带来的技术困难。
We prove a priori estimates for the compressible Euler equations modeling the motion of a liquid with moving physical vacuum boundary in an unbounded initial domain. The liquid is under influence of gravity but without surface tension. Our fluid is not assumed to be irrotational. But the physical sign condition needs to be assumed on the free boundary. We generalize the method used in \cite{LL} to prove the energy estimates in an unbounded domain up to arbitrary order. To our knowledge, this result appears to be the first that concerns a priori energy bounds for the compressible water wave. In…
We modify the nonlinear shallow water equations, the Korteweg-de Vries equation, and the Whitham equation, to permit constant vorticity, and examine wave breaking, or the lack thereof. By wave breaking, we mean that the solution remains bounded but its slope becomes unbounded in finite time. We show that a solution of the vorticity-modified shallow water equations breaks if it carries an increase of elevation; the breaking time decreases to zero as the size of vorticity increases. We propose a full-dispersion shallow water model, which combines the dispersion relation of water waves and the no…
Fully localised three-dimensional solitary water waves on Beltrami flows with strong surface tension
本文研究了在Beltrami流中强表面张力条件下，三维孤立水波的局域化存在性及KP-I方程的扰动解
In a recent work [1] the authors studied the dynamics of the interface separating a vacuum from an inviscid incompressible fluid, subject to the self-gravitational force and neglecting surface tension, in two space dimensions. The fluid is additionally assumed to be irrotational, and we proved that for data which are size $ε$ perturbations of an equilibrium state, the lifespan $T$ of solutions satisfies $T \gtrsim ε^{-2}$. The key to the proof is to find a nonlinear transformation of the unknown function and a coordinate change, such that the equation for the new unknown in the new coordinate …
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