Adrian Constantin, Konstantinos Kalimeris, Otmar Scherzer
We provide high-order approximations to periodic travelling wave profiles and to the velocity field and the pressure beneath the waves, in flows with constant vorticity over a flat bed.
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We prove that for a large class of vorticity functions the crest of a corresponding travelling water wave is necessarily a point of maximal horizontal velocity. We also show that for waves with nonpositive vorticity the pressure in the flow is everywhere larger than the atmospheric pressure. A related a priori estimate for waves with nonnegative vorticity is also given.
We prove that the streamlines and the profile of periodic gravity water waves traveling over a flat bed with wavespeed which exceeds the horizontal velocity of all fluid particles are real-analytic graphs if the vorticity function is merely integrable.
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.
In this paper we construct periodic capillarity-gravity water waves with a piecewise constant vorticity distribution. They describe water waves traveling on superposed linearly sheared currents that have different vorticities. This is achieved by associating to the height function formulation of the water wave problem a diffraction problem where we impose suitable transmission conditions on each line where the vorticity function has a jump. The solutions of the diffraction problem, found by using local bifurcation theory, are the desired solutions of the hydrodynamical problem.
In this paper, we prove the existence of two-dimensional, traveling, capillary-gravity, water waves with compactly supported vorticity. Specifically, we consider the cases where the vorticity is a $δ$-function (a point vortex), or has small compact support (a vortex patch). Using a global bifurcation theoretic argument, we construct a continuum of finite-amplitude, finite-vorticity solutions for the periodic point vortex problem. For the non-periodic case, with either a vortex point or patch, we prove the existence of a continuum of small-amplitude, small-vorticity solutions.
Steady periodic water waves bifurcating for fixed-depth rotational flows with discontinuous vorticity
该研究运用局部分叉理论，证明了平坦床面、固定平均深度且涡量分布间断的旋转流动中存在小振幅稳态周期水波。
We construct small-amplitude steady periodic gravity water waves arising as the free surface of water flows that contain stagnation points and possess a discontinuous distribution of vorticity in the sense that the flows consist of two layers of constant but different vorticities. We also describe the streamline pattern in the moving frame for the constructed flows.
On the amplitude of steady water waves with positive constant vorticity
该研究针对具有恒定有利涡量的单向二维稳态纯重力水波，推导得出波振幅的显式界（随涡量趋于无穷衰减至零），且该结论适用于任意水波，无需限定周期、孤立或对称情形。
On the capillary water waves with constant vorticity
本文研究实数轴上二维常涡度深水波的局部适定性，通过抛物线性化建立拟线性色散方程组，并利用能量估计与Strichartz估计证明了该重力-毛细水波系统在$\mathcal{H}^{s}(\mathbb{R})$中局部适定。
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.
We prove a long-term regularity result for three-dimensional gravity water waves with small initial data but nonzero initial vorticity. We consider solutions whose vorticity vanishes on the free boundary and use this to derive a system for the evolution of the free boundary which reduces to the Zakharov/Craig-Sulem formulation in the irrotational case. We are able to continue the solution until a time determined by the size of the initial vorticity in such a way that if the vorticity is zero, one recovers a lifespan $T\sim ε^{-N}$ where $N$ can be taken arbitrarily large if the initial data is…
In this paper we derive a new formulation of the water waves equations with vorticity that generalizes the well-known Zalkarov-Craig-Sulem formulation used in the irrotational case. We prove the local well-posedness of this formulation, and show that it is formally Hamiltonian. This new for- mulation is cast in Eulerian variable, and in finite depth; we show that it can be used to provide uniform bounds on the lifespan and on the norms of the solutions in the singular shallow water regime. As an application to these re- sults, we derive and provide the first rigorous justification of a shallow…
We consider two-dimensional solitary water waves on a shear flow with an arbitrary distribution of vorticity. Assuming that the horizontal velocity in the fluid never exceeds the wave speed and that the free surface lies everywhere above its asymptotic level, we give a very simple proof that a suitably defined Froude number $F$ must be strictly greater than the critical value $F=1$. We also prove a related upper bound on $F$, and hence on the amplitude, under more restrictive assumptions on the vorticity.
Equivalent formulations for steady periodic water waves of fixed mean-depth with discontinuous vorticity
针对不连续涡量的固定平均深度稳态周期水波问题，证明了广义欧拉公式、广义流函数公式与弱修正高度公式三种弱公式在赫尔德空间中的等价性。
In this paper, we consider steady Euler flows in a planar bounded domain in which the vorticity is sharply concentrated in a finite number of disjoint regions of small diameter. Such flows are closely related to the point vortex model and can be regarded as desingularization of point vortices. By an adaption of the vorticity method, we construct a family of steady Euler flows in which the vorticity is concentrated near a global minimum point of the Robin function of the domain, and the corresponding stream function satisfies a semilinear elliptic equation with a given profile function. Further…
Sharp Basic Velocity Estimates for the Plane Steady Navier--Stokes Equations through Vorticity
本文针对平面稳态Navier-Stokes方程，利用涡量的L^2范数，给出了速度在两个同心圆上平均值之差的最优估计，并通过反例证明了其尖锐性。
We prove the existence of small amplitude time quasi-periodic solutions of the pure gravity water waves equations with constant vorticity, for a bidimensional fluid over a flat bottom delimited by a space periodic free interface. Using a Nash-Moser implicit function iterative scheme we construct traveling nonlinear waves which pass through each other slightly deforming and retaining forever a quasiperiodic structure. These solutions exist for any fixed value of depth and gravity and restricting the vorticity parameter to a Borel set of asymptotically full Lebesgue measure.
In this paper we consider the steady water wave problem for waves that possess a merely $L_r-$integrable vorticity, with $r\in(1,\infty)$ being arbitrary. We first establish the equivalence of the three formulations--the velocity formulation, the stream function formulation, and the height function formulation-- in the setting of strong solutions, regardless of the value of $r$. Based upon this result and using a suitable notion of weak solution for the height function formulation, we then establish, by means of local bifurcation theory, the existence of small amplitude capillary and capillary…
This article is concerned with infinite depth gravity water waves with constant vorticity in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good and stable approximation to the system on the natural cubic time scale. The proof relies on refined cubic energy estimates and perturbative analysis.
We study stationary capillary-gravity waves in a two-dimensional body of water that rests above a flat ocean bed and below vacuum. This system is described by the Euler equations with a free surface. Our main result states that there exist large families of such waves that carry finite energy and exhibit an exponentially localized distribution of (nontrivial) vorticity. This is accomplished by combining ideas drawn from the theory of spike-layer solutions to singularly perturbed elliptic equations, with techniques from the study of steady solutions of the water wave problem.
We prove real analyticity of all the streamlines, including the free surface, of a gravity- or capillary-gravity-driven steady flow of water over a flat bed, with a Hölder continuous vorticity function, provided that the propagating speed of the wave on the free surface exceeds the horizontal fluid velocity throughout the flow. Furthermore, if the vorticity possesses some Gevrey regularity of index $s$, then the stream function admits the same Gevrey regularity throughout the fluid domain; in particular if the Gevrey index $s$ equals to 1, then we obtain analyticity of the stream function. The…
The vortex-wave system describes the motion of a two-dimensional ideal fluid in which the vorticity includes continuously distributed vorticity, which is called the background vorticity, and a finite number of concentrated vortices. In this paper we restrict ourselves to the case of a single point vortex in bounded domains. We prove the existence of steady vortex patch solutions to this system with prescribed distribution for the background vorticity. Moreover, we show that the supports of these solutions "shrink" to a minimum point of the Kirchhoff-Routh function as the strength parameter of …
This paper studies the classical water wave problem with vorticity described by the Euler equations with a free surface under the influence of gravity over a flat bottom. Based on fundamental work \cite{ConstantinStrauss}, we first obtain two continuous bifurcation curves which meet the laminar flow only one time by using modified analytic bifurcation theorem. They are symmetric waves whose profiles are monotone between each crest and trough. Furthermore, we find that there is at least one inflection point on the wave profile between successive crests and troughs and the free surface is strict…
This paper studies periodic traveling gravity waves at the free surface of water in a flow of constant vorticity over a flat bed. Using conformal mappings the free-boundary problem is transformed into a quasilinear pseudodifferential equation for a periodic function of one variable. The new formulation leads to a regularity result and, by use of bifurcation theory, to the existence of waves of small amplitude even in the presence of stagnation points in the flow.
Stability of two-dimensional steady Euler flows with concentrated vorticity
本文研究二维稳态欧拉流在集中涡量下的非线性稳定性，通过证明流为等涡面上动能局部最大化子的紧且孤立集合，并利用流函数分离性质，得出涡量集中在一个或两个异号小区域时流稳定。
Low regularity well-posedness for two-dimensional deep gravity water waves with constant vorticity
本文研究无限深、非零常涡量二维重力水波的低正则性局部适定性，利用低正则能量与坐标/变量方法证明初值在H^s（s≥3/4）中适定，将此前结果提升1/4阶，并对应无涡情形的突破。
In this paper we construct periodic capillarity-gravity water waves with an arbitrary bounded vorticity distribution. This is achieved by reexpressing, in the height function formulation of the water wave problem, the boundary condition obtained from Bernoulli's principle as a nonlocal differential equation. This enables us to establish the existence of weak solutions of the problem by using elliptic estimates and bifurcation theory. Secondly, we investigate the a priori regularity of these weak solutions and prove that they are in fact strong solutions of the problem, describing waves with a …
Shock formation for the 2D rotating shallow water equations with non-zero vorticity
本文构造了导致二维旋转浅水系统有限时间爆破的初始数据，首次完整描述了带涡量情形下激波形成的几何机制，并推广了相关方法。
Two-dimensional water waves with constant vorticity and general bottom topography
针对恒定涡度、有限深度和一般底部地形的二维水波，提出一种推广的Zakharov-Craig-Sulem和Constantin-Ivanov-Prodanov的公式，并研究其中出现的算子，证明局部适定性。
This paper considers two-dimensional steady solitary waves with constant vorticity propagating under the influence of gravity over an impermeable flat bed. Unlike in previous works on solitary waves, we allow for both internal stagnation points and overhanging wave profiles. Using analytic global bifurcation theory, we construct continuous curves of large-amplitude solutions. Along these curves, either the wave amplitude approaches the maximum possible value, the dimensionless wave speed becomes unbounded, or a singularity develops in a conformal map describing the fluid domain. We also show t…
Two-dimensional solitary water waves with constant vorticity, Part II: the deep capillary case
本文研究无限深度常涡量二维毛细水波，通过将Babenko方程约化为聚焦三次非线性Schrödinger方程，证明存在无量纲参数临界值，并据此得到小振幅孤立波解的存在性。
Two dimensional solitary water waves with constant vorticity, Part I: the deep gravity case
该研究在无限深常涡量二维纯重力水波场景下，采用全纯坐标与不动点论证，构造出临界速度附近、类缩放Benjamin-Ono孤子的光滑孤立波，并给出其渐近展开式。
We consider the angle $θ$ of inclination (with respect to the horizontal) of the profile of a steady 2D inviscid symmetric periodic or solitary water wave subject to gravity. Although $θ$ may surpass 30$^\circ$ for some irrotational waves close to the extreme wave, Amick [Ami87] proved that for any irrotational wave the angle must be less than 31.15$^\circ$. Is the situation similar for periodic or solitary waves that are not irrotational? The extreme Gerstner wave has infinite depth, adverse vorticity and vertical cusps ($θ= 90^\circ$). Moreover, numerical calculations show that even waves of…
Large-Amplitude Steady Solitary Water Waves with General Vorticity
研究有限深度流体表面稳态孤立重力波，处理一般涡度分布和悬垂波形，通过共形映射将问题转化为椭圆系统和边界值问题，利用中心 manifold 方法证明小振幅波存在，基于解析全局分支定理获得大振幅波。
Large-amplitude steady gravity water waves with general vorticity and critical layers
本文研究了有限深度水面上的二维稳态周期重力波，允许悬垂水面、停滞点和临界层存在。通过保角映射和Bernoulli方程重构，利用Rabinowitz全局分岔定理构造了从层流分岔出的全局连通解集，并进行了节点分析及顺流波研究。
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…
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…
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。
Global bifurcation of capillary-gravity water waves with overhanging profiles and arbitrary vorticity
针对平坦河床上方带任意涡量的二维周期性毛细重力水波问题，提出基于共形映射与Babenko型伯努利方程重构的等价转化方法，无需限制剖面几何与假设无驻点，构造出从平表层流分岔的局部及全局解曲线。
We prove real analyticity of all the streamlines, including the free surface, of a steady stratified flow of water over a flat bed in the absence of stagnation points, with a Hölder continuous Bernoulli function and a Hölder continuously differentiable density function. Furthermore, we show that if the Bernoulli function and the density function possess some Gevrey regularity of index s, then the stream function admits the same Gevrey regularity throughout the fluid domain; in particular if the Gevrey index s equals to 1, then we obtain analyticity of the stream function. The regularity result…
Two-component system modelling shallow-water waves with constant vorticity under the Camassa-Holm scaling
该研究从带线性剪切的 Green-Naghdi 方程推导出 Camassa-Holm 标度下恒定涡量浅水波二分量系统，并证明其 Besov 空间局部适定性、爆破准则及特殊情形全局强解条件。
Analytical study of a generalized Dirichlet-Neumann operator for three-dimensional water waves with vorticity
本文研究了重力作用下带涡量三维水波中由 Castro 和 Lannes 提出的广义 Dirichlet-Neumann 算子，将其在无旋水波中的经典结果（如 Taylor 展开、微分计算和仿线性化）进行了推广，且未对速度场或涡量施加任何几何假设。
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。
We investigate the existence of solitary gravity waves traversing a two-dimensional body of water that is bounded below by a flat impenetrable ocean bed and above by a free surface of constant pressure. Our main interest is constructing waves of this form that exhibit a discontinuous distribution of vorticity. More precisely, this means that the velocity limits both upstream and downstream to a laminar flow that is merely Lipschitz continuous. We prove that, for any choice of background velocity with this regularity, there exists a global curve of solutions bifurcating from a critical laminar …
A new highly nonlinear equation modelling shallow-water waves with constant vorticity
本文通过形式渐近展开和摄动理论，从常涡量不可压缩流体方程导出一个新的高度非线性浅水模型，采用比Camassa-Holm更大的最优尺度以避免非局部项，并在Besov空间中建立局部适定性和爆破判据。
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。
In this paper, we investigate the singularity near the degenerate points of the steady axisymmetric flow with general vorticity of an inviscid incompressible fluid acted on by gravity and with a free surface. We called the points on the free boundary at which the gradient of the stream function vanishes as the degenerate points. The main results in this paper give the different classifications of the singularity near the degenerate points on the free surface. More precisely, we obtained that at the stagnation points, the possible profiles must be a Stokes corner, or a horizontal cusp, or a hor…
On two-dimensional steady compactly supported Euler flows with constant vorticity
本文研究了二维稳态有限支撑不可压缩欧拉方程的自由边界问题，探讨了常涡度流与环形平衡流的扰动，通过形状导数和局部分支理论证明了非平凡可接受域的存在，并展示了标准环形流在小扰动下的稳定性。
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。
This paper is a study of the water wave problem in a two-dimensional domain of infinite depth in the presence of nonzero constant vorticity. A goal is to describe the effects of uniform shear flow on the modulation of weakly nonlinear quasi-monochromatic surface gravity waves. Starting from the Hamiltonian formulation of this problem and using techniques from Hamiltonian transformation theory, we derive a Hamiltonian Dysthe equation for the time evolution of the wave envelope. Consistent with previous studies, we observe that the uniform shear flow tends to enhance or weaken the modulational i…
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.
Gravity water waves over constant vorticity flows: From laminar flows to touching waves
本文扩展了Hur & Wheeler关于常涡度流上重力水波的存在性结果，证明了在固定非零重力下存在连接层流与接触波的连续曲线，并揭示了破波和临界层行为。
Full-Density Degenerate Stagnation Points for Water Waves with General Vorticity
本文针对带一般涡量的二维行进重力水波，利用改进的Weiss型单调性公式与Almgren型频率公式，证明了严格双侧线性增长区域外不存在全密度退化驻点，并推广了旋转波的Stokes猜想。
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。
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…
The problem for two-dimensional steady gravity driven water waves with vorticity is investigated. Using a multidimensional bifurcation argument, we prove the existence of small-amplitude periodic steady waves with an arbitrary number of crests per period. The role of bifurcation parameters is played by the roots of the dispersion equation.
We prove the existence of three-dimensional steady gravity-capillary waves with vorticity on water of finite depth. The waves are periodic with respect to a given two-dimensional lattice and the relative velocity field is a Beltrami field, meaning that the vorticity is collinear to the velocity. The existence theory is based on multi-parameter bifurcation theory.
In this paper we mainly investigate the traveling wave solution of the two dimensional Euler equations with gravity at the free surface over a flat bed. We assume that the free surface is almost periodic in the horizontal direction. Using conformal mappings, one can change the free boundary problem into a fixed boundary problem with some unknown functions in the boundary condition. By virtue of the Hilbert transform, the problem is equivalent to a quasilinear pseudodifferential equation for a almost periodic function of one variable. The bifurcation theory ensures us to obtain a existence resu…
The quasi-geostrophic equation or the Euler equation with dissipation studied in the present paper is a simplified form of the atmospheric circulation model introduced by Charney and DeVore [J. Atmos. Sci. 36(1979), 1205-1216] on the existence of multiple steady states to the understanding of the persistence of atmospheric blocking. The fluid motion defined by the equation is driven by a zonal thermal forcing and an Ekman friction forcing measured by $κ>0$. It is proved that the steady-state solution is unique for $κ>1$ while multiple steady-state solutions exist for $κ<κ_{crit}$ with respect …
Smooth transonic flows with nonzero vorticity to a quasi two dimensional steady Euler flow model
研究准二维定常欧拉流模型中非零涡量光滑跨音速流的存在性、唯一性与结构稳定性，通过退化约化和变形-旋度分解等方法，证明了正加速度情形下解的存在唯一性。
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