Samuel Walsh
This paper considers two-dimensional stratified water waves propagating under the force of gravity over an impermeable flat bed and with a free surface. We prove the existence of a global continuum of classical solutions that are periodic and traveling. These waves, moreover, can exhibit large density variation, speed and amplitude.
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Spectral properties of the Frechet derivatives of stratified steady Stokes waves
研究层状稳态斯托克斯波的弗雷歇导数频谱性质，证明斯托克斯波附近不存在次谐波水波及多重周期波。
Consider a two-dimensional stratified solitary wave propagating through a body of water that is bounded below by an impermeable ocean bed. In this work, we study how such a wave can be reconstructed from data consisting of the wave speed, upstream and downstream density profile, and the trace of the pressure on the bed. First, we prove that this data uniquely determines the wave, both in the (real) analytic and Sobolev regimes. Second, for waves that consist of multiple layers of constant density immiscible fluids, we provide an exact formula describing each of the interfaces in terms of the d…
In this paper, we mainly consider two class of travelling stratified periodic water waves, one with negative (or without) surface tension and the other with constant Bernoulli's function and stagnation points. We first establish the symmetry result for stratified water waves with negative (or without) surface tension, but without stagnation by using the modified maximum principle. Furthermore, the symmetry property of stratified water waves with constant Bernoulli's function and stagnation points is also obtained provided the monotonic property is known.
Variational formulation for stratified steady water wave in two-layer flows
本文给出了两层流中稳定周期分层水波的变分公式，证明了自然能量泛函的临界点即控制方程的解，并给出了该泛函的二阶变分。
Sharp stability and instability of stratified steady states for the incompressible porous media equation
本文研究二维不可压缩多孔介质方程分层稳态的稳定性，证明周期通道上均匀递减稳态在小扰动下非线性稳定，而具有正导数上确界的稳态则非线性不稳定，并给出收敛速率与不稳定性机制。
We construct an explicit steady stratified purely azimuthal flow for the governing equations of geophysical fluid dynamics. These equations are considered in a setting that applies to the Antarctic Circumpolar Current, accounting for eddy viscosity and forcing terms.
We prove the existence of small steady periodic capillary-gravity water waves for general stratified flows, where we allow for stagnation points in the flow. We establish the existence of both laminar and non-laminar flow solutions for the governing equations. This is achieved by using bifurcation theory and estimates based on the ellipticity of the system, where we regard, in turn, the mass-flux and surface tension as bifurcation parameters.
We consider Euler's equations for free surface waves traveling on a body of density stratified water in the scenario when gravity and surface tension act as restoring forces. The flow is continuously stratified, and the water layer is bounded from below by an impermeable horizontal bed. For this problem we establish three equivalent classical formulations in a suitable setting of strong solutions which may describe nevertheless waves with singular density gradients. Based upon this equivalence we then construct two-dimensional symmetric periodic traveling waves that are monotone between each c…
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…
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.
The existence of stratified linearly steady two-mode water waves with stagnation points
本文通过共形映射与多维核分岔定理，首次构造了分层线性流中具有内部驻点的稳态双模态毛细-重力水波，并证明其存在性与形式稳定性。
This paper considers two-dimensional steady continuous stratified periodic water waves. Firstly, we prove that each streamline must be symmetric about the crest line when it is strictly monotonous between troughs and crests by exploiting the maximum principle and analysis of surface profile. Then, standard Schauder estimates are exploited on the uniform oblique derivative problems to show that all streamlines are real analytic (including the free surface). Based on above symmetry and regularity of streamlines, finally we provide an analytic expansion method to recover the water waves from hori…
We construct small-amplitude periodic water waves with multiple critical layers. In addition to waves with arbitrarily many critical layers and a single crest in each period, two-dimensional sets of waves with several crests and troughs in each period are found. The setting is that of steady two-dimensional finite-depth gravity water waves with vorticity.
Approximating a continuously stratified hydrostatic system by the multi-layer shallow water system
该研究建立了带Gent-McWilliams附加项的多层浅水系统与连续分层流体静力学不可压缩Euler方程的关联，证明稳定分层下后者正则解可被前者近似且收敛速率为1/N²，依赖离散化能量估计是证明关键。
We study small-amplitude steady water waves with multiple critical layers. Those are rotational two-dimensional gravity-waves propagating over a perfect fluid of finite depth. It is found that arbitrarily many critical layers with cat's-eye vortices are possible, with different structure at different levels within the fluid. The corresponding vorticity depends linearly on the stream function.
This paper considers two-dimensional gravity solitary waves moving through a body of density stratified water lying below vacuum. The fluid domain is assumed to lie above an impenetrable flat ocean bed, while the interface between the water and vacuum is a free boundary where the pressure is constant. We prove that, for any smooth choice of upstream velocity field and density function, there exists a continuous curve of such solutions that includes large-amplitude surface waves. Furthermore, following this solution curve, one encounters waves that come arbitrarily close to possessing points of…
Our aim is to study the effect of a continuous prescribed density variation on the propagation of ocean waves. More precisely, we derive KdV-type shallow water model equations for unidirectional flows along the Equator from the full governing equations by taking into account a prescribed but arbitrary depth-dependent density distribution. In contrast to the case of constant density, we obtain for each fixed water depth a different model equation for the horizontal component of the velocity field. We derive explicit formulas for traveling wave solutions of these model equations and perform a de…
We present a large-amplitude existence theory for two-dimensional solitary waves propagating through a two layer body of water. The domain of the fluid is bounded below by an impermeable flat ocean floor and above by a free boundary at constant pressure. For any piecewise smooth upstream density distribution and laminar background current, we construct a global curve of solutions. This curve bifurcates from the background current and, following along the curve, we find waves that are arbitrarily close to having horizontal stagnation points. The small-amplitude waves are constructed using a cen…
There are two distinct regimes commonly used to model traveling waves in stratified water: continuous stratification, where the density is smooth throughout the fluid, and layer-wise continuous stratification, where the fluid consists of multiple immiscible strata. The former is the more physically accurate description, but the latter is frequently more amenable to analysis and computation. By the conservation of mass, the density is constant along the streamlines of the flow; the stratification can therefore be specified by prescribing the value of the density on each streamline. We call this…
Amplitude bounds of steady rotational water waves
本文研究恒定涡量下稳定旋转水波的振幅界，分别在 $γ>0$ 且足够小以及 $γ\leq0$ 的情形证明绝对上界，并揭示振幅随 $γ\to-\infty$ 趋于零。
Liouville-type results for time-dependent stratified water flows over variable bottom in the $β$-plane approximation
本研究证明在$\eta$-平面近似下，变底地形上两层常涡度三维分层水流的有界解存在条件，并揭示界面平坦、自由表面行进特征及静水压力等性质。
Unstable Manifolds of Stratified Euler Equations
本文针对d维区域上不可压缩分层欧拉方程的谱不稳定定态，利用李雅普诺夫-佩尔顿方法构造其局部稳定与不稳定流形，并探讨了相关应用。
In this article, we mainly investigate the properties of vertical velocity v for two dimensional steady water waves over a flat bed. Firstly we prove the existence of the inflection point for each streamline, then we find the behavior of v along each streamline depends strictly on concavity and convexity of streamline, which contributes to complete Constantin conjecture on v in Stokes wave. And the location of maximum vertical fluid velocity is also proven to be at the inflection point. Besides, we also extend our results to the cases with monotonous vorticity.
We construct three-dimensional families of small-amplitude gravity-driven rotational steady water waves on finite depth. The solutions contain counter-currents and multiple crests in each minimal period. Each such wave generically is a combination of three different Fourier modes, giving rise to a rich and complex variety of wave patterns. The bifurcation argument is based on a blow-up technique, taking advantage of three parameters associated with the vorticity distribution, the strength of the background stream, and the period of the wave.
Mathematical and numerical study of a model for navigation in stratified waters
本文提出了一种用于分层水域中导航的线性模型，并设计了基于离散傅里叶变换和指数积分器的数值方案，分析了临界速度对两种状态的影响。
We prove the equivalence of three weak formulations of the steady water waves equations, namely the velocity formulation, the stream function formulation, and the Dubreil-Jacotin formulation, under weak Holder regularity assumptions on their solutions.
We prove local higher-order asymptotics for extreme water waves with vorticity near stagnation points. We obtain that the behaviour of solutions and their regularity depend substantially on the vorticity. In particular, we show that extreme waves with a negative vorticity distribution have concave profiles near the crest. Our approach is based on new regularity results and asymptotic analysis of the corresponding nonlinear problem in a half-strip. Our main result is local and therefore is valid for a broad range of problems, such as for waves with a piecewise constant vorticity, stratified wav…
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.
In this paper, we investigate steady states of shallow water moment equations including bottom topographies. We derive a new hyperbolic shallow water moment model based on linearized moment equations that allows for a simple assessment of the steady states. After proving hyperbolicity of the new model, the steady states are fully identified. A well-balanced scheme is adopted to the specific structure of the new model and allows to preserve the steady states in numerical simulations.
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 consider a singular limit problem for the complete compressible Euler system in the low Mach and strong stratification regime. We identify the limit problem - the anelastic Euler system - in the case of well prepared initial data. The result holds in the large class of the dissipative measure-valued solutions of the primitive system. Applications are discussed to the driven shallow water equations.
We consider steady three-dimensional gravity-capillary water waves with vorticity propagating on water of finite depth. We prove a variational principle for doubly periodic waves with relative velocities given by Beltrami vector fields, under general assumptions on the wave profile.
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。
In this paper we develop an existence theory for small amplitude, steady, two-dimensional water waves in the presence of wind in the air above. The presence of the wind is modeled by a Kelvin--Helmholtz type discontinuity across the air--water interface, and a corresponding jump in the circulation of the fluids there. We consider both fluids to be inviscid, with the water region being irrotational and of finite depth. The air region is considered with constant vorticity in the case of infinite depth and with a general vorticity profile in the case of a finite, lidded atmosphere.
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。
We prove the Benjamin and Lighthill conjecture for all two-dimensional steady water waves with an arbitrary vorticity distribution. We show that the flow force constant of an arbitrary smooth wave is bounded by the corresponding flow force constants for conjugate laminar flows. We prove these inequalities without any assumptions on the geometry of the surface profile and put no restrictions on wave's amplitude. Furthermore, we give a complete description of cases when equalities can occur. Our results are new already for Stokes waves with vorticity, while the case of equalities is new even in …
In this note, we show the existence of regular solutions to the stationary version of the Navier-Stokes system for compressible fluids with a density dependent viscosity, known as the shallow water equations. For arbitrary large forcing we are able to construct a solution, provided the total mass is sufficiently large. The main mathematical part is located in the construction of solutions. Uniqueness is impossible to obtain, since the gradient of the velocity is of magnitude of the force. The investigation is connected to the corresponding singular limit as Mach number goes to zero and methods…
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…
In this paper, we present results about the existence and uniqueness of solutions of elliptic equations with transmission and Wentzell boundary conditions. We provide Schauder estimates and existence results in Hölder spaces. As an application, we develop an existence theory for small-amplitude two-dimensional traveling waves in an air-water system with surface tension. The water region is assumed to be irrotational and of finite depth, and we permit a general distribution of vorticity in the atmosphere.
We construct large families of two-dimensional travelling water waves propagating under the influence of gravity in a flow of constant vorticity over a flat bed. A Riemann-Hilbert problem approach is used to recast the governing equations as a one-dimensional elliptic pseudo-differential equation with a scalar constraint. The structural properties of this formulation, which arises as the Euler-Lagrange equation of an energy functional, enable us to develop a theory of analytic global bifurcation.
In this paper we consider two-dimensional, stratified, steady water waves propagating over an impermeable flat bed and with a free surface. The motion is assumed to be driven by capillarity (that is, surface tension) on the surface and a gravitational force acting on the body of the fluid. We prove the existence of global continua of classical solutions that are periodic and traveling. This is accomplished by first constructing a 1-parameter family of laminar flow solutions, $\mathcal{T}$, then applying bifurcation theory methods to obtain local curves of small amplitude solutions branching fr…
We study wave-current interactions in two-dimensional water flows of constant vorticity over a flat bed. For large-amplitude periodic traveling waves that propagate at the water surface in the same direction as the underlying current (downstream waves), we prove explicit uniform bounds for their amplitude. In particular, our estimates show that the maximum amplitude of the waves becomes vanishingly small as the vorticity increases without limit. We also prove that the downstream waves on a global bifurcating branch are never overhanging, and that their mass flux and Bernoulli constant are unif…
This is a study of two-dimensional steady periodic travelling waves on the surface of an infinitely deep irrotational ocean, when the top streamline is in contact with a membrane which has a nonlinear response to stretching and bending, and the pressure in the air above is constant. It is not supposed that the waves have small amplitude. The problem of existence of such waves is addressed using methods from the calculus of variations. The analysis involves the Hilbert transform and a Riemann-Hilbert formulation.
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.
Stratified equatorial flows in cylindrical coordinates with surface tension
本文研究了柱坐标下具有自由表面的无粘不可压缩流体在赤道区域的稳定流动，考虑了重力和表面张力作用下的波传播，并证明了自由表面的正则性结果。
This paper considers steady surface waves `riding' a Beltrami flow (a three-dimensional flow with parallel velocity and vorticity fields). It is demonstrated that the hydrodynamic problem can be formulated as two equations for two scalar functions of the horizontal spatial coordinates, namely the elevation $η$ of the free surface and the potential $Φ$ defining the gradient part (in the sense of the Hodge-Weyl decomposition) of the horizontal component of the tangential fluid velocity there. These equations are written in terms of a nonlocal operator $H(η)$ mapping $Φ$ to the normal fluid veloc…
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。
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…
In this paper we present a characterization of the symmetric rotational periodic gravity water waves of finite depth and without stagnation points in terms of the underlying flow. Namely, we show that such a wave is symmetric and has a single crest and trough per period if and only if there exists a vertical line within the fluid domain such that all the fluid particles located on that line minimize there simultaneously their distance to the fluid bed as they move about. Our analysis uses the moving plane method, sharp elliptic maximum principles, and the principle of analytic continuation.
The present work is devoted to the derivation of a fully well-balanced and positivepreserving numerical scheme for the shallow water equations with Coriolis force. The first main issue consists in preserving all the steady states, including the geostrophic equilibrium. Our strategy relies on a Godunov-type scheme with suitable source term and steady state discretisations. The second challenge lies in improving the order of the scheme while preserving the fully well-balanced property. A modification of the classical methods is required since no conservative reconstruction can preserve all the s…
Global branches of Stokes waves of variable period on stratified fluids
研究二维通道中分层稳态水波，考虑从层流分岔的Stokes波分支，以波周期为参数，提出新密度和Bernoulli函数类，使层流产生全局分岔分支，且分岔周期可任意大。
On the vorticity threshold for steady water waves
本文针对定常水波，证明了高于明确阈值的恒定反向涡量下第一个停滞点必在波峰正下方床面，还建立强反向涡量单向波的相关不等式与新振幅上界。
We consider a spectral problem associated with steady water waves of extreme form on the free surface of a rotational flow. It is proved that the spectrum of this problem contains arbitrary large negative eigenvalues and they are simple. Moreover, the asymptotics of such eigenvalues is obtained.
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…
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…
Compressible steady water waves
本文通过分岔方法，以马赫数为参数，从不可压缩解构造可压缩水波，涵盖周期波、孤立波及两相前驱波，并给出詹森-瑞利展开的严格版本。
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…
A new model for two-layer liquid-gas stratified flows in pipes with general cross sections
本文提出了一种适用于通用横截面管道中两层液气分层流的新模型，通过非保守项连接底层不可压缩流体和顶层可压缩气体，实现了质量、动量和能量守恒，并通过数值测试验证了模型的平衡性和收敛性。
The paper considers Euler-Poisson equations which govern the steady state of a self gravitating, rotating, axi-symmetric fluid under the additional assumption that it is incompressible and stratified. In this setting we show that the original system of six nonlinear partial differential equations can be reduced to two equations, one for the mass density and the other for gravitational field. This reduction is carried out in cylindrical coordinates. As a result we are able to derive also expressions for the pressure as a function of the density. The resulting equations are then solved analytica…
In this paper we construct small amplitude periodic internal waves traveling at the boundary region between two rotational and homogeneous fluids with different densities. Within a period, the waves we obtain have the property that the gradient of the stream function associated to the fluid beneath the interface vanishes, on the wave surface, at exactly two points. Furthermore, there exists a critical layer which is bounded from above by the wave profile. Besides, we prove, without excluding the presence of stagnation points, that if the vorticity function associated to each fluid in part is r…
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