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 velocity at the free surface, and are shown to arise from a variational principle. In the irrotational limit the equations reduce to the Zakharov-Craig-Sulem formulation of the classical three-dimensional steady water-wave problem, while $H(η)$ reduces to the familiar Dirichlet-Neumann operator.
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.
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 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 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.
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.
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.
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…
Understanding complexity in fluid mechanics is a major problem that has attracted the attention of physicists and mathematicians during the last decades. Using the concept of renormalization in dynamics, we show the existence of a locally dense set $\mathscr G$ of stationary solutions to the Euler equations in $\mathbb R^3$ such that each vector field $X\in \mathscr G$ is universal in the sense that any area preserving diffeomorphism of the disk can be approximated (with arbitrary precision) by the Poincaré map of $X$ at some transverse section. We remark that this universality is approximate …
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.
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 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.
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…
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.
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…
University of Alabama（阿拉巴马大学）；CONSERVE Center, Alabama Water Institute, The University of Alabama（阿拉巴马大学阿拉巴马水研究所CONSERVE中心）；University of Missouri（密苏里大学）
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…
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…
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.
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 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.
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…
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.
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 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…
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.
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.
We prove the existence of knotted and linked thin vortex tubes for steady solutions to the incompressible Euler equation in R^3. More precisely, given a finite collection of (possibly linked and knotted) disjoint thin tubes in R^3, we show that they can be transformed with a C^m-small diffeomorphism into a set of vortex tubes of a Beltrami field that tends to zero at infinity. The structure of the vortex lines in the tubes is extremely rich, presenting a positive-measure set of invariant tori and infinitely many periodic vortex lines. The problem of the existence of steady knotted vortex tubes…
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.
In this paper, we prove the nonlinear orbital stability of vortex dipoles for the quasi-geostrophic shallow-water (QGSW) equations. The vortex dipoles are explicit travelling wave solutions to the QGSW equations, which are analogues of the classical circular vortex of Lamb and Chaplygin for the steady planar Euler equations. We establish a variational characterization of these vortex poles, which provides a basis for the stability result.
This survey covers the mathematical theory of steady water waves with an emphasis on topics that are at the forefront of current research. These areas include: variational characterizations of traveling water waves; analytical and numerical studies of periodic waves with critical layers that may overhang; existence, nonexistence, and qualitative theory of solitary waves and fronts; traveling waves with localized vorticity or density stratification; and waves in three dimensions.
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.
In this article we study the so-called water tank system. In this system, the behavior of water contained in a 1-D tank is modelled by Saint-Venant equations, with a scalar distributed control. It is well-known that the linearized systems around uniform steady-states are not controllable, the uncontrollable part being of infinite dimension. Here we will focus on the linearized systems around non-uniform steady states, corresponding to a constant acceleration of the tank. We prove that these systems are controllable in Sobolev spaces, using the moments method and perturbative spectral estimates…
