We rigorously justify in 3D the main asymptotic models used in coastal oceanography, including: shallow-water equations, Boussinesq systems, Kadomtsev-Petviashvili (KP) approximation, Green-Naghdi equations, Serre approximation and full-dispersion model. We first introduce a ``variable'' nondimensionalized version of the water-waves equations which vary from shallow to deep water, and which involves four dimensionless parameters. Using a nonlocal energy adapted to the equations, we can prove a well-posedness theorem, uniformly with respect to all the parameters. Its validity ranges therefore from shallow to deep-water, from small to large surface and bottom variations, and from fully to weakly transverse waves. The physical regimes corresponding to the aforementioned models can therefore be studied as particular cases; it turns out that the existence time and the energy bounds given by the theorem are always those needed to justify the asymptotic models. We can therefore derive and justify them in a systematic way.
In this paper we consider the initial value problem for a family of shallow water equations on the line $\R$ with various asymptotic conditions at infinity. In particular we construct solutions with prescribed asymptotic expansion as $x\to\pm\infty$ and prove their invariance with respect to the solution map.
Consider the capillary water waves equations, set in the whole space with infinite depth, and consider small data (i.e. sufficiently close to zero velocity, and constant height of the water). We prove global existence and scattering. The proof combines in a novel way the energy method with a cascade of energy estimates, the space-time resonance method and commuting vector fields.
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 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.
We consider two- and three-dimensional gravity and gravity-capillary solitary water waves in infinite depth. Assuming algebraic decay rates for the free surface and velocity potential, we show that the velocity potential necessarily behaves like a dipole at infinity and obtain a related asymptotic formula for the free surface. We then prove an identity relating the "dipole moment" to the kinetic energy. This implies that the leading-order terms in the asymptotics are nonvanishing and in particular that the angular momentum is infinite. Lastly we prove a related integral identity which rules ou…
In arXiv:0908.1031, Varvaruca and Weiss eliminate the existence of cusps for a free-boundary problem for two-dimensional water waves under assumptions that hold for solutions such that $\{u>0\}$ is a "strip-like" domain in the sense of arXiv:0708.4371. In this paper it is proven that cusps do not exists in the natural setting for these free-boundary problems. In particular, non strip-like domains are also allowed. This builds upon recent work on non-existence of cusps in arXiv:2202.00616.
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 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.
In this work we introduce a well-posed mathematical model for the processes involved in the artificial circulation of water, in order to avoid eutrophication phenomena, for instance, in a lake. This novel and general formulation is based on the modified Navier-Stokes equations following the Smagorinsky model of turbulence, and presenting a suitable nonhomogeneous Dirichlet boundary condition. For the analytical study of the problem, we prove several theoretical results related to existence, uniqueness and smoothness for the solution of this recirculation model.
This paper considers the existence and stability properties of two-dimensional solitary waves traversing an infinitely deep body of water. We assume that above the water is vacuum, and that the waves are acted upon by gravity with surface tension effects on the air--water interface. In particular, we study the case where there is a finite dipole in the bulk of the fluid, that is, the vorticity is a sum of two weighted $δ$-functions. Using an implicit function theorem argument, we construct a family of solitary waves solutions for this system that is exhaustive in a neighborhood of $0$. Our mai…
In this paper we consider the Cauchy problem for 2D viscous shallow water system in $H^s(\mathbb{R}^2)$, $s>1$. We first prove the local well-posedness of this problem by using the Littlewood-Paley theory, the Bony decomposition, and the theories of transport equations and transport diffusion equations. Then, we get the global existence of the system with small initial data in $H^s(\mathbb{R}^2)$, $s>1$. Our obtained result improves the recent result in \cite{W}
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 …
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 consider the gravity-capillary waves in any dimension and in fluid domains with general bottoms. Using the paradiferential reduction established in the companion paper, we prove Strichartz estimates for solutions to this problem, at a low regularity level such that initially, the velocity field can be non-Lipschitz up to the free surface. We then use those estimates to solve the Cauchy problem at this level of regularity.
In this paper we consider the Cauchy problem for 2D viscous shallow water system in Besov spaces. We firstly prove the local well-posedness of this problem in $B^s_{p,r}(\mathbb{R}^2)$, $s>max\{1,\frac{2}{p}\}$, $1\leq p,r\leq \infty$ by using the Littlewood-Paley theory, the Bony decomposition and the theories of transport equations and transport diffusion equations. Then we can prove the global existence of the system with small enough initial data in $B^s_{p,r}(\mathbb{R}^2)$, $1\leq p\leq2$, $1\leq r<\infty$ and $s>\frac{2}{p}$. Our obtained results generalize and cover the recent results …
We are interested in asymptotic models for the propagation of internal waves at the interface between two shallow layers of immiscible fluid, under the rigid-lid assumption. We review and complete existing works in the literature, in order to offer a unified and comprehensive exposition. Anterior models such as the shallow water and Boussinesq systems, as well as unidirectional models of Camassa-Holm type, are shown to descend from a broad Green-Naghdi model, that we introduce and justify in the sense of consistency. Contrarily to earlier works, our Green-Naghdi model allows a non-flat topogra…
We show existence of global strong solutions with large initial data on the irrotational part for the shallow-water system in dimension $N\geq 2$. We introduce a new notion of \textit{quasi-solutions} when the initial velocity is assumed to be irrotational, these last one exhibit regularizing effects both on the velocity and in a very surprising way also on the density (indeed the density is a priori governed by an hyperbolic equation). We would like to point out that this smoothing effect is purely non linear and is absolutely crucial in order to deal with the pressure term as it provides new…
In this paper, we prove the existence of relative equilibria with holes for quasi-geostrophic shallow-water equations. More precisely, using bifurcation techniques, we establish for any m large enough the existence of two branches of m-fold doubly-connected V-states bifurcating from any annulus of arbitrary size.
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 consider a Dirichlet to Neumann operator $\mathcal{L}_a$ arising in a model for water waves, with a nonlocal parameter $a\in(-1,1)$. We deduce the expression of the operator in terms of the Fourier transform, highlighting a local behavior for small frequencies and a nonlocal behavior for large frequencies. We further investigate the $ Γ$-convergence of the energy associated to the equation $ \mathcal{L}_a(u)=W'(u) $, where $W$ is a double-well potential. When $a\in(-1,0]$ the energy $Γ$-converges to the classical perimeter, while for $a\in(0,1)$ the $Γ$-limit is a new nonlocal operator, tha…
We study classical solutions of one dimensional rotating shallow water system which plays an important role in geophysical fluid dynamics. The main results contain two contrasting aspects. First, when the solution crosses certain threshold, we prove finite-time singularity formation for the classical solutions by studying the weighted gradients of Riemann invariants and utilizing conservation of physical energy. In fact, the singularity formation will take place for a large class of ${C}^1$ initial data whose gradients and physical energy can be arbitrarily small and higher order derivatives s…
This paper is devoted to the proof of a global existence result for the water waves equation with smooth, small, and decaying at infinity Cauchy data. We obtain moreover an asymptotic description in physical coordinates of the solution, which shows that modified scattering holds. The proof is based on a bootstrap argument involving $L^2$ and $L^\infty$ estimates. The $L^2$ bounds are proved in a companion paper of this article. They rely on a normal forms paradifferential method allowing one to obtain energy estimates on the Eulerian formulation of the water waves equation. We give here the pr…
In this paper, we derive asymptotic models for the propagation of two and three-dimensional gravity waves at the free surface and the interface between two layers of immiscible fluids of different densities, over an uneven bottom. We assume the thickness of the upper and lower fluids to be of comparable size, and small compared to the characteristic wavelength of the system (shallow water regimes). Following a method introduced by Bona, Lannes and Saut based on the expansion of the involved Dirichlet-to-Neumann operators, we are able to give a rigorous justification of classical models for wea…
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 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 the recent result [8] concerning the existence of quasi-periodic in time traveling waves for the 2d pure gravity water waves system in infinite depth. We provide the first existence result of quasi-periodic water waves solutions bifurcating from a completely resonant elliptic fixed point. The proof is based on a Nash-Moser scheme, Birkhoff normal form methods and pseudo-differential calculus techniques. We deal with the combined problems of small divisors and the fully-nonlinear nature of the equations.
In this article we address some issues related to the initial value problems for a rotating shallow water hyperbolic system of equations and the diffusive regularization of this system. For initial data close to the solution at rest, we establish the local existence and the uniqueness of a solution to the hyperbolic system, as well as the global existence of a solution to the regularized system. In order to prove this, we use suitable variables that symmetrize the system.
In a recent paper, we showed that the large $(x,t)$ behavior of a class of physically relevant solutions of Boussinesq's equation for water waves is described by ten main asymptotic sectors. In the sector adjacent to the positive $x$-axis, referred to as Sector I, we stated without proof an exact expression for the leading asymptotic term together with an error estimate. Here we provide a proof of this asymptotic formula.
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…
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.
We study asymptotic stability of solitary wave solutions in the one-dimensional Benney-Luke equation, a formally valid approximation for describing two-way water wave propagation. For this equation, as for the full water wave problem, the classic variational method for proving orbital stability of solitary waves fails dramatically due to the fact that the second variation of the energy-momentum functional is infinitely indefinite. We establish nonlinear stability in energy norm under the spectral stability hypothesis that the linearization admits no non-zero eigenvalues of non-negative real pa…
The main result of this work is the construction of multi-solitons solutions that is to say solutions that are time asymptotics to a sum of decoupling solitary waves for the full water waves system with surface tension.
We consider the gravity water waves system in the case of a one dimensional interface, for sufficiently smooth and localized initial data, and prove global existence of small solutions. This improves the almost global existence result of Wu \cite{WuAG}. We also prove that the asymptotic behavior of solutions as time goes to infinity is different from linear, unlike the three dimensional case \cite{GMS2,Wu3DWW}. In particular, we identify a suitable nonlinear logarithmic correction and show modified scattering. The solutions we construct in this paper appear to be the first global smooth nontri…
This paper deals with the dead-water phenomenon, which occurs when a ship sails in a stratified fluid, and experiences an important drag due to waves below the surface. More generally, we study the generation of internal waves by a disturbance moving at constant speed on top of two layers of fluids of different densities. Starting from the full Euler equations, we present several nonlinear asymptotic models, in the long wave regime. These models are rigorously justified by consistency or convergence results. A careful theoretical and numerical analysis is then provided, in order to predict the…
We prove the existence and the linear stability of small amplitude time {\it quasi-periodic} standing wave solutions (i.e. periodic and even in the space variable $ x $) of a $ 2 $-dimensional ocean with infinite depth under the action of gravity and surface tension. Such an existence result is obtained for all the values of the surface tension belonging to a Borel set of asymptotically full Lebesgue measure.
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…
We consider the existence of periodic traveling waves in a bidirectional Whitham equation, combining the full two-way dispersion relation from the incompressible Euler equations with a canonical shallow water nonlinearity. Of particular interest is the existence of a highest, cusped, traveling wave solution, which we obtain as a limiting case at the end of the main bifurcation branch of $2π$-periodic traveling wave solutions continuing from the zero state. Unlike the unidirectional Whitham equation, containing only one branch of the full Euler dispersion relation, where such a highest wave beh…
This work is devoted to the study of a viscous shallow-water system with friction and capillarity term. We prove in this paper the existence of global strong solutions for this system with some choice of large initial data when $N\geq 2$ in critical spaces for the scaling of the equations. More precisely, we introduce as in \cite{Hprepa} a new unknown,\textit{a effective velocity} $v=u+μ\n\ln h$ ($u$ is the classical velocity and $h$ the depth variation of the fluid) with $μ$ the viscosity coefficient which simplifies the system and allow us to cancel out the coupling between the velocity $u$ …
The goal of this monograph is to prove that any solution of the Cauchy problem for the capillarity-gravity water waves equations, in one space dimension, with periodic, even in space, initial data of small size $ε$, is almost globally defined in time on Sobolev spaces, i.e. it exists on a time interval of length of magnitude $ε^{-N}$ for any $N$, as soon as the initial data are smooth enough, and the gravity-capillarity parameters are taken outside an exceptional subset of zero measure. In contrast to the many results known for these equations on the real line, with decaying Cauchy data, one c…
We prove an extended lifespan result for the full gravity-capillary water waves system with a $2$ dimensional periodic interface: for initial data of sufficiently small size $\varepsilon$, smooth solutions exist up to times of the order of $\varepsilon^{-5/3+}$, for almost all values of the gravity and surface tension parameters. Besides the quasilinear nature of the equations, the main difficulty is to handle the weak small divisors bounds for quadratic and cubic interactions, growing with the size of the largest frequency. To overcome this difficulty we use (1) the (Hamiltonian) structure of…
We prove the local existence for the Water Waves equations with large bathymetric variations on a time interval of size 1/ε, where $ε$ measures the amplitude of the wave. We just need the presence of surface tension.
We investigate the Riemann Problem for a shallow water model with porosity and terrain data. Based on recent results on the local existence, we build the solution in the large settings (the magnitude of the jump in the initial data is not supposed to be ``small enough''). One difficulty for the extended solution arises from the double degeneracy of the hyperbolic system describing the model. Another difficulty is given by the fact that the construction of the solution assumes solving an equation which has no global solution. Finally, we present some cases to illustrate the existence and non-ex…
In this article, we consider the compressible Navier-Stokes equation with density dependent viscosity coefficients and a term of capillarity introduced by Coquel et al in \cite{5CR}. This model includes at the same time the barotropic Navier-Stokes equations with variable viscosity coefficients, shallow-water system and the model of Rohde. We first study the well-posedness of the model in critical regularity spaces with respect to the scaling of the associated equations. In a functional setting as close as possibleto the physical energy spaces, we prove global existence of solutions close to a…
Considered in this paper is a bi-directional model for the propagation of interfacial capillary-gravity waves in a two-layer system of fluids with rigid lid condition for the upper layer and lower layer with a much larger or infinite depth. The system is derived from a reformulation of the Euler equations for internal waves with nonnegligible surface tension effects in the interface and the corresponding asymptotic model under the Benjamin-Ono regime. Another unidirectional model, so-called regularized Benjamin equation, generalizing the Benjamin equation, is also introduced. Well-posedness of…
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…
In this article we consider the inviscid two-dimensional shallow water equations in a rectangle. The flow occurs near a stationary solution in the so called supercritical regime and we establish short term existence of smooth solutions for the corresponding initial and boundary value problem.
In this paper, we prove the existence and uniqueness of the solutions for the 2D viscous shallow water equations with low regularity assumptions on the initial data as well as the initial height bounded away from zero.
We study the inviscid multilayer Saint-Venant (or shallow-water) system in the limit of small density contrast. We show that, under reasonable hyperbolicity conditions on the flow and a smallness assumption on the initial surface deformation, the system is well-posed on a large time interval, despite the singular limit. By studying the asymptotic limit, we provide a rigorous justification of the widely used rigid-lid and Boussinesq approximations for multilayered shallow water flows. The asymptotic behaviour is similar to that of the incompressible limit for Euler equations, in the sense that …
In this paper we give a global existence and uniqueness theorem for an initial and boundary value problem (IBVP) relative to the coagulation equation of water droplets and we show the convergence of the global solution to the stationary solution. The coagulation equation is an integro-differential equation that describes the variation of the density $σ$ of water droplets in the atmosphere. Furthermore, IBVP is considered on a strip limited by two horizontal planes and its boundary condition is such that rain fall from the strip. To obtain this result of global existence of the solution $σ$ in …
We prove an asymptotic stability result for the water wave equations linearized around small solitary waves. The equations we consider govern irrotational flow of a fluid with constant density bounded below by a rigid horizontal bottom and above by a free surface under the influence of gravity neglecting surface tension. For sufficiently small amplitude waves, with waveform well-approximated by the well-known sech-squared shape of the KdV soliton, solutions of the linearized equations decay at an exponential rate in an energy norm with exponential weight translated with the wave profile. This …
This paper is dedicated to the study of both viscous compressible barotropic fluids and Navier-Stokes equation with dependent density, when the viscosity coefficients are variable, in dimension $d\geq2$. We aim at proving the local and global well-posedness for respectively {\it large} and \textit{small} initial data having critical Besov regularity and more precisely we are interested in extending the class of initial data velocity when we consider the shallow water system, improving the results in \cite{CMZ1,H2} and \cite{arma}. Our result relies on the fact that the velocity $u$ can be writ…
In this paper we prove the existence of water waves with sign-changing Taylor sign coefficients, that is, the strong Taylor sign holds initially, while breaks down at a later time, and vice versa. Such a phenomenon can be regarded as the transition between the stable and unstable regime in the sense of Rayleigh-Taylor of water waves. As a byproduct, we prove the sharp wellposedness of 2d water waves in Gevrey-2 spaces.
