The zero surface tension limit two‐dimensional water waves

We consider two‐dimensional water waves of infinite depth, periodic in the horizontal direction. It has been proven by Wu (in the slightly different nonperiodic setting) that solutions to this initial value problem exist in the absence of surface tension. Recently Ambrose has proven that solutions exist when surface tension is taken into account. In this paper, we provide a shorter, more elementary proof of existence of solutions to the water wave initial value problem both with and without surface tension. Our proof requires estimating the growth of geometric quantities using a renormalized arc length parametrization of the free surface and using physical quantities related to the tangential velocity of the free surface. Using this formulation, we find that as surface tension goes to 0, the water wave without surface tension is the limit of the water wave with surface tension. Far from being a simple adaptation of previous works, our method requires a very original choice of variables; these variables turn out to be physical and well adapted to both cases. © 2005 Wiley Periodicals, Inc.     

The NLS and FWI approximation make wrong predictions for the water wave problem in case of small surface tension

In has been shown in [5] that the NLS approximation makes wrong predictions for the water wave problem in case of small surface tension. We explain that the ideas used in [5] allow to show a similar result for the FWI approximation, too. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A remark on the two dimensional water wave problem with surface tension

We consider the motion of a two-dimensional interface between air (above) and an irrotational, incompressible, inviscid, infinitely deep water (below), with surface tension present. We propose a new way to reduce the original problem into an equivalent quasilinear system which is related to the interface's tangent angle and a quantity related to the difference of tangential velocities of the interface in the Lagrangian and the arc-length coordinates. The new way is relatively simple because it involves only taking differentiation and the real and the imaginary parts. Then if assuming that waves are periodic, we establish a priori energy inequality.     

Classical verigin problem as a limit case of verigin problem with surface tension at free boundary

In this psper we consider Verigin problem with surface tension st free     

Variational problem of the two-phase medium elasticity theory for the zero surface tension coefficient

In this article, we give a proof of the existence theorem for an equilibrium state for the surface tension coefficient σ=0 and investigate the behavior of the equilibrium state for small σ. Bibliography: 4 titles. Dedicated to N. N. Uraltseva on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 221, 1900, pp. 208–225. Translated by S. Yu. Pilyugin.     

A shallow‐water approximation to the full water wave problem

We demonstrate that the system of the Green‐Naghdi equations as a two‐directional, nonlinearly dispersive wave model is a close approximation to the two‐dimensional full water wave problem. Based on the energy estimates and the proof of the well‐posedness for the Green‐Naghdi equations and the water wave problem, we compare solutions of the two systems, showing that without restrictions on the wave amplitude, any two solutions of the two systems remain close, at least in some finite time within the shallow‐water regime, provided that their initial data are close in the Banach space H^s × H^s+1 for some s > . As a consequence, we show that if the depth of the water compared with the wavelength is sufficiently small, the two solutions exist for the same finite time using the uniformly bounded energies defined in the paper. © 2006 Wiley Periodicals, Inc.     

A new solitary wave solution for water waves with surface tension

In this paper we show that when the Froude number is less than but close to 1 and the Bond number is greater than but close to 1/3 there exists a new solitary wave solution for surface waves on water with surface tension. An approximate expression for the new solitary wave solution, which satisfies a fourth order ordinary differential equation and represents a wave of depression is presented.     

The cauchy problem for a semilinear wave equation. I

The Cauchy problem for a semilinear wave equation on the torus Tⁿ, n3:

The existence theorem and exact solutions of a variational problem on high-temperature phase transition with zero surface tension

A model of a two-phase elastic medium with classical energy density is considered. In the case under consideration the energy functional depends on the temperature, which is assumed to be constant along the whole body. The question on the equilibrium state for σ=0 is studied, and a family of exact solutions to the problem is constructed. Bibliography: 8 titles. Translated fromProblemy Matematicheskogo Analiza. No. 15, 1995, pp. 201–212.     

Existence and uniqueness of solution in Sobolev space for an unsteady crystal growth problem with zero surface tension

We study an initial value problem for a two-dimensional dendritic crystal growth model with zero surface tension. If the initial data is in Sobolev space H²(R), it is proved that an unique local solution exists in proper Sobolev space.     

Hamiltonian long‐wave expansions for free surfaces and interfaces

The theory of internal waves between two bodies of immiscible fluid is important both for its interest to ocean engineering and as a source of numerous interesting mathematical model equations that exhibit nonlinearity and dispersion. In this paper we derive a Hamiltonian formulation of the problem of a dynamic free interface (with rigid lid upper boundary conditions), and of a free surface and a free interface, this latter situation occurring more commonly in experiment and in nature. From the formulation, we develop a Hamiltonian perturbation theory for the long‐wave limits, and we carry out a systematic analysis of the principal long‐wave scaling regimes. This analysis provides a uniform treatment of the classical works of Peters and Stoker (28), Benjamin (3, 4), Ono (26), and many others. Our considerations include the Boussinesq and Korteweg–de Vries (KdV) regimes over finite‐depth fluids, the Benjamin‐Ono regimes in the situation in which one fluid layer is infinitely deep, and the intermediate long‐wave regimes. In addition, we describe a novel class of scaling regimes of the problem, in which the amplitude of the interface disturbance is of the same order as the mean fluid depth, and the characteristic small parameter corresponds to the slope of the interface. Our principal results are that we highlight the discrepancies between the case of rigid lid and of free surface upper boundary conditions, which in some circumstances can be significant. Motivated by the recent results of Choi and Camassa (6, 7), we also derive novel systems of nonlinear dispersive long‐wave equations in the large‐amplitude, small‐slope regime. Our formulation of the dynamical free‐surface, free‐interface problem is shown to be very effective for perturbation calculations; in addition, it holds promise as a basis for numerical simulations. © 2005 Wiley Periodicals, Inc.     

Stability of one-phase equilibrium states of a two-phase elastic medium with zero surface tension coefficient. One-dimensional case

All the equilibrium states of a one-dimensional variational phase-transition problem are explicitly found. The temperature-dependence of the stability of one-phase equilibrium states is studied. Bibliography: 5 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 32, 2006, pp. 3–19.     

The Petersson conjecture for the weight zero. I

In the present paper, a known result by Eichler-Deligne concerning the Petersson conjecture for finite-dimensional classical spaces is proved for infinite-dimensional Hilbert spaces of weight 0. In this work, the techniques of spectral decompositions of convolutions are used. The work is divided into two parts. In this (first) part, an explicit representation of an eigenvalue of the Hecke operator in terms of the spectral components of the convolution is obtained. On the basis of this representation, the Petersson conjecture will be proved in the second part. Bibliography: 9 titles. Dedicated to O. A. Ladyzhenskaya Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 249, 1997, pp. 118–152. Translated by N. A. Karazeeva.     

Local smoothing effects for the water-wave problem with surface tension

The water-wave problem with a one-dimensional free surface of infinite depth is considered, based on the formulation as a second-order nonlinear dispersive equation. The local smoothing effects are established under the influence of surface tension, stating that on average in time solutions acquire locally 1/4 derivative of smoothness as compared to the initial state. The analysis combines energy methods with techniques of Fourier integral operators. To cite this article: H. Christianson et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

On the hydrodynamic limit of Ginzburg‐Landau wave vortices

In this paper, we study the hydrodynamic limit of the finite Ginzburg‐Landau wave vortices, which was established in [16]. Unlike the classical vortex method for incompressible Euler equations, we prove here that the densities approximated by the vortex blob method associated with the Ginzburg‐Landau wave vortices tend to the solutions of the pressure‐less compressible Euler‐Poisson equations. The convergence of such approximation is proved before the formation of singularities in the limit system as the blob sizes and the grid sizes tend to zero in appropriate rates. © 2002 John Wiley & Sons, Inc.     

The behavior of the area of the boundary of phase interface in the problem about phase transitions as the surface tension coefficient tends to zero

We prove that the area of the boundary of the phase interface monotonically increases as the surface tension coefficient tends to zero and study the limit behavior of equilibrium states. Bibliography: 7 titles.     

Existence and uniqueness of analytic solution for unsteady crystals with zero surface tension

We study the initial value problem for two-dimensional dendritic crystal growth with zero surface tension. If the initial data is analytic and close to Ivantsov steady solution, it is proved that unique analytic solution exists locally in time. The analysis is based on a Nirenberg Theorem on abstract Cauchy-Kovalevsky problem in properly chosen Banach spaces.