growth rates for the linearized motion of fluid interfaces away from equilibrium we consider the motion of a two dimensional interface separating an inviscid incompressible irrotational fluid influenced by gravity from a region of zero density we show that under certain conditions the equations of motion linearized about a presumed time dependent solution are wellposed that is linear disturbances have a bounded rate of growth if surface tension is neglected the linear equations are well posed provided the underlying exact motion satisfies a condition on the acceleration of the interface relative to gravity similar to the criterion formulated by g i taylor if surface tension is included the linear equations are well posed without qualifications whether the fluid is above or below the interface an interesting qualitative structure is found for the linear equations a lagrangian approach is used like that of numerical work such as 3 except that the interface is assumed horizontal at infinity certain integral equations which occur involving double layer potentials are shown to be solvable in the present case 1993 john wiley sons inc

a remark on the two dimensional water wave problem with surface tension abstract 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

the clifford hilbert transform and riemann hilbert problems associated to the dirac operator in lebesgue spaces abstract in this paper we establish some properties of the hilbert transform in clifford analysis setting and mainly show the weak type 1 1 inequality for the clifford hilbert transform in euclidean space r m m 3 using the clifford algebra and analysis generalizations of the one dimensional proofs furthermore we prove kolmogorov s inequality for clifford hilbert transform with the help of the properties for the hilbert transform and clifford analytic techniques we establish the riemann hilbert problems in clifford value lebesgue p integrable spaces based on the newton embedding method we prove an existence and uniqueness for the nonlinear riemann hilbert problem and also give an error estimation for the approximate solutions in the newton embedding procedure in lebesgue spaces

normal forms and quadratic nonlinear klein gordon equations on construit une transformation qui change l equation a non linearite cubique que l on sait resoudre a 3 1 dimensions on demontre que la solution du probleme original a le meme comportement asymptotique que l equation de klein gordon non lineaire

a global theory of internal solitary waves in two fluid systems abstract the study of single crested progressing gravity waves was initiated over a century ago with the observations by russell of what he termed solitary waves which progressed without change of form over a considerable distance on the glasgow edinburgh canal the mathematical analysis of this wave motion on the surface of water begun in the nineteenth century has undergone a rapid development in the last three decades due to the scattering theory for the korteweg de vries equation which models the motion of long waves due to the development of techniques in nonlinear analysis allowing for the analysis of finite amplitude motions the work on surfce waves has many parallels in the study of waves in fluids with variable density in the case of a heterogeneous fluid with a free upper surface gravity waves still occur in analogy with surface waves in a fluid of constant density what is distinctive about a fluid with density stratification however is the presence of waves which are predominantly due to the stratification and not to the free surface these waves called internal waves exist in a heterogeneous fluid even when it is confined between horizontal boundaries a configuration which precludes gravity waves in a fluid of constant density this paper is concerned with progressing solitary gravity waves in a system consisting of two fluids of differing densities confined in a channel of unit depth and infinite horizontal extent

solitary internal waves in continuously stratified fluids of great depth abstract this paper considers the existence of solitary internal waves moving with a constant speed in a continuously stratified fluid of great depth the fluid is inviscid incompressible and bounded below by a rigid bottom only the density of fluid decreases continuously upwards to a certain level and then the fluid above has a constant density and extends to the infinity vertically it is rigorously shown that the exact governing equations of this flow have a solitary wave solution that decays algebraically to a uniform state at infinity when the wave speed is near its first critical value the first order approximation of the solution is the solution obtained from the benjamin ono equation time independent

small internal waves in two fluid systems this paper treats travelling waves in a heterogeneous inviscid non diffusive fluid bounded between two horizontal boundaries the fluid has two incompressible components of different but constant density and is acted on by gravity the flow is steady when viewed in a moving reference frame and gives rise to a quasilinear elliptic problem with an eigenvalue parameter related to the wave speed the small amplitude solutions are analyzed using a dynamical systems approach a center manifold reduction in combination with a conserved quantity for the flow is used to parametrise all small solutions of the full elliptic system in terms of solutions of an autonomous first order ordinary differential equation for a principal component of the wave amplitude the result is a characterization of all small waves irrotational in each fluid near the critical speed for the system they are solitary waves surges connecting distinct conjugate flows at extreme ends of the channel conjugate flows and periodic waves

on the regularity of the flow map for the gravity capillary equations abstract we prove via explicitly constructed initial data that solutions to the gravity capillary wave system in r 3 representing a 2 d air water interface immediately fail to be c 3 with respect to the initial data if the initial data h 0 0 h s 1 2 h s for s 3 similar results hold in r 2 domains with a 1 d interface furthermore we discuss the related threshold for the pure gravity water wave system

structure formation in a zonal barotropic current a treatment via the centre manifold reduction formation of spatial structures in zonal barotropic currents is a classical problem of dynamic meteorology it is of fundamental importance in the forecasting of lowfrequency variability of climate

weighted l and l1 estimates for solutions to the classical wave equation in three space dimensions on considere le probleme aux valeurs initiales u 0 u x 0 0 u t x 0 g x on donne des estimations l et l 1 pour les solutions dans le cas general a symetrie non spherique

well posedness for the motion of an incompressible liquid with free surface boundary we study the motion of an incompressible perfect liquid body in vacuum this can be thought of as a model for the motion of the ocean or a star the free surface moves with the velocity of the liquid and the pressure vanishes on the free surface this leads to a free boundary problem for euler s equations where the regularity of the boundary enters to highest order we prove local existence in sobolev spaces assuming a physical condition related to the fact that the pressure of a fluid has to be positive

symmetry of free surface flows certain geometrical properties of steady two dimensional potential flows in the presence of gravity are considered the two main problems are that of a solitary wave in an interface between two immiscible fluids of different densities and that of steady waves in a free surface of a fluid flowing past an object such as a ship hull hydrofoil or surfboard our results include statements of positivity or negativity monotonicity symmetry and the asymptotic behavior at infinity of such flows the methods are extensions of those presented in earlier work on the solitary wave 7

perturbed homoclinic solutions in reversible 1 1 resonance vector fields we consider a smooth reversible vector field in r 4 such that the origin is a fixed point the differential at the origin has two double pure imaginary eigenvalues iq for the critical value 0 of the parameter we show by a normal form analysis that the vector field can be approximated by an integrable field in r 4 for which we know all solutions specially interesting ones are the homoclinics to 0 and homoclinics to periodic solutions depending on the sign of a leading nonlinear coefficient we prove in particular the persistence of these homoclinics for the full vector field

global solutions for the gravity water waves equation in dimension 3 we show existence of global solutions for the gravity water waves equation in dimension 3 in the case of small data the proof combines energy estimates which yield control of l 2 related norms with dispersive estimates which give decay in l 8 to obtain these dispersive estimates we use an analysis in fourier space the study of space and time resonances is then the crucial point

existence of a generalized solitary wave solution for water with positive bond number less than 13 abstract the objective of this paper is to give a rigorous answer to the question whether there exists a generalized solitary wave of elevation on water when the nondimensional surface tension coefficient the so called bond number is positive but less than 1 3 and the froude number is near 1 but greater than 1 the main difficulties to overcome are the small amplitude oscillations at infinity and the choice of appropriate banach spaces for the solution of the problem it is shown that an asymptotic expansion developed for this problem indeed yields an approximate solitary wave solution to the exact equations

existence and symmetry of ground states to the boussinesq abcd systems we consider a four parameter family of boussinesq systems derived by bona et al j nonlinear sci 12 283 318 2002 we establish the existence of the ground states which are solitary waves minimizing the action functional of the systems we further show that in the presence of large surface tension the ground states are even up to translation

distributional solutions of the beltrami equation abstract we study the distributional solutions to the generalized beltrami equation under sobolev assumptions on the beltrami coefficients in this setting we prove that these distributional solutions are true quasiregular maps and they are smoother than expected that is they have second order derivatives in l l o c 1 e for some e 0

zero surface tension limit of the free surface incompressible euler equations with damping abstract we consider the free boundary problem for a layer of an incompressible inviscid fluid in a uniform gravitational field lying above a rigid bottom and below the atmosphere in a horizontally periodic setting we establish the global in time zero surface tension limit of the problem with damping for the small initial data

global existence of quasi stratified solutions for the confined ipm equation in this paper we consider a confined physical scenario to prove the global existence of smooth solutions with bounded density and finite energy for the inviscid incompressible porous media ipm equation the result is proved using the stability of stratified solutions combined with an additional structure of our initial perturbation which allows us to get rid of the boundary terms in the energy estimates

symmetry of steady periodic gravity water waves with vorticity we prove that steady periodic two dimensional rotational gravity water waves with a monotone surface profile between troughs and crests have to be symmetric about the crest irrespective of the vorticity distribution within the fluid

asymptotic expansions for solitary gravity capillary waves in two and three dimensions high order asymptotic series are obtained for gravity capillary solitary waves where the first term in the series is the well known sech2 solution of the kdv equation the asymptotic series is used with nine terms included to investigate the effects of surface tension on the height and energy of large amplitude waves and waves close to the solitary version of stokes extreme wave in particular for surface tension below a critical value the solitary wave with the maximum energy is obtained for large surface tension the series is also used to study the energy related to the solitary waves of depression energy considerations suggest that for large enough surface tension there are solitary waves that can get close to the fluid bottom comparisons are also made with recent experiments

well posedness of the free boundary problem in incompressible elastodynamics under the mixed type stability condition abstract we consider the free boundary problem for the flow of an incompressible inviscid elastic fluid the columns of the deformation gradient are tangent and the pressure vanishes along the free interface we prove the local existence of a unique smooth solution to the free boundary problem under the mixed type stability condition provided that the rayleigh taylor sign condition is satisfied at all the points of the initial interface where the non collinearity condition for the deformation gradient among three columns of the deformation gradient there are two non collinear vectors fails in particular we solve an open problem proposed by y trakhinin in the paper 29 under the incompressible setting

center manifolds for a class of degenerate evolution equations and existence of small amplitude kinetic shocks abstract we construct center manifolds for a class of degenerate evolution equations including the steady boltzmann equation and related kinetic models establishing in the process existence and behavior of small amplitude kinetic shock and boundary layers notably for boltzmann s equation we show that elements of the center manifold decay in velocity at near maxwellian rate in accord with the formal chapman enskog picture of near equilibrium flow as evolution along the manifold of maxwellian states or grad moment approximation via hermite polynomials in velocity our analysis is from a classical dynamical systems point of view with a number of interesting modifications to accommodate ill posedness of the underlying evolution equation

on the boussinesq burgers equations driven by dynamic boundary conditions abstract we study the qualitative behavior of the boussinesq burgers equations on a finite interval subject to the dirichlet type dynamic boundary conditions assuming h 1 h 2 initial data which are compatible with boundary conditions and utilizing energy methods we show that under appropriate conditions on the dynamic boundary data there exist unique global in time solutions to the initial boundary value problem and the solutions converge to the boundary data as time goes to infinity regardless of the magnitude of the initial data

the free boundary euler equations with large surface tension abstract we study the free boundary euler equations with surface tension in three spatial dimensions showing that the equations are well posed if the coefficient of surface tension is positive then we prove that under natural assumptions the solutions of the free boundary motion converge to solutions of the euler equations in a domain with fixed boundary when the coefficient of surface tension tends to infinity

fully localised solitary wave solutions of the three dimensional gravity capillary water wave problem a model equation derived by kadomtsev petviashvili sov phys dokl 15 539 541 1970 suggests that the hydrodynamic problem for three dimensional water waves with strong surface tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal spatial direction this prediction is rigorously confirmed for the full water wave problem in the present paper the theory is variational in nature a simple but mathematically unfavourable variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle with significantly better mathematical properties the reduced functional is related to the functional associated with the kadomtsev petviashvili equation and a nontrivial critical point is detected using the direct methods of the calculus of variations

capillary gravity waves on the free surface of an inviscid fluid of infinite depth existence of solitary waves permanent capillary gravity waves on the free surface of a two dimensional inviscid fluid of infinite depth are investigated an application of the hodograph transform converts the free boundary value problem into a boundary value problem for the cauchy riemann equations in the lower halfplane with nonlinear differential boundary conditions this can be converted to an integro differential equation with symbol k2 4 k 4 1 where is a bifurcation parameter a normal form analysis is presented which shows that the boundary value problem can be reduced to an integrable system of ordinary differential equations plus a remainder term containing nonlocal terms of higher order for small this normal form system has been studied thoroughly by several authors iooss kirchgassner 8 iooss peroueme 10 dias iooss 5 it admits a pair of solitary wave solutions which are reversible in the sense ofkirchgassner 11 by applying a method introduced in 11 it is shown that this pair of reversible solitary waves persists for the boundary value problem and that the decay at infinity of these solitary waves is at least like 1 x

on the existence of small amplitude solitary waves with strong surface tension an existence theory for small amplitude solitary waves with surface tension effects included is developed for large values of the surface tension parameter 13 using ideas of beale the nash moser implicit function theorem is applied to justify the well known approximation of korteweg and devries some of the recent results of amick and kirchgassner are thereby achieved more directly and additional insight obtained for the open case 13

asymptotic expansions for solitary gravity capillary waves in two and three dimensions high order asymptotic series are obtained for two and three dimensional gravity capillary solitary waves in two dimensions the first term in the asymptotic series is the well known sech 2 solution of the korteweg de vries equation in three dimensions the first term is the rational lump solution of the kadomtsev petviashvili equation i the two dimensional series is used with nine terms included to investigate how small surface tension affects the height and energy of large amplitude waves and waves close to the solitary version of stokes extreme wave in particular for small surface tension the solitary wave with the maximum energy is obtained for large surface tension the two dimensional series is also used to study the energy of depression solitary waves energy considerations suggest that for large enough surface tension there are solitary waves that can get close to the fluid bottom in three dimensions analytic solutions for the high order perturbation terms are computed numerically and the resulting asymptotic series to three terms is used to obtain the speed versus maximum amplitude curve for solitary waves subject to sufficiently large surface tension finally the above asymptotic method is applied to the benney luke bl equation and the resulting asymptotic series to three terms is verified to agree with the solitary wave solution of the bl equation

solitary waves with damped oscillatory tails an analysis of the fifth order korteweg de vries equation abstract we construct oscillatory solitary wave solutions of a fifth order korteweg de vries equation where the oscillations decay at infinity these waves arise as a bifurcation from the linear dispersion curve at that wavenumber where the linear phase speed and group velocity coincide our approach is a wave packet analysis about this wavenumber which leads in the first instance to a higher order nonlinear schrodinger equation from which we then obtain the steady solitary wave solution we then describe a complementary normal form analysis which leads to the same result in addition we derive the nonlinear schrodinger equation for all wavenumbers and list all the various anomalous cases

long time existence for the boussinesq full dispersion systems abstract we establish the long time existence of solutions for the boussinesq full dispersion systems modeling the propagation of internal waves in a two layer system for the two dimensional hamiltonian case b d 0 a 0 c 0 we study the global existence of small solutions of the corresponding system

global solutions of the gravity capillary water wave system in three dimensions in this paper we prove global regularity for the full water wave system in three dimensions for small data under the influence of both gravity and surface tension this problem presents essential difficulties which were absent in all of the earlier global regularity results for other water wave models to construct global solutions we use a combination of energy estimates and matching dispersive estimates there is a significant new difficulty in proving energy estimates in our problem namely the combination of slow pointwise decay of solutions no better than lvert t rvert 5 6 and the presence of a large codimension 1 set of quadratic time resonances to deal with such a situation we propose here a new mechanism which exploits a non degeneracy property of the time resonant hypersurfaces and some special structure of the quadratic part of the non linearity connected to the conserved energy of the system the dispersive estimates rely on analysis of the duhamel formula in the fourier space the main contributions come from the set of space time resonances which is a large set of dimension 1 to control the corresponding bilinear interactions we use harmonic analysis techniques such as orthogonality arguments in the fourier space and atomic decompositions of functions most importantly we construct and use a refined norm which is well adapted to the geometry of the problem

global analysis of a model for capillary water waves in two dimensions in this paper we prove a global regularity result for a quadratic quasilinear model associated to the water waves system with surface tension and no gravity in dimension 2 the capillary waves system the model we consider here retains most of the difficulties of the full capillary water waves system including the delicate time resonance structure and modified scattering it is slightly simpler however at the technical level and our goal here is to present our method in this simplified situation the analysis of the full system is presented in a preprint by the authors 2014 2016 wiley periodicals inc

long wave approximations for water waves in this paper we obtain new nonlinear systems describing the interaction of long water waves in both two and three dimensions these systems are symmetric and conservative rigorous convergence results are provided showing that solutions of the complete free surface euler equations tend to associated solutions of these systems as the amplitude becomes small and the wavelength large using this result as a tool a rigorous justification of all the two dimensional approximate systems recently put forward and analysed by bona chen and saut is obtained in the two dimensional context our methods also allows a significant improvement of the convergence estimate obtained by schneider and wayne in their justification of the decoupled korteweg de vries approximation of the two dimensional euler equations it also follows from our theory that coupled models provide a better description than the decoupled ones over short time scales results are obtained both on an unbounded domain for solutions that evanesce at infinity as well as for solutions that are spatially periodic

the zero surface tension limit of 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 c

poisson structure of the reversible 1 1 resonance abstract vector fields that are reversible but non hamiltonian are considered in the neighborhood of a 1 1 resonance it is found that the normalized vector field has a hidden poisson structure on the orbit space that is reminiscent of the equations governing the rigid body in mechanics the theory is used to give a geometric characterization of the phase space in the neighborhood of a reversible 1 1 resonance certain similarities and also essential differences with the hamiltonian 1 1 resonance are found

some analytical properties of capillary gravity waves in two fluid flows of infinite depth capillary gravity waves of finite amplitude at the interface of a two fluid flow are considered the fluids of constant densities extended to infinity in each direction are inviscid and incompressible the flow is two dimensional and irrotational in each fluid assume that there is a permanent wave at the interface moving with a constant speed and the wave decays to zero at infinity with a certain rate it is shown that the wave must decay at infinity quadratically and the solution of the exact governing equations corresponding to such a wave must satisfy two integral identities for the net displacement and the energy of the wave while the former implies non existence of waves of pure depression or pure elevation in the flow

two dimensional travelling waves over a moving bottom abstract two dimensional travelling waves on an ideal fluid with gravity and surface tension over a periodically moving bottom with a small amplitude are studied the bottom and the wave travel with a same speed the exact euler equations are formulated as a spatial dynamic system by using the stream function a manifold reduction technique is applied to reduce the system into one of ordinary differential equations with finite dimensions a homoclinic solution to the normal form of this reduced system persists when higher order terms are added which gives a generalized solitary wave the homoclinic solution connecting a periodic solution

area distortion of quasiconformal mappings in recent years quasiconformal mappings have been an efficient tool in the study of dynamical systems of the complex plane we show here that in turn methods or ideas from dynamical systems can be used to solve a number of open questions in the theory of planar quasiconformal mappings it has been known since the work of ahlfors a and mori mo that k quasiconformal mappings are locally h51der continuous with exponent 1 k the function

global wellposedness of the 3 d full water wave problem we consider the problem of global in time existence and uniqueness of solutions of the 3 d infinite depth full water wave problem in the setting that the interface tends to the horizontal plane the velocity and acceleration on the interface tend to zero at spatial infinity we show that the nature of the nonlinearity of the water wave equation is essentially of cubic and higher orders for any initial interface that is sufficiently small in its steepness and velocity we show that there exists a unique smooth solution of the full water wave problem for all time and the solution decays at the rate 1 t

capillary gravity solitary waves with damped oscillations abstract capillary gravity solitary waves with damped oscillations are studied analytically the analysis follows the work of iooss and kirchgassner who proved that these waves exist for all values of the froude number smaller than one the water wave problem is reduced to a system of ordinary differential equations by using the center manifold theorem the normal form of this reduced system can be obtained and a good approximation to these waves for small amplitude is constructed the limit as the water depth becomes infinite is considered as a special case a comparison with existing numerical results is made for small amplitude waves

bilinear dispersive estimates via space time resonances dimensions two and three consider a bilinear interaction between two linear dispersive waves with a generic resonant structure roughly speaking space and time resonant sets intersect transversally we derive an asymptotic equivalent of the solution for data in the schwartz class and bilinear dispersive estimates for data in weighted lebesgue spaces an application to water waves with infinite depth gravity and surface tension is also presented

validity of the korteweg de vries approximation for the two dimensional water wave problem in the arc length formulation we consider the two dimensional water wave problem in an infinitely long canal of finite depth both with and without surface tension it has been proven by several authors that long wavelength solutions to this problem can be approximated over a physically relevant timespan by solutions of the korteweg de vries equation or for certain values of the surface tension by solutions of the kawahara equation these proofs are formulated either in lagrangian or in eulerian coordinates in this paper we provide a new proof which is simpler more elementary and shorter moreover the rigorous justification of the kdv approximation can be given for the cases with and without surface tension together by one proof in our proof we parametrize the free surface by arc length and use some geometrically and physically motivated variables with good regularity properties this formulation of the water wave problem has already been of great usefulness for ambrose and masmoudi to simplify the proof of the local well posedness of the water wave problem in sobolev spaces 2011 wiley periodicals inc

global existence of small amplitude solutions to nonlinear klein gordon equations in four space time dimensions consider the following class of nonlinear wave equations square text u u text text f u u u n k g where t 2 1 2 2 2 3 2 is the d alembertian of the 4 dimensional minkowski space time and f a smooth function of u u t x and its first and second partial derivates vanishing together with its first derivatives at u u u 0 in other words f is at least quadratic in u u u in a neighborhood of the trivial solution u 0 we subject u to the pure initial value problem text u mkern 1mu varepsilon text f x u t varepsilon g x at t 0 i v p with f g c 0 3 and e 0 a small parameter

almost global wellposedness of the 2 d full water wave problem we consider the problem of global in time existence and uniqueness of solutions of the 2 d infinite depth full water wave equation it is known that this equation has a solution for a time period 0 t e for initial data of the form e where t depends only on in this paper we show that for such data there exists a unique solution for a time period 0 e t e this is achieved by better understandings of the nature of the nonlinearity of the full water wave equation

existence of solitary internal waves in a two layer fluid of infinite depth abstract this paper concerns the existence of internal solitary waves moving with a constant speed at the interface of a two layer fluid with infinite height the fluids are immiscible inviscid and incompressible with constant but different densities assume that the height of the upper fluid is infinite and the depth of the lower fluid is finite it has been formally derived before that under long wave assumption the first order approximation of the interface satisfies the benjamin ono equation which has algebraic solitary wave solutions this paper gives a rigorous proof of the existence of solitary wave solutions of the exact equations governing the fluid motion whose first order approximations are the algebraic solitary wave solutions of the benjamin ono equation the proof relies on estimates of integral operators using fourier transforms in l2 r space and is different from the previous existence proof of solitary waves in a two layer fluid with finite depth

beltrami operators in the plane we determine optimal lp properties for the solutions of the general nonlinear elliptic system in the plane of the form fz h z fz h l c where h is a measurable function satisfying h z w1 h z w2 k w1 w2 and k is a constant k 1 we will also establish the precise invertibility and spectral properties in lp c for the operators i t i t and t where t is the beurling transform these operators are basic in the theory of quasiconformal mappings and in linear and nonlinear elliptic partial differential equations in two dimensions in particular we prove invertibility in lp c whenever 1 p 1 1 we also prove related results with applications to the regularity of weakly quasiconformal mappings

large time existence for 3d water waves and asymptotics 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

well posedness in sobolev spaces of the full water wave problem in 2 d we consider the motion of the interface of 2 d irrotational incompressible inviscid water wave with air above water and surface tension zero we show that the interface is always not accelerating into the water region normal to itself as rapidly as the normal acceleration of gravity as long as the interface is nonself intersect we therefore obtain the existence and uniqueness of solutions of the full water wave problem locally in time for any initial interface which is nonself intersect including the case that the interface is of multiple heights

well posedness of the water waves equations 1 1 presentation of the problem the water waves problem for an ideal liquid consists of describing the motion of the free surface and the evolution of the velocity field of a layer of perfect incompressible irrotational fluid under the influence of gravity in this paper we restrict our attention to the case when the surface is a graph parameterized by a function x where t denotes the time variable and x xi xd rd the horizontal spatial variables the method developed here works equally well for any integer d 1 but the only physically relevant cases are of course d 1 and d 2 the layer of fluid is also delimited from below by a not necessarily flat bottom parameterized by a time independent function b x we denote by q t the fluid domain at time t the incompressibility of the fluids is expressed by 1 1 div v 0 in qu t 0

galloping instability of viscous shock waves abstract motivated by physical and numerical observations of time oscillatory galloping spinning and cellular instabilities of detonation waves we study poincare hopf bifurcation of traveling wave solutions of viscous conservation laws the main difficulty is the absence of a spectral gap between oscillatory modes and essential spectrum preventing standard reduction to a finite dimensional center manifold we overcome this by direct lyapunov schmidt reduction using detailed pointwise bounds on the linearized solution operator to carry out a nonstandard implicit function construction in the absence of a spectral gap the key computation is a space time stability estimate on the transverse linearized solution operator reminiscent of duhamel estimates carried out on the full solution operator in the study of nonlinear stability of spectrally stable traveling waves

critical states and minima for an energy with second order gradients coleman marcus mizel studied the thermodynamical equilibrium of second order materials i e materials for which the free energy density depends not only on the concentration but also on its first and second gradients they showed that in certain regimes such materials must exhibit equilibrium states that are non uniform by proving that space periodic solutions can have lower energy than space uniform solutions a dynamical systems approach is used here and it is shown that there is a critical value of the mean concentration when the mean concentration crosses this critical value the uniform state can lose its minimizing character and periodic or quasi periodic states can have lower energy mathematically speaking the presence of that critical value corresponds to a 1 1 resonance the reversible 1 1 resonance has been studied extensively by iooss peroueme who showed that a reversible system having a 1 1 resonance singularity can be approximated as closely as desired by an integrable vector field on this integrable field all bounded solutions can be easily found and they are described by two integrals however in the present application a zero eigenvalue is present in addition to the double imaginary eigenvalues bounded solutions are then described by three integrals moreover since the average concentration must remain constant a non local constraint is added which changes the results significantly finally a selection principle for second order materials stemming from energy considerations is introduced the energy of all bounded solutions is compared with the energy of the uniform state in some cases non uniform solutions have a lower energy and lead to non uniform equilibrium states more precisely it is shown that space periodic or space quasi periodic states are minima quite surprisingly it may occur that non uniform states appear by dilution i e by decreasing the concentration

well posedness of the free surface incompressible euler equations with or without surface tension we provide a new method for treating free boundary problems in perfect fluids and prove local in time well posedness in sobolev spaces for the free surface incompressible 3d euler equations with or without surface tension for arbitrary initial data and without any irrotationality assumption on the fluid this is a free boundary problem for the motion of an incompressible perfect liquid in vacuum wherein the motion of the fluid interacts with the motion of the free surface at highest order

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

justification of the nonlinear schrodinger equation for the evolution of gravity driven 2d surface water waves in a canal of finite depth in 1968 v e zakharov derived the nonlinear schrodinger equation for the two dimensional water wave problem in the absence of surface tension that is for the evolution of gravity driven surface water waves in order to describe slow temporal and spatial modulations of a spatially and temporarily oscillating wave packet in this paper we give a rigorous proof that the wave packets in the two dimensional water wave problem in a canal of finite depth can be approximated over a physically relevant timespan by solutions of the nonlinear schrodinger equation

uniform decay estimates and the lorentz invariance of the classical wave equation on etudie le comportement asymptotique des solutions de u 0 ou t 2 1 2 n 2 pour des conditions initiales u 0 u t g x en t 0 avec g reguliere a support compact dans r n

on the free boundary problem of three dimensional incompressible euler equations in this paper we consider in three dimensions the motion of a general inviscid incompressible fluid with a free interface that separates the fluid region from the vacuum we assume that the fluid region is below the vacuum and that there is no surface tension on the free surface then we prove the local well posedness of the free boundary problem in sobolev space provided that there is no self intersection point on the initial surface and under the stability assumption that with being restricted to the initial surface 2007 wiley periodicals inc

well posedness in sobolev spaces of the full water wave problem in 3 d we consider the motion of the interface separating an inviscid incompressible irrotational fluid from a region of zero density in three dimensional space we assume that the fluid region is below the vacuum the fluid is under the influence of gravity and the surface tension is zero assume that the density of mass of the fluid is one the gravitational field is 0 0 1 the free interface is t at time t 0 and the fluid occupies the region t the motion of the fluid is described by vt v v 0 0 1 p on t t 0 euler s equation 1 1 div v 0 on t t 0 incompressible 1 2 curl v 0 on t t 0 irrotational 1 3 where v v1 v2 v3 is the fluid velocity p is the fluid pressure since we neglect the surface tension the pressure is zero on the interface so on the interface p 0 on t 1 4 1 v is tangent to the free surface t t 1 5 we want to find solutions of system 1 1 1 5 taking prescribed initial data such that for every fixed t 0 t approaches the xy plane at infinity and v x y z t 0 vt x y z t 0 as x y z since the fluid is assumed irrotational incompressible we can reduce the study of the entire motion to the motion of the free surface the above model is a 3 d water wave model it is generally known that when surface tension is neglected the motion of the interface between an inviscid fluid and vacuum under the influence of gravity can be subject to taylor instability 8 22 in a previous work 24 we studied the 2 dimensional water wave model we showed that for a 2 d water wave the sign condition relating to taylor instability always holds for nonself intersecting interface that is the motion of the interface is not subject to taylor instability we showed further that the 2 d full nonlinear water wave problem is uniquely solvable in sobolev spaces locally in time for any initially nonself intersecting interface earlier works on the well posedness in sobolev spaces of the 2 d water wave problem include nalimov 18 yosihara 25 and walter craig 6 where the main results concern the well posedness of the

geometry and a priori estimates for free boundary problems of the euler s equation in this paper we derive estimates to the free boundary problem for the euler equation with surface tension and without surface tension provided the rayleigh taylor sign condition holds we prove that as the surface tension tends to 0 when the rayleigh taylor condition is satisfied solutions converge to the euler flow with zero surface tension 2007 wiley periodicals inc

long time existence results for bore type initial data for bbm boussinesq systems abstract in this paper we deal with the long time existence for the cauchy problem associated to bbm type boussinesq systems of equations which are asymptotic models for long wave small amplitude gravity surface water waves as opposed to previous papers devoted to the long time existence issue we consider initial data with nontrivial behavior at infinity which may be used to model bore propagation

multifrequency nls scaling for a model equation of gravity capillary waves this paper is the first in a series papers devoted to the study of the rigorous derivation of the nonlinear schrodinger nls equation as well as other related systems starting from a model coming from the gravity capillary water wave system in the long wave limit our main goal is to understand resonances and their effects on having the nonlinear schrodinger approximation or modification of it or having other models to describe the limit equation in this first paper our goal is not to derive nls but to allow the presence of an arbitrary sequence of frequencies around which we have a modulation and prove local existence on a uniform time this yields a new class of large data for which we have a large time of existence 2012 wiley periodicals inc

deep water internal solitary waves near critical density ratio we study both supercritical and subcritical bifurcations of internal solitary waves propagating along the interface between two ideal fluids we derive a generalized nonlinear schrodinger equation that describes solitons near the critical density ratio corresponding to the transition from a subcritical to a supercritical bifurcation this equation takes into account gradient terms associated with the four wave interactions the so called lifshitz term and a nonlocal term analogous to that first found by dysthe for pure gravity waves as well as the term representing six wave nonlinear interactions within this model we find two branches of solitons and analyze their lyapunov stability a stability analysis shows that solitons below the critical ratio are stable in the lyapunov sense in the wide range of soliton parameters above the critical density ratio solitons are shown to be unstable with respect to finite perturbations

the long wave limit for the water wave problem i the case of zero surface tension the korteweg de vries equation boussinesq equation and many other equations can be formally derived as approximate equations for the two dimensional water wave problem in the limit of long waves here we consider the classical problem concerning the validity of these equations for the water wave problem in an infinitely long canal without surface tension we prove that the solutions of the water wave problem in the long wave limit split up into two wave packets one moving to the right and one to the left where each of these wave packets evolves independently as a solution of a korteweg de vries equation our result allows us to describe the nonlinear interaction of solitary waves 2000 john wiley sons inc

a simple global characterization for normal forms of singular vector fields we derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems using an appropriate scalar product in the space of homogeneous vector polynomials we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator this leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form explicit examples and results obtained when there are additional symmetries are also presented

existence and conditional energetic stability of three dimensional fully localised solitary gravity capillary water waves in this paper we show that the hydrodynamic problem for three dimensional water waves with strong surface tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal direction the proof is based upon the classical variational principle that a solitary wave of this type is a critical point of the energy which is given in dimensionless coordinates by epsilon eta phi integral r2 1 2 integral 1 eta 0 phi 2 x phi 2 y phi 2 z dy 1 2 eta 2 beta root 1 eta 2 x eta 2 z 1 dxdz subject to the constraint that the momentum i eta phi integral r2 eta x phi vertical bar y 1 eta dzdz is fixed here x y z x z is an element of r y is an element of 0 1 eta x z is the fluid domain phi is the velocity potential and beta 1 3 is the bond number these functionals are studied locally for eta in a neighbourhood of the origin in h 3 r 2 we prove the existence of a minimiser of epsilon subject to the constraint i 2 mu where 0 mu 1 the existence of a small amplitude solitary wave is thus assured and since epsilon and i are both conserved quantities a standard argument may be used to establish the stability of the set d mu of minimisers as a whole stability is however understood in a qualified sense due to the lack of a global well posedness theory for three dimensional water waves we show that solutions to the evolutionary problem starting near d mu remain close to d mu in a suitably defined energy space over their interval of existence they may however explode in finite time due to higher order derivatives becoming unbounded c 2012 elsevier inc all rights reserved