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

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

global infinite energy solutions for the 2d gravity water waves system we prove global existence and a modified scattering property for the solutions of the 2d gravity water waves system in the infinite depth setting for a class of initial data which is only required to be small above the level h 1 5 h 1 5 1 2 no assumption is made below this level therefore the nonlinear solution can have infinite energy as a direct consequence the momentum condition assumed on the physical velocity in all previous small energy results by ionescu pusateri alazard delort and ifrim tataru is removed 2017 wiley periodicals inc

particle method solution of two dimensional convection diffusion equations abstract we present a new method for solving two dimensional convection dominated convection diffusion equations containing spatially and temporally localized source terms the method uses grid free transport particles which carry point values of the concentration gradient move by convection and undergo random walk to simulate diffusion no numerical diffusion is introduced equations for the evolution of the gradient values are solved the concentration is recovered from the particle data by solving poisson equations the method is applied to problems in rotational and elongational flow fields numerical results demonstrating convergence of the method are presented

wellposedness of the 2d full water wave equation in a regime that allows for non c 1 c 1 interfaces we consider the two dimensional gravity water wave equation in a regime where the free interface is allowed to be non c 1 in this regime only a degenerate taylor inequality frac partial p partial mathbf n ge 0 holds with degeneracy at the singularities in kinsey and wu camb j math 6 2 93 181 2018 an energy functional mathcal e t was constructed and an a priori estimate was proved the energy functional mathcal e t is not only finite for interfaces and velocities in sobolev spaces but also finite for a class of non c 1 interfaces with angled crests in this paper we prove the existence uniqueness and stability of the solution of the 2d gravity water wave equation in the class where mathcal e t infty locally in time for any given data satisfying mathcal e 0 infty

finite time singularities for the free boundary incompressible euler equations in this paper we prove the existence of smooth initial data for the 2d free boundary incompressible euler equations also known for some particular scenarios as the water wave problem for which the smoothness of the interface breaks down in nite time into a splash singularity or a splat singularity

energy estimates and local well posedness of 3d interfacial hydroelastic waves between two incompressible fluids abstract in the current paper we are concerned with waves propagating through the deformation of a thin elastic sheet between two incompressible and inviscid fluids which are usually called hydroelastic waves in the literature to model deformable sheets interacting with surrounding fluids the main purpose of the present study is to solve a basic question on the theoretical side i e the local well posedness issue the problem is formulated by the full euler equations without the assumption of irrotationality for fluids combined with the plotnikov toland model for the elastic sheet based on geometric considerations we derive energy estimates and prove the local existence and uniqueness of solution for this system in n 2 dimensions even if velocity fields are rotational

accuracy of the random vortex method for a problem with non smooth initial conditions abstract the accuracy of the random vortex method in approximating the solution of the navier stokes equation is investigated using the model problem of a circular vortex as suggested by milinazzo and saffman 13 the method consists of partitioning the vorticity into vortex blobs these blobs are moved via two actions first a blob is deterministically moved under the action of the velocity field associated with the other blobs then to simulate viscosity a random component is added to the position of the blob for this model problem the nonlinear terms of the navier stokes equation vanish thus the major error inherent in the deterministic component of the method vanishes consequently for this model problem concentration is on the interaction of the deterministic and random components of the method results show that the accuracy of the method depends heavily on the initial distribution and strength of the computational elements i e the vortex blobs with the right choice of initial conditions it is found that e1 t l t a t a t is o r 1 2 n 1 2 where l t and a t are respectively the exact and computed angular moment of vorticity distribution at time t for n vortex blobs at a reynolds number r

time quasi periodic gravity water waves in finite depth we prove the existence and the linear stability of cantor families of small amplitude time quasi periodic standing water wave solutions namely periodic and even in the space variable x of a bi dimensional ocean with finite depth under the action of pure gravity such a result holds for all the values of the depth parameter in a borel set of asymptotically full measure this is a small divisor problem the main difficulties are the fully nonlinear nature of the gravity water waves equations the highest order x derivative appears in the nonlinear term but not in the linearization at the origin and the fact that the linear frequencies grow just in a sublinear way at infinity we overcome these problems by first reducing the linearized operators obtained at each approximate quasi periodic solution along a nash moser iterative scheme to constant coefficients up to smoothing operators using pseudo differential changes of variables that are quasi periodic in time then we apply a kam reducibility scheme which requires very weak melnikov non resonance conditions which lose derivatives both in time and space despite the fact that the depth parameter moves the linear frequencies by just exponentially small quantities we are able to verify such non resonance conditions for most values of the depth extending degenerate kam theory

shortest spanning trees and a counterexample for random walks in random environments we construct forests that span d d 2 that are stationary and directed and whose trees are infinite but for which the subtrees attached to each vertex are as short as possible for d 3 two independent copies of such forests pointing in opposite directions can be pruned so as to become disjoint from this we construct in d 3 a stationary polynomially mixing and uniformly elliptic environment of nearest neighbor transition probabilities on d for which the corresponding random walk disobeys a certain zero one law for directional transience

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

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

on the absence of splash singularities in the case of two fluid interfaces we show that splash singularities cannot develop in the case of locally smooth solutions of the two fluid interfaces in two dimensions more precisely we show that the scenario of formation of singularities discovered by castro cordoba fefferman gancedo gomez serrano 6 in the case of the water waves system in which the interface remains locally smooth but self intersects in finite time is completely prevented in the case of two fluid interfaces with positive densities

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

on the water wave equations with surface tension the purpose of this article is to clarify the cauchy theory of the water waves equations as well in terms of regularity indexes for the initial conditions as for the smoothness of the bottom of the domain namely no regularity assumption is assumed on the bottom our main result is that following the approach developped by t alazard and g metivier in 1 after suitable paralinearizations the system can be arranged into an explicit symmetric system of schrodinger type we then show that the smoothing effect for the one dimensional surface tension water waves proved by h christianson v m hur and g staffilani in 9 is in fact a rather direct consequence of this reduction which allows also to lower the regularity indexes of the initial data and to obtain the natural weights in the estimates

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

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

boundary conditions for stochastic solutions of the convection diffusion equation stochastic methods offer an attractively simple solution to complex transport controlled problems and have a wide range of physical chemical and biological applications stochastic methods do not suffer from the numerical diffusion that plagues grid based methods but they typically lose accuracy in the vicinity of interfacial boundaries in this work we introduce some ideas and algorithms that can be used to implement boundary conditions in stochastic simulations of the convection diffusion equation with accuracies comparable to the bulk phase the algorithms have been tested in two dimensional channel flows over a range of peclet numbers and compared with independent finite difference calculations

asymptotic stability of solitons for mkdv abstract we prove a full asymptotic stability result for solitary wave solutions of the mkdv equation we consider small perturbations of solitary waves with polynomial decay at infinity and prove that solutions of the cauchy problem evolving from such data tend uniformly on the real line to another solitary wave as time goes to infinity we describe precisely the asymptotics of the perturbation behind the solitary wave showing that it satisfies a nonlinearly modified scattering behavior this latter part of our result relies on a precise study of the asymptotic behavior of small solutions of the mkdv equation

stochastic homogenization of hamilton jacobi bellman equations we study the homogenization of some hamilton jacobi bellman equations with a vanishing second order term in a stationary ergodic random medium under the hyperbolic scaling of time and space imposing certain convexity growth and regularity assumptions on the hamiltonian we show the locally uniform convergence of solutions of such equations to the solution of a deterministic effective first order hamilton jacobi equation the effective hamiltonian is obtained from the original stochastic hamiltonian by a minimax formula our homogenization results have a large deviations interpretation for a diffusion in a random environment c 2006 wiley periodicals inc

error estimates and convergence rates for the stochastic homogenization of hamilton jacobi equations we present exponential error estimates and demonstrate an algebraic convergence rate for the homogenization of level set convex hamilton jacobi equations in i i d random environments the first quantitative homogenization results for these equations in the stochastic setting by taking advantage of a connection between the metric approach to homogenization and the theory of first passage percolation we obtain estimates on the fluctuations of the solutions to the approximate cell problem in the ballistic regime away from flat spot of the effective hamiltonian in the sub ballistic regime on the flat spot we show that the fluctuations are governed by an entirely different mechanism and the homogenization may proceed without further assumptions at an arbitrarily slow rate we identify a necessary and sufficient condition on the law of the hamiltonian for an algebraic rate of convergence to hold in the sub ballistic regime and show under this hypothesis that the two rates may be merged to yield comprehensive error estimates and an algebraic rate of convergence for homogenization our methods are novel and quite different from the techniques employed in the periodic setting although we benefit from previous works in both first passage percolation and homogenization the link between the rate of homogenization and the flat spot of the effective hamiltonian which is related to the nonexistence of correctors is a purely random phenomenon observed here for the first time

nonlinear fractional schrodinger equations in one dimension abstract we consider the question of global existence of small smooth and localized solutions of a certain fractional semilinear cubic nls in one dimension i t u u c 0 u 2 u c 1 u 3 c 2 u u 2 c 3 u 3 x x 1 2 where c 0 r and c 1 c 2 c 3 c this model is motivated by the two dimensional water wave equation which has a somewhat similar structure in the eulerian formulation in the case of irrotational flows we show that one cannot expect linear scattering even in this simplified model more precisely we identify a suitable nonlinear logarithmic correction and prove global existence and modified scattering of solutions

global solutions of the euler maxwell two fluid system in 3d the fundamental two uid model for describing plasma dynamics is given by the euler maxwell system in which compressible ion and electron uids interact with their own self consistent electromagnetic eld we prove global stability of a constant neutral background in the sense that irrotational smooth and localized perturbations of a constant background with small amplitude lead to global smooth solutions in three space dimensions for the euler maxwell system our construction is robust in dimension 3 and applies equally well to other plasma models such as the euler poisson system for two uids and a relativistic euler maxwell system for two uids our solutions appear to be the rst nontrivial global smooth solutions in all of these models

long range scattering for nonlinear schrodinger equations in one space dimension we consider the scattering problem for the nonlinear schrodinger equation in 1 1 dimensions open image in new window where x r 0 r p 3 we show that modified wave operators for exist on a dense set of a neighborhood of zero in the lebesgue spacel2 r or in the sobolev spaceh1 r the modified wave operators are introduced in order to control the long range nonlinearity u 2u

homogenization for stochastic hamilton jacobi equations homogenization asks whether average behavior can be discerned from partial differential equations that are subject to high frequency fluctuations when those fluctuations result from a dependence on two widely separated spatial scales we prove homogenization for certain stochastic hamilton jacobi partial differential equations the idea is to use the subadditive ergodic theorem to establish the existence of an average in the infinite scale separation limit in some cases we also establish a central limit theorem

3d quadratic nls equation with electromagnetic perturbations abstract in this paper we study the asymptotic behavior of a quadratic schrodinger equation with electromagnetic potentials we prove that small solutions scatter the proof builds on earlier work of the author for quadratic nls with a non magnetic potential the main novelty is the use of various smoothing estimates for the linear schrodinger flow in place of boundedness of wave operators to deal with the loss of derivative as a byproduct of the proof we obtain boundedness of the wave operator of the linear electromagnetic schrodinger equation on an l 2 weighted space for small potentials as well as a dispersive estimate for the corresponding flow

strichartz estimates and local existence for the gravity capillary waves with non lipschitz initial velocity abstract we consider the gravity capillary waves in any dimension and in fluid domains with general bottoms using the paradifferential reduction established in 19 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

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

rayleigh taylor breakdown for the muskat problem with applications to water waves the muskat problem models the evolution of the interface between two different fluids in porous media the rayleigh taylor condition is natural to reach linear stability of the muskat problem we show that the rayleigh taylor condition may hold initially but break down in finite time as a consequence of the method used we prove the existence of water waves turning

global well posedness of the free surface incompressible euler equations with damping abstract we consider a layer of an incompressible inviscid fluid bounded below by a fixed solid boundary and above by a free moving boundary in a horizontally periodic setting the fluid dynamics is governed by the gravity driven incompressible euler equations with damping and the effect of surface tension is neglected on the free surface we prove that the problem is globally well posed for the small initial data and that solutions decay to the equilibrium at an almost exponential rate

splitting mixed hodge structures over affine invariant manifolds we prove that affine invariant manifolds in strata of flat surfaces are algebraic varieties the result is deduced from a generalization of a theorem of moller namely we prove that the image of a certain twisted abel jacobi map lands in the torsion of a factor of the jacobians this statement can be viewed as a splitting of certain mixed hodge structures

analysis of chebyshev collocation methods for parabolic equations time discretizations of both the heat equation and the advection diffusion equation in two space variables are analyzed for space approximation the pseudospectral chebyshev method which enjoys infinite order of accuracy for smooth solutions is used the equation is collocated at the gauss chebyshev points and derivatives are computed by the pseudospectral differencing technique at any time interval the pseudospectral solution which is a polynomial of degree n is advanced in time using the implicit theta method for the diffusive part of the equation while the advective term is dealt with explicitly we prove unconditional stability and optimal error bounds depending on both n and delta t the time step in the norms of the weighted sobolev spaces the method considered here is the most commonly used spectral method for parabolic equations to solve the linear system arising at any time interval efficient iterative techniques with scaling are presented

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

random vortex methods for the navier stokes equations two random vortex methods of runge kutta type are presented for solving the two dimensional navier stokes equations we intesgitate the accuracy of these methods by considering the model problem of a rotating flow with initial vorticity concentrated uniformly on a disk of finite radius functionals of the numerical solution are computed by monte carlo estimates with efficient variance reduction and the results are compared to those obtained from euler s method the numerical results show that both the methods produce errors smaller by one power of the time step size than euler s method one seemingly even better than the other these runge kutta methods are derivations of similar schemes proposed by us in an earlier time for solving stochastic differential equations with constant diffusion coefficients copyright 1988 academic press inc

on the finite time splash and splat singularities for the 3 d free surface euler equations we prove that the 3 d free surface incompressible euler equations with regular initial geometries and velocity fields have solutions which can form a finite time splash or splat singularity first introduced in castro et al splash singularity for water waves http arxiv org abs 1106 2120v2 2011 wherein the evolving 2 d hypersurface the moving boundary of the fluid domain self intersects at a point or on surface such singularities can occur when the crest of a breaking wave falls unto its trough or in the study of drop impact upon liquid surfaces our approach is founded upon the lagrangian description of the free boundary problem combined with a novel approximation scheme of a finite collection of local coordinate charts as such we are able to analyze a rather general set of geometries for the evolving 2 d free surface of the fluid we do not assume the fluid is irrotational and as such our method can be used for a number of other fluid interface problems including compressible flows plasmas as well as the inclusion of surface tension effects

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

grid free simulation of diffusion using random walk methods abstract the simulation of diffusion of a continuum field by the random walk displacement of a set of particles is investigated in detail computational particles are used to transport elements of the gradients of the diffusive concentration one dimensional and quasi one dimensional cases are treated for a generalized diffusion variable different types of boundary conditions of the diffusion equation are considered as well as the extensions to a system of coupled diffusion equations the reaction diffusion equations and the convection diffusion set of equations it is shown that by using concentration gradients in the random walk process statistical errors are reduced and each realization of the numerical solution is a representation of the exact solution moreover transport of higher order derivatives can be utilized to improve the smoothness the algorithm is grid free and the computational elements move to follow the gradients thus it is self adaptive and uniform resolution is attained for all times the method is particularly suitable for the simulation of diffusion in systems which involve more than one transport mechanism when the diffusivity is small and when lagrangian elements are used to model the other mechanisms

stochastic homogenization of nonconvex hamilton jacobi equations a counterexample this paper provides a counterexample to hamilton jacobi homogenization in the nonconvex case for general stationary ergodic environments

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

space time resonances and the null condition for first order systems of wave equations in this manuscript we prove global existence and linear asymptotic behavior of small solutions to nonlinear wave equations we assume that the quadratic part of the nonlinearity satisfies a nonresonance condition that is a generalization of the null condition given by klainerman 2013 wiley periodicals inc

global solutions for the gravity water waves system in 2d 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 invent math 177 1 45 135 2009 we also prove that the asymptotic behavior of solutions as time goes to infinity is different from linear unlike the three dimensional case germain et al ann math 175 2 691 754 2012 wu invent math 184 1 125 220 2011 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 nontrivial solutions of the gravity water waves system in 2d

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

asymptotics for large time of solutions to the nonlinear schrodinger and hartree equations we study the asymptotic behavior in time of solutions to the cauchy problems for the nonlinear schrodinger equation with a critical power nonlinearity and the hartree equation we prove the existence of modified scattering states and the sharp time decay estimate in the uniform norm of solutions to the cauchy problem with small initial data this estimate is very important for the proof of the existence of modified scattering states to the nonlinear schrodinger equations with a critical nonlinearity and the hartree equation in order to derive the desired estimates we introduce a certain phase function since the previous methods based solely on a priori estimates of the operator x it acting on the solution without specifying any phase function do not work for the critical case under consideration the well known nonexistence of the usual l 2 scattering states shows that our result is sharp

numerical solution of stochastic differential equations with constant diffusion coefficients we present runge kutta methods of high accuracy for stochastic differential equations with constant diffusion coefficients we analyze l sub 2 convergence of these methods and present convergence proofs for scalar equations a second order method is derived and for systems a method of order one and one half is derived we further consider a variance reduction technique based on hermite expansions for evaluating expectations of functions of sample solutions numerical examples in two dimensions are presented

gradient particle solutions of fokker planck equations for noisy delay bifurcations a gradient particle method is used to compute probability densities for processes with delay bifurcations that are sensitive to very small noise the method is particularly useful for computing the probability density in the transition region where asymptotic approximations may not be valid for a one dimensional steady bifurcation problem and a two dimensional fitzhugh nagumo model we solve the fokker planck equation fpe and compare the results with direct simulations and asymptotic approximations

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

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

local well posedness for fluid interface problems in this paper we prove the local well posedness of the fluid interface problem with surface tension where the velocity fields are not assumed to be irrotational and the fluid domains are not assumed to be simply connected viewed as a lagrangian system with the configuration space being an infinite dimensional manifold possessing many symmetries this problem is reduced to the evolution of the interface determined by its mean curvature and the evolution of the rotational part of the velocity fields determined by the symmetries this framework also applies to several other fluid surface problems which are outlined in the paper

two dimensional water waves in holomorphic coordinates this article is concerned with the infinite depth water wave equation in two space dimensions we consider this problem expressed in position velocity potential holomorphic coordinates viewing this problem as a quasilinear dispersive equation we establish two results i local well posedness in sobolev spaces and ii almost global solutions for small localized data neither of these results are new they have been recently obtained by alazard burq zuily invent math 198 71 163 2014 respectively by wu invent math 177 45 135 2009 using different coordinates and methods instead our goal is to improve the understanding of this problem by providing a single setting for both problems by proving sharper versions of the above results as well as presenting new simpler proofs this article is self contained

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

numerical simulation of gravity waves abstract we present a new spectral method to simulate numerically the waterwave problem in a channel for a fluid of finite or infinite depth it is formulated in terms of the free surface elevation and the velocity potential the numerical method is based on the reduction of this problem to a lower dimensional computation involving surface variables alone to accomplish this we describe the taylor expansion of the dirichlet neumann operator in homogeneous powers of the surface elevation each term is a concatenation of fourier multipliers with powers of and its derivatives and is valid uniformly in wavenumber these are easily calculated using the fast fourier transform the method is illustrated by computing the long time evolution of modulated wave packets and of approximations to the stokes steady wave train by imposing a surface pressure we observe surface steepening in large amplitude evolution and wake and bow wave development for flows with a close to critical froude number finally we give an example of nonlinear evolution of the distribution of energy among normal modes

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

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

global existence for capillary water waves 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 2015 wiley periodicals inc

a sublinear variance bound for solutions of a random hamilton jacobi equation we estimate the variance of the value function for a random optimal control problem the value function is the solution w of a hamilton jacobi equation with random hamiltonian h p x k p v x in dimension d 2 it is known that homogenization occurs as 0 but little is known about the statistical fluctuations of w our main result shows that the variance of the solution w is bounded by o log the proof relies on a modified poincare inequality of talagrand