We review recent progress on the long-time regularity of solutions of the Cauchy problem for the water waves equations, in two and three dimensions.
We begin by introducing the free boundary Euler equations and discussing the local existence of solutions using the paradifferential approach, as in [7, 1, 2]. We then describe in a unified framework, using the Eulerian formulation, global existence results for three dimensional
and two dimensional gravity waves, see [70, 146, 145, 87, 5, 6, 79, 80, 136], and our joint
result with Deng and Pausader [60] on global regularity for the 3D gravity-capillary model.
We conclude this review with a short discussion about the formation of singularities, and
give a few additional references to other interesting topics in the theory.
In this paper we prove global regularity for the full water waves system in
3 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 |t| −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 nonlinearity, 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.
In this paper and its companion [32] we prove global regularity for the full
water waves system in 3 dimensions for small data, under the influence of both gravity
and surface tension. The main difficulties are the weak, and far from integrable, pointwise
decay of solutions, together with the presence of a full codimension one set of quadratic
resonances. To overcome these difficulties we use a combination of improved energy
estimates and dispersive analysis.
In this paper we prove the energy estimates, while the dispersive estimates are proved
in [32]. These energy estimates depend on several new ingredients, such as a key nondegeneracy
property of the resonant hypersurfaces and some special structure of the
quadratic part of the nonlinearity.
In this paper and its companion [32] we prove global regularity for the full
water waves system in 3 dimensions for small data, under the influence of both gravity
and surface tension. The main difficulties are the weak, and far from integrable, pointwise
decay of solutions, together with the presence of a full codimension one set of quadratic
resonances. To overcome these difficulties we use a combination of improved energy
estimates and dispersive analysis.
In this paper we prove the dispersive estimates, while the energy estimates are proved
in [32]. The dispersive estimates rely on analysis of the Duhamel formula in a carefully
constructed weighted norm, taking into account the nonlinear contribution of special
frequencies, such as the space-time resonances, and the slowly decaying frequencies.
We consider the full irrotational water waves system with surface tension and no
gravity in dimension two (the capillary waves system), and prove global regularity and modified
scattering for suitably small and localized perturbations of a flat interface. An important point
of our analysis is to develop a sufficiently robust method which
allows us to deal with strong singularities arising from time resonances in the applications of
the normal form method. .
As a result, we are able to consider
a suitable class of perturbations with finite energy, but no other momentum conditions.
Part of our analysis relies on a new treatment of the Dirichlet-Neumann operator in dimension
two which is of independent interest. As a consequence, the results in this paper are
self-contained.
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 two (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 full system is the subject of a forthcoming paper.
In this note we explain how to derive an asymptotic formula for solutions of the 2d gravity water waves system expressed in Eulerian coordinates, using some of the bounds obtained by the authors in [IPu13]. The main ingredients of the proof are: 1) the uniform estimates for the Fourier transform of the profile of solutions obtained in [IPu13], 2) a refined linear estimate for the propagator $exp(it|\partial_x|^1/2)$, and 3) an argument similar to the one used by Hayashi and Naumkin in [HN98] in the context of NLS type equations.
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.
This paper proposes a fairly general new point of view on the question of asymptotic stability of (topological) solitons. Our approach is based on the use of the distorted Fourier transform at the nonlinear level; it does not rely on Strichartz or virial estimates and is therefore able to treat low power nonlinearities (hence also non-localized solitons) and capture the global (in space and time) behavior of solutions. More specifically, we consider quadratic nonlinear Klein-Gordon equations with a potential in one space dimension. The potential is assumed to be regular, decaying, and either generic or exceptional (with some additional parity assumptions). Assuming that the associated Schrödinger operator has no negative eigenvalues, we obtain global-in-time bounds, including sharp pointwise decay and modified asymptotics, for small solutions. These results have implications for the asymptotic stability of solitons, or topological solitons, for a variety of problems. For instance, we obtain full asymptotic stability of kinks with respect to odd perturbations for the double Sine-Gordon problem (in an appropriate range of the deformation parameter). For the $\phi^4$ problem, we obtain asymptotic stability of the kink (with respect to odd perturbations) when the coupling to the internal mode appearing in the linearization around it is neglected. Our results also go beyond these examples since our approach allows for the presence of a fully coherent phenomenon at the level of quadratic interactions, which creates a degeneracy in distorted Fourier space. We devise a suitable framework that incorporates this, and use multilinear harmonic analysis in the distorted setting to control all nonlinear interactions.
We propose an approach to nonlinear evolution equations with large and decaying external potentials that addresses the question of controlling globally-in-time the nonlinear interactions of localized waves in this setting. This problem arises when studying localized perturbations around (possibly non-decaying) special solutions of evolution PDEs, and trying to control the projection onto the continuous spectrum of the nonlinear radiative interactions. One of our main tools is the Fourier transform adapted to the Schrödinger operator $H=-\Delta+V$, which we employ at a nonlinear level. As a first step we analyze the spatial integral of the product of three generalized eigenfunctions of $H$, and determine the precise structure of its singularities. This leads to study bilinear operators with certain singular kernels, for which we derive product estimates of Coifman-Meyer type. This analysis can then be combined with multilinear harmonic analysis tools and the study of oscillations to obtain (distorted Fourier space analogues of) weighted estimates for dispersive and wave equations. As a first application we consider the nonlinear Schrödinger equation in $3$d in the presence of large decaying potential with no bound states, and with a $u^2$ non-linearity. The main difficulty is that a quadratic nonlinearity in $3$d is critical with respect to the Strauss exponent; moreover, this nonlinearity has non-trivial fully coherent interactions even when $V=0$. We prove quantitative global-in-time bounds and scattering for small solutions.
We consider the 1d cubic nonlinear Schrödinger equation with a large external potential V with no bound states. We prove global regularity and quantitative bounds for small solutions under mild assumptions on V. In particular, we do not require any differentiability of V, and make spatial decay assumptions that are weaker than those found in the literature (see for example \cite{Del,N,GPR}). We treat both the case of generic and non-generic potentials, with some additional symmetry assumptions in the latter case. Our approach is based on the combination of three main ingredients: the Fourier transform adapted to the Schrödinger operator, basic bounds on pseudo-differential operators that exploit the structure of the Jost function, and improved local decay and smoothing-type estimates. An interesting aspect of the proof is an "approximate commutation" identity for a suitable notion of a vectorfield, which allows us to simplify the previous approaches and extend the known results to a larger class of potentials. Finally, under our weak assumptions we can include the interesting physical case of a barrier potential as well as recover the result of \cite{MMS} for a delta potential.
We consider the cubic nonlinear Schrödinger equation with a potential in one space dimension. Under the assumptions that the potential is generic, sufficiently localized, and does not have bound states, we obtain the long time asymptotic behavior of small solutions. In particular, we prove that, as time goes to infinity, solutions exhibit nonlinear phase corrections that depend on the scattering matrix associated to the potential. The proof of our result is based on the use of the distorted Fourier transform - the so-called Weyl-Kodaira-Titchmarsh theory -, a precise understanding of the "nonlinear spectral measure" associated to the equation, and nonlinear stationary phase arguments as well as multilinear estimates in this distorted setting
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.
For Schrödinger equations with both time-independent and time-dependent Kato potentials, we give a simple proof of the maximal speed bound. The latter states that the probability to find the quantum system outside the ball of radius proportional to the time lapsed decays as an inverse power of time. We give an explicit expression for the constant of proportionality in terms of the maximal energy available to the initial condition. For the time-independent part of the interaction, we require neither decay at infinity nor smoothness.
The density functional theory (DFT) is a remarkably successful theory of electronic structure of matter. At the foundation of this theory lies the Kohn-Sham (KS) equation. In this paper, we describe the long-time behaviour of the time-dependent KS equation. Assuming weak self-interactions, we prove global existence and scattering in (almost) the full "short-range" regime. This is achieved with new and simple techniques, naturally compatible with the structure of the DFT and involving commutator vector fields and non-abelian versions of Sobolev-Klainerman-type spaces and inequalities.
We consider non-gauge-invariant cubic nonlinear Schröodinger equations in one space dimension. We show that initial data of size ε in a weighted Sobolev space lead to solutions with sharp L^\infty decay up to time $exp(Cε^{-2})$. We also exhibit norm growth beyond this time for a specific choice of nonlinearity.
We consider the Chern–Simons–Schrödinger model in 1+2 dimensions, and prove scattering for small solutions of the Cauchy problem in the Coulomb gauge. This model is a gauge covariant Schrödinger equation, with a potential decaying like r^{−1} at infinity. To overcome the difficulties due to this long-range decay, we perform L^2 -based estimates covariantly. This procedure gives favorable commutation identities so that only curvature terms, which decay faster than r^{−1}, appear in our weighted energy estimates. We then select the Coulomb gauge to reveal a genuinely cubic null structure, which allows us to establish scattering to linear solutions by Fourier methods
We consider the question of scattering for the boson star equation in three space dimensions. This is a semi-relativistic Klein-Gordon equation with a cubic nonlinearity of Hartree type. We combine weighted estimates, obtained by exploiting a special null structure present in the equation, and a refined asymptotic analysis performed in Fourier space, to obtain global solutions evolving from small and localized Cauchy data. We describe the behavior at infinity of such solutions by identifying a suitable nonlinear asymptotic correction to scattering. As a byproduct of the weighted energy estimates alone, we also obtain global existence and (linear) scattering for solutions of semi-relativistic Hartree equations with potentials decaying faster than Coulomb.
We consider the question of global existence of small, smooth, and localized solutions of a certain fractional semilinear cubic NLS in one dimension, i\partial_t u − Λ u = c_0|u|^2u + c_1 u^3 + c_2 u\bar{u}^2 + c_3 \bar{u}^3, Λ=Λ(\partial_x)=|\partial_x|^(1/2), where c_0 \in R and c_1,c_2,c_3 \in 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.
We present a new proof of global existence and long range scattering, from small initial data, for the one-dimensional cubic gauge invariant nonlinear Schröodinger equation, and for Hartree equations in dimension n≥2. The proof relies on an analysis in Fourier space, related to the recent works of Germain, Masmoudi and Shatah on space-time resonances. An interesting feature of our approach is that we are able to identify the long range phase correction term through a very natural stationary phase argument
We study the motion of an incompressible, inviscid two-dimensional fluid in a rotating frame of reference. There the fluid experiences a Coriolis force, which we assume to be linearly dependent on one of the coordinates. This is a common approximation in geophysical fluid dynamics and is referred to as beta-plane. In vorticity formulation the model we consider is then given by the Euler equation with the addition of a linear anisotropic, non-degenerate, dispersive term. This allows us to treat the problem as a quasilinear dispersive equation whose linear solutions exhibit decay in time at a critical rate. Our main result is the global stability and decay to equilibrium of sufficiently small and localized solutions. Key aspects of the proof are the exploitation of a "double null form" that annihilates interactions between spatially coherent waves and a lemma for Fourier integral operators which allows us to control a strong weighted norm.
We consider the free-boundary motion of two perfect incompressible fluids with different densities ρ+ and ρ−, separated by a surface of discontinuity along which the pressure experiences a jump proportional to the mean curvature by a factor ϵ^2. Assuming the Raileigh-Taylor sign condition and ρ− ≤ ϵ^{3/2} we prove energy estimates uniform in ρ− and ϵ. As a consequence we obtain convergence of solutions of the interface problem to solutions of the free-boundary Euler equations in vacuum without surface tension as ϵ and ρ− tend to zero.
We consider the interface problem between two incompressible and inviscid fluids in the presence of surface tension. Following the geometric approach of [Shatah,J.;Zeng,C. A priori estimates for Fluid Interface Problems. CPAM, vol.16, no.6, June 2008] we show that solutions to this problem converge to solutions of the free-boundary Euler equations in vacuum as one of the densities goes to zero.
We consider a class of quasilinear wave equations in $3+1$ space-time dimensions that satisfy the ``weak null condition'' as defined by Lindblad and Rodnianski \cite{LR1}, and study the large time behavior of solutions to the Cauchy problem. The prototype for the class of equations considered is $-\partial_t^2 u + (1+u) \Delta u = 0$. Global solutions for such equations have been constructed by Lindblad \cite{Lindblad1,Lindblad2} and Alinhac \cite{Alinhac1}. Our main results are the derivation of a precise asymptotic system with good error bounds, and a detailed description of the behavior of solutions close to the light cone, including the blow-up at infinity.
We consider a class of wave-Schrödinger systems with a Zakharov-Schulman type coupling. This class of systems is indexed by a parameter gamma which measures the strength of the null form in the nonlinearity of the wave equation. The case gamma = 1 corresponds to the well-known Zakharov system, while the case gamma = -1 corresponds to the Yukawa system. Here we show that sufficiently smooth and localized Cauchy data lead to pointwise decaying global solutions which scatter, for any gamma in (0,1].
We prove global existence and scattering for small localized solutions of the Cauchy problem for the Zakharov system in 3 space dimensions. The wave component is shown to decay pointwise at the optimal rate of t^{-1}, whereas the Schrödinger component decays almost at a rate of t^{-7/6}.
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 non-resonant condition which is a generalization of the null condition given by Klainerman.
We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth, and prove a rigorous reduction of these equations to Bikhoff normal form up to degree four. This proves a conjecture of Zakharov-Dyachenko [62] based on the formal Birkhoff integrability of the water waves Hamiltonian truncated at order four. As a consequence, we also obtain a long-time stability result: periodic perturbations of a flat interface that are of size $\varepsilon$ in a sufficiently smooth Sobolev space lead to solutions that remain regular and small up to times of order $\varepsilon^{-3}$. Main difficulties in the proof are the quasilinear nature of the equations, the presence of small divisors arising from near-resonances, and non-trivial resonant four-waves interactions, the so-called Benjamin-Feir resonances. The main ingredients that we use are: (1) various reductions to constant coefficient operators through flow conjugation techniques; (2) the verification of key algebraic properties of the gravity water waves system which imply the integrability of the equations at non-negative orders; (3) smoothing procedures and Poincar\'e-Birkhoff normal form transformations; (4) a normal form identification argument that allows us to handle Benajamin-Feir resonances by comparing with the formal computations of [62,22,30,20].
We prove an extended lifespan result for the full gravity-capillary water waves system with a 2 dimensional periodic interface: for initial data of sufficiently small size ε, smooth solutions exist up to times of the order of ε^{−5/3+}, for almost all values of the gravity and surface tension parameters. Besides the quasilinear nature of the equations, the main difficulty is to handle the weak small divisors bounds for quadratic and cubic interactions, growing with the size of the largest frequency. To overcome this difficulty we use (1) the (Hamiltonian) structure of the equations which gives additional smoothing close to the resonant hypersurfaces, (2) another structural property, connected to time-reversibility, that allows us to handle "trivial" cubic resonances, (3) sharp small divisors lower bounds on three and four-waves modulation functions based on counting arguments, and (4) partial normal form transformations and symmetrization arguments in the Fourier space. Our theorem appears to be the first extended lifespan result for quasilinear equations with non-trivial resonances on a multi-dimensional torus.
In 2004, F\'ejoz [D\'emonstration du 'th\'eor\'eme d'Arnold' sur la stabilit\'e du syst\`eme plan\'etaire (d'apr\`es M. Herman). Ergod. Th. & Dynam. Sys. 24(5) (2004), 1521-1582], completing investigations of Herman's [D\'emonstration d'un th\'eor\'eme de V.I. Arnold. S\'eminaire de Syst\'emes Dynamiques et manuscripts, 1998], gave a complete proof of 'Arnold's Theorem' [V. I. Arnol'd. Small denominators and problems of stability of motion in classical and celestial mechanics. Uspekhi Mat. Nauk. 18(6(114)) (1963), 91-192] on the planetary many-body problem, establishing, in particular, the existence of a positive measure set of smooth (C\infty) Lagrangian invariant tori for the planetary many-body problem. Here, using R\"u{\ss}mann's 2001 KAM theory [H. R\"u{\ss}mann. Invariant tori in non-degenerate nearly integrable Hamiltonian systems. R. & C. Dynamics 2(6) (2001), 119-203], we prove the above result in the real-analytic class.