Recent advances on the global regularity for irrotational water waves

Abstract

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. We then describe in a unified framework, using the Eulerian formulation, global existence results for three- and two-dimensional gravity waves, and our joint result (with Deng and Pausader) on global regularity for the gravity–capillary model in three dimensions. 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.

This article is part of the theme issue ‘Nonlinear water waves’.

1. Introduction

The study of the motion of water waves, such as those on the surface of the ocean, is a classical question, and one of the main problems in fluid dynamics. The origins of water waves theory can be traced back at least to the work of Laplace and Lagrange, Cauchy [1] and Poisson, and then Russel, Green and Airy, among others (see [2]). Classical studies include those by Stokes [3], Levi-Civita [4] and Struik [5] on progressing waves, the instability analysis of Taylor [6], the works on solitary waves by Friedrichs & Hyers [7] and on steady waves by Gerber [8].

The main questions one can ask about water waves are the typical ones for any physical evolution problem: the existence of solutions of the initial value problem, their regularity, the possible formation of various singularities in the flow, the existence of special solutions (such as solitary waves, standing waves, periodic/quasi-periodic waves) and their stability, and the long-time existence and asymptotic behaviour of the flow. There is a vast body of literature dedicated to all of these aspects.

The main focus of this article is to review the local and global existence theory for the initial value problem associated to the water waves equations, and give an overview of the recent progress in this area. We will refer the reader to various books, research papers and surveys for other aspects of the theory.

We will concentrate on the motion of an inviscid and irrotational two- or three-dimensional fluid occupying a region of infinite depth and infinite extent below the graph of a function. These are models for the motion of waves on the surface of the deep ocean, where the two-dimensional case corresponds to waves whose motion is assumed to be constant in one direction on the interface. We will consider both two- and three-dimensional dynamics under the influence of the gravitational force and/or surface tension effects acting on particles at the interface. Our main goal is to present, in a unified framework, several results about the global existence of solutions which are initially small, that is, sufficiently close to a flat and still interface in a suitable sense.

(a). Structure of the paper

In §1b, we introduce the free boundary Euler equations in the standard Eulerian formulation and the Zakharov–Craig–Schanz–Sulem Hamiltonian formulation for irrotational flows. In §2, we discuss the short-time existence of solutions following the paradifferential approach in [9–11]. Section 3 is dedicated to global existence results. We discuss three different problems, in increasing order of difficulty: the three-dimensional gravity water waves, the two-dimensional gravity water waves and the three-dimensional gravity–capillary water waves. Section 4 contains a brief discussion about the formation of singularities, and few additional references to other interesting topics in the theory.

(b). Free boundary Euler equations

The evolution of an inviscid perfect fluid that occupies a domain , for n≥2, at time , is described by the free boundary incompressible Euler equations. We let v and p denote, respectively, the velocity and the pressure of the fluid, at time t and position x∈Ω_t, and assume that the fluid has constant density equal to 1. If the fluid evolves in a gravitational field, the equations of motion are

1.1a

where e_n is the nth standard unit vector of and g is the gravitational constant. The first equation in (1.1a) is the conservation of momentum equation, while the second is the incompressibility condition. When gravitational effects are neglected, one sets g=0 in (1.1a).

The boundary of the fluid evolves with time and is part of the unknowns in the problem. In particular, the free surface S_t:=∂Ω_t moves with the normal component of the velocity according to the kinematic boundary condition

1.1b

The atmospheric pressure outside the fluid domain is assumed to be constant, and set to zero for convenience. On the interface the pressure is given by

1.1c

where κ is the mean curvature of S_t and σ≥0 is the surface tension coefficient. At liquid–air interfaces, the surface tension force results from the greater attraction of water molecules to each other than to the molecules in the air.

One can consider the free boundary Euler equations above in various types of domains Ω_t (bounded, periodic, unbounded) and study flows with different characteristics (rotational or irrotational, with gravity and/or surface tension), or even more complicated scenarios where the moving interface separates two fluids.

There are several difficulties in treating the system (1.1a)–(1.1c) which are due to the quasi-linear nature of the equations, i.e. the highest derivatives appear nonlinearly, and, above all, to the free moving boundary and its interaction with the fluid. As we will discuss below, the system (1.1a)–(1.1c) has a ‘hyperbolic’ structure, which can only be captured owing to great insights into the nature of the equations, as was done, for example, in [10,12–18]. This structure leads to a priori control and local existence of solutions for sufficiently regular initial data in the case of non-self-intersecting interfaces, provided that

1.2

where N is the outer unit normal, which is the so-called Rayleigh–Taylor sign condition. In general, when (1.2) is violated, instabilities might occur [19–21].

We will discuss in more detail these local regularity issues in §2 below by restricting our attention to the case of irrotational flows, and following the paradifferential approach of Alazard–Métivier [9] and Alazard–Burq–Zuily [10,11]. We choose to present this approach among the various possible ones, because it is well suited as a starting point for the discussion of the long-time regularity results in §3.

(c). The Hamiltonian formulation

In the case of irrotational flows, that is when

1.3

one can reduce (1.1) to a system of two equations on the boundary. Indeed, assume also that is the region below the graph of a function , that is

1.4

Let Φ denote the velocity potential,

1.5

and let

1.6

denote the restriction of Φ to the boundary S_t (figure 1).

Figure 1.

The water waves problem in Eulerian coordinates.

Then, the equations of motion can be reduced to the following system for the unknowns :

1.7

Here

1.8

where n is the outward unit normal to S_t and denotes the Dirichlet–Neumann (DN) map associated to the domain Ω_t. Roughly speaking, one can think of G(h) as a first-order, non-local, linear operator that depends nonlinearly on the domain. We refer the reader to [22, ch. 11; 23] or the book of Lannes [24] for the derivation of (1.7), which is the so-called Zakharov–Craig–Schanz–Sulem formulation [25].

The system (1.7) admits the conserved energy

1.9

which is the sum of the kinetic energy corresponding to the L²-norm of the velocity field and the potential energy due to gravity and surface tension. It was first observed by Zakharov [26] that (1.7) is the Hamiltonian¹ flow associated with (1.9). For sufficiently small and smooth solutions one has

1.10

The formal linearization of (1.7)–(1.8) around a flat and still interface is

1.11

By defining the linear dispersion relation

1.12

the identities (1.11) can be written as a single equation for a complex-valued unknown,

1.13

One generally refers to (1.7) as the gravity water waves system when g>0 and σ=0, as the capillary water waves system when g=0 and σ>0, and as the gravity–capillary water waves system when g>0 and σ>0.

We remark that the presence of the various forces (gravity and/or surface tension) does have an impact on the existence theory of solutions. In the local existence theory this impact is mostly quantitative and the techniques developed for a specific scenario are likely to be adaptable to others. On the other hand, when considering the long-time existence of solutions, the presence of gravity and/or surface tension has a major impact on the evolution. This can be seen already at the level of the linearized equations (1.13), and is even more apparent when looking at three waves’ resonant interactions (quadratic time resonances), and at fully coherent interactions (space–time resonances). We explain these concepts in §3d.

(d). Other approaches and formulations

There are of course other possible descriptions of the equations. In the series of works [12,13,28–30] (see also the survey [31]), Wu uses a combination of Lagrangian coordinates and tools from complex analysis such as the Riemann mapping theorem and the theory of holomorphic functions (Clifford analysis in three dimensions). Lagrangian coordinates and variations have been used in [14,15,17], and complex coordinates in the works of Nalimov [32], in Zakharov et al. [33], in various theoretical and numerical works (e.g. [34,35] and references therein) and in the series of papers [36–39].

2. Local well-posedness

Because of the complicated nature of the equations, the development of a basic local well-posedness theory (existence and uniqueness of smooth solutions for the Cauchy problem) has proved to be highly non-trivial. Early results on the local well-posedness of the water waves system include those by Nalimov [32], Shinbrot [40], Yosihara [41], Kano & Nishida [42] and Craig [43]. All these results deal with small perturbations of a flat interface for which the Rayleigh–Taylor sign condition (1.2) always holds. It was first observed by Wu [13] that, in the irrotational case, (1.2) holds without smallness assumptions, and that local-in-time solutions can be constructed with initial data of arbitrary size in Sobolev spaces [12,13]. Following the breakthrough of Wu, the question of local well-posedness of the water waves and free boundary Euler equations has been addressed by several authors. Christodoulou & Lindblad [14] and Lindblad [15] considered the problem with vorticity, Beyer & Gunther [44] took into account the effects of surface tension and Lannes [16] treated the case of non-trivial bottom topography. The works by Ambrose & Masmoudi [45], Coutand & Shkoller [17] and Shatah & Zeng [18] extended these results to more general scenarios with vorticity and surface tension, including two-fluid systems [46–48] where surface tension is necessary for well-posedness. Other important papers that include surface tension and/or low regularity analysis are those by Christianson et al. [49] and Alazard et al. [10,11]; see also [50–56].

Owing to all the contributions mentioned above, the local well-posedness theory is presently well understood in a variety of different scenarios. In short, one can say that, for sufficiently nice initial configurations, it is possible to find classical smooth solutions on a small time interval. See theorem 2.3 for a typical result in the case of irrotational flows.

To explain some aspects of the local well-posedness theory, we follow the approach in Lannes [16], Alazard & Métivier [9] and the series of works by Alazard et al. [10,11,57,58], based on the use of paradifferential calculus.² We choose this path mostly because it is a good starting point for the long-time regularity theory, which we discuss in the next section.

(a). Paradifferential analysis

The main objective of the paradifferential analysis of the water waves system is to formulate the Hamiltonian system (1.7)–(1.8) for the unknowns h and ϕ in terms of new unknowns h and ω, so that the quasi-linear structure of the system is apparent. In other words, one wants to identify the terms that are responsible for the loss of derivatives in the nonlinearity, and write the equations in a convenient form, so that it is possible to obtain a priori energy estimates by a relatively simple procedure. A key point is to achieve a good understanding of the DN operator G(h)ϕ in (1.8).

(i). Elements of paradifferential calculus

Given a symbol a=a(x,ζ), and a function , we define the paradifferential operator T_af according to³

2.1

where is the Fourier transform of g, denotes the Fourier transform of a in the first coordinate, and is a smooth function supported in [0,2⁻²⁰] and equal to 1 on [0,2⁻²¹].

Note that when a=a(ζ), (2.1) reduces to a standard Fourier multiplier. If instead a=a(x), then T_af is the product of a and f where the frequencies of a are restricted to have size comparable to or smaller than the frequencies of f. In particular, T_af has the same regularity of f. Moreover, one has the basic paradifferential decomposition of Bony [59]:

There are various ways to measure symbols a=a(x,ζ) and the norms of the associated paradifferential operators. Without going into technical details, one should think of the homogeneity of a with respect to the variable ζ, when |ζ|≥1, as representing the order of the paradifferential operator, i.e. the number of derivatives acting on the function f, while the dependence on x is somewhat less relevant as far as regularity is concerned (which is what one cares about to establish local existence). A simple choice of a norm for symbols is⁴

2.2

where is the order of the symbol (and of the associated paradifferential operator), and measures the smoothness in x and the integrability.

As in the case of differential operators, it is possible to establish several algebra properties for suitable classes of paradifferential operators. In particular, one has the mapping property

2.3

and the formulae

2.4

where ≈ denotes an equality up to terms of lower order, [⋅,⋅] is the commutator and {a,b}:=∇_xa∇_ζb−∇_ζa∇_xb is the Poisson bracket.

(ii). The ‘good unknown’ and the Dirichlet–Neumann operator

To describe the action of the DN operator, one introduces

2.5

Here, is the restriction to the interface of the velocity field v of the fluid, and function ω is the so-called ‘good-unknown’ of Alinhac [9,65,66]. The origin of ω is in the paracomposition formula f°g≈T_f′°gg, which holds when g is rougher than f. As a result, the variable ω in (2.5) has better smoothness properties than ϕ, when h has limited regularity. One of the most important outcomes of the paradifferential analysis in the context of water waves is the following key formula for the DN operator.

Proposition 2.1 (Paralinearization of the DN operator). —

Let G(h) be the operator defined in (1.8), with (1.5)–(1.6), and let ω,B,V be given by (2.5). Then we have

2.6

where G₂ denotes smoother terms which are also quadratic in the unknowns, and the symbol of the operator is given by

2.7

This proposition shows the relevance of the good unknown in the fact that the main action of the DN map can be expressed in terms of a paradifferential operator acting on ω, plus a simple transport-like term div(T_Vh).

(iii). Diagonalization and energy estimates

Using (2.6) and standard paralinearization arguments, one can reduce (1.7) to the following system:

2.8

where

2.9

and G₂ and Ω₂ denote smoother terms which are quadratic in the unknowns. One can then arrive at the following result.

Proposition 2.2 (Diagonalization and a priori energy estimates). —

Let (h,ω) be solutions of (2.8)–(2.9), and recall the notation (2.1) and (2.5). Define the diagonal variable⁵

2.10

where

2.11

is the diagonal symbol. Then, the following hold:

• (i) U satisfies the equation

  2.12

  where denotes nonlinear terms of lower order.
• (ii) For any k≥0 consider

  2.13

  then⁶

  2.14

  where P is a polynomial with positive coefficients.

The procedure leading to (2.12) is the nonlinear analogue of the basic diagonalization in (1.13). For the purpose of the local existence theory one can think of the nonlinear terms having the form . Then, the paralinearized equation (2.12) has a fairly simple structure which allows one to derive the energy estimates (2.14) in a relatively straightforward fashion, for example, by applying multiple times the operator T_Σ. We see that (2.14) gives a priori control on a short time interval on the function U in the space H^3k/2, and hence control on (g−σΔ)h and |∇|^1/2ϕ in the same space. Once these a priori estimates are established, local well-posedness can then be obtained by standard procedures. In conclusion, one obtains the following result.

Theorem 2.3 (Local well-posedness of (1.7)). —

Consider the system of equations (1.7). Let an initial data h₀=h(t=0), ϕ₀=ϕ(t=0), be given so that with , |∇|^1/2ϕ∈ H^s, for s>d/2+1 large enough. Then, there exist T>0 and a unique solution

of the system (1.7) with the assigned initial data.

We notice that, at the linear level,

2.15

This is used to construct suitable models for global analysis; see (3.4)–(3.5) and (3.33).

(b). Conclusion

We have summarized the main ingredients in the local existence theory using Eulerian coordinates. This is a natural description, which is also tied to the Hamiltonian nature of the equations, and is a good starting point for the global theory. The other formulations described in §1d can also be used to develop the local theory, as in the references mentioned at the beginning of the section. This includes, of course, well-posedness results in the presence of vorticity analogous to theorem 2.3. We also refer to recent work of Lannes on the interaction with floating structures [67] and de Poyferré [68] on emerging bottom.

3. Global regularity and asymptotic behaviour

The problem of global existence of solutions for water waves models is more challenging, and much fewer results have been obtained so far. As in many other quasi-linear problems, global regularity has been studied in a perturbative and dispersive setting. Large initial data can lead to breakdown in finite time; see, for example, the papers on ‘splash’ singularities [69,70].

In three dimensions (two-dimensional interfaces), the first global regularity results were proved by Germain et al. [71] and Wu [30] for the gravity problem (g>0, σ=0). Global regularity in three dimensions was also proved for the capillary problem (g=0, σ>0) by Germain et al. [72] and for the full gravity–capillary problem (g>0, σ>0) by Deng et al. [62]. In the case of a finite flat bottom, global regularity was proved recently by Wang [73–75] in both the gravity and the capillary problems in three dimensions.

In two dimensions (one-dimensional interfaces), the first long-time result for the water waves system (1.7) is due to Wu [29], who showed almost-global existence for the gravity problem (g>0, σ=0). This was improved to global regularity by the authors in [76] and, independently, by Alazard & Delort [63,64]. A different proof of Wu's two-dimensional almost-global existence result was later given by Hunter et al. [36], and then complemented to a proof of global regularity in [37]. See also Wang [77] for a global regularity result for a class of small data of infinite energy. For the capillary problem in two dimensions, global regularity was proved by the authors in [78] and, independently, by Ifrim & Tataru [38] in the case of data satisfying an additional momentum condition.

We remark that all the global regularity results that have been proved so far require three basic assumptions: small data (small perturbations of the rest solution), trivial vorticity inside the fluid and flat Euclidean geometry. More subtle properties are also important, such as the Hamiltonian structure of the equations, the rate of decay of the linearized waves and the resonance structure of the bilinear wave interactions.

(a). Main ideas

The classical mechanism to establish global regularity for quasi-linear equations has two main components:

• (1) Propagate control of high-order energy functionals (Sobolev norms and weighted norms).

• (2) Prove dispersion and decay of the solution over time.

The interplay of these two aspects has been present since the seminal work of Klainerman [79,80] on nonlinear wave equations and vector fields, Shatah [81] on three-dimensional Klein–Gordon and normal forms, Christodoulou–Klainerman [82] on the stability of Minkowski space–time and Delort [83] on one-dimensional Klein–Gordon equations.

In the last few years, new methods have emerged in the study of global solutions of quasi-linear evolutions, inspired by the advances in semilinear theory. The basic idea is to combine the classical energy and vector-fields methods with refined analysis of the Duhamel formula, using the Fourier transform and carefully constructed ‘designer norms’. This is the essence of the ‘method of space–time resonances’ of Germain, Masmoudi and Shatah [71,72,84] and Gustafson et al. [85], and of the work on plasma models and water waves in [62,76,78,86–91].

In the rest of this section, we illustrate the development of these ideas in the setting of water waves by analysing three systems, in increasing order of difficulty: gravity water waves in three dimensions, gravity water waves in two dimensions and gravity–capillary water waves in three dimensions.

For the sake of exposition, in all three cases we take the following approach: we replace the full water waves systems with suitable simplified quasi-linear models, and then outline the main ideas needed to analyse these models. The quasi-linear models constructed here have two main properties: (i) they capture the essential difficulties of the global theory of the full systems and (ii) they are technically simpler than the full systems, mainly because they bypass all the difficulties of the local theory, such as the use of paradifferential calculus.

One should keep in mind that there are certain difficulties in transferring the global analysis from the model equations to the real water waves systems, mostly at the level of the energy estimates. Nevertheless, our simplified models are very useful to explain some of the key ideas of the global analysis, in problems that are more algebraically transparent.

(b). Gravity water waves in three dimensions

We consider first the system (1.7) in two dimensions in the case (g,σ)=(1,0). Global regularity in this case was proved in [30,71]. Here, we follow essentially the exposition and the proof of Germain et al. [71]; Wu's theorem in [30] is essentially equivalent, but involves slightly different hypothesis on the data and a very different proof (in Lagrangian coordinates, using also the Clifford algebra).

Theorem 3.1 —

Assume that are small and smooth initial data, satisfying

3.1

where N is sufficiently large, is sufficiently small, , and

3.2

Then there is a unique global solution of the initial-value problem (1.7) with (g,σ)=(1,0). Moreover, the solution satisfies the global bounds

3.3

for any and l∈{1,2}, where δ=10⁻⁸ is a small constant and u(t):=e^itΛU(t) is the associated linear profile of the solution U.

We describe now some of the main ingredients of the proof. We highlight two main ideas: (i) the proof of high-order energy estimates by symmetrization and (ii) the proof of dispersive estimates using the method of space–time resonances.

To simplify the exposition, we replace system (1.7) with the quasi-linear evolution equation

3.4

The quadratic nonlinearity is defined by

3.5

Here, and in the rest of the section, we use smooth cut-off functions defined as follows: we fix an even smooth function supported in [−2,2] and equal to 1 in [−1,1], and let

for any and interval . We define also the Littlewood–Paley projections P_k and P_I as the operators induced by the Fourier multipliers φ_k and φ_I, respectively. Equations (3.4)–(3.5) are a good substitute for the full system (1.7); see the discussion in §2.

The proof relies on a bootstrap argument: we assume that is a solution of (3.4)–(3.5) satisfying the bootstrap hypothesis

3.6

for any t∈[0,T], where ε₁:=ε^2/3, and we would like to prove the improved bounds

3.7

This suffices, by a simple continuity argument, because the stronger bounds (3.7) hold at time t=0 due to the initial-data assumptions (3.1).

We remark that the bootstrap norms used in (3.6) capture the main features of the nonlinear solution, namely smoothness, localization in space and sharp pointwise decay matching the decay of linear gravity waves.

(i). Energy estimates

These are very simple in our model (3.4)–(3.5): we define

3.8

Then we calculate, using the equation and symmetrization (or integration by parts),

3.9

where

3.10

The symmetrization in the symbol m avoids the potential loss of derivative, and the identity above can be easily used to show that

3.11

This leads to the desired improved energy bound in (3.7). Note how this step relies in a crucial way on the sharp pointwise decay of 〈t〉⁻¹ for U(t).

We remark that, in the real water waves systems analysed in [30,71], the final result is similar (an energy inequality similar to (3.11)), but the proof is substantially more complicated because of the quasi-linear structure of the problem. In particular, the proof has to address all the difficulties of the local regularity theory of the water waves models.

(ii). Dispersion and decay

It remains to control the other terms in (3.7). The idea is to write the equation in terms of the linear profile u(t):=e^itΛU(t),

3.12

where u₊:=u, , the sum is taken over choices of the signs +,− and m_±± are suitable smooth multipliers. In integral form this becomes

3.13

One would like to estimate u by integrating by parts either in s or in η. According to [71], the main contribution is expected to come from the set of quadratic space–time resonances (the stationary points of the integral)

3.14

where m=m_±± and the phases Φ are defined by

3.15

As , the first main observation is that the phases Φ only vanish when either ξ=0, or η=0, or ξ−η=0. In this case, however, the multipliers m also vanish. In other words, there are no quadratic time resonances and one can use normal forms (integration by parts in time) to transform the quadratic terms into cubic terms.

Loss of derivative is not important at this stage of the argument, so one can integrate by parts in time and use (3.12). It remains to estimate the contribution of cubic terms of the form

3.16

where .

The resulting multiplier (m(ξ,η)/Φ(ξ,η))m′(η,σ) is regular, so one can now analyse cubic integrals of this type using again the method of space–time resonances. An important algebraic observation, which is used in the analysis of the phases to control the weighted norms, is the identity

3.17

This is a slow propagation of iterated resonances property; more subtle versions of this property are also important in the three-dimensional gravity–capillary model described below (e.g. (3.53)).

The dispersive analysis in [71] is simplified by the fact that there are no quadratic space–time resonances in the problem. However, the basic idea of the method of space–time resonances, namely to identify these points and centre the analysis around them, plays a crucial role in many other global regularity results on plasma models and water waves models. Further developments of these ideas, and much more sophisticated arguments, are used in the proof global regularity for the three-dimensional gravity–capillary model, where one has to deal with a full set of quadratic space–time resonances (see §3d).

(c). Gravity water waves in two dimensions

We consider now the system (1.7) in one dimension in the case (g,σ)=(1,0). Global regularity in this case was proved in [29,36,37,63,64,76,77]. The precise assumptions on the initial data (low frequencies, high frequencies and the number of vector fields involved) are not identical in these papers. We will follow mostly the set-up in [76].

Theorem 3.2 —

Assume that are small and smooth initial data, satisfying

3.18

where N is sufficiently large, is a sufficiently small constant, U₀=h₀+i|∇|^1/2ϕ₀ and

3.19

• (i) Then there is a unique global solution of the initial-value problem (1.7) with (g,σ)=(1,0). The solution U satisfies the global bounds

  3.20

  for any , where is the scaling vector field and δ=10⁻⁸ is small.
• (ii) The solution U(t) undergoes modified (nonlinear) scattering as , i.e.

  3.21

  where and

  3.22

As before, we discuss two main ideas of the proof: (i) the quartic energy inequality which is needed to prove energy estimates and (ii) the construction of nonlinear profiles, to prove modified scattering and dispersion. As before, we use the simplified model

3.23

which is the one-dimensional analogue of the model (3.4)–(3.5). We use again a bootstrap argument, with the bootstrap hypothesis

3.24

for a solution on some time interval [0,T].

(i). Energy estimates

One can start proving energy estimates as in the two-dimensional model; see (3.8)–(3.11). These identities still hold, but the bound (3.11) does not suffice to close the energy estimate, because the optimal -type decay is 〈t〉^−1/2 in one dimension, which is far from integrable.

The idea, which was introduced by Wu [29], is to refine the energy method by proving instead a quartic energy inequality of the form

3.25

for a suitable functional satisfying . The point is to get two factors of in the right-hand side, in order to have almost-integrable decay.

In our model (3.23), a quartic energy inequality can be proved easily using the identities (3.9)–(3.10).⁷ The idea is to write the bulk integrals in the right-hand side of (3.10) in terms of the linear profiles u=e^itΛU and w=e^itΛW and integrate by parts in time. More precisely, the bulk term can be written as a linear combination of integrals of the form

3.26

where m is the multiplier defined in (3.10). The key observation is that the phases Λ(η)−Λ(ξ)±Λ(ξ−η) do not vanish, except when one of the frequencies vanishes. In this case, however, also the multipliers m vanish as well.

The profiles w satisfy transport equations similar to (3.12). Integration by parts in time and changes of variables show that the integrals in (3.26) can be written as (i) sums of cubic boundary terms of the form

3.27

where s∈{0,t} and (ii) sums of quartic space–time integrals of the form

3.28

All the quartic space–time integrals contain two copies of w and two copies of u, and, most importantly, the multipliers are regular and do not lose high-order derivatives (after suitable symmetrization). The desired inequality (3.25) follows: the boundary cubic expressions in (3.27) can be combined with the quadratic energies to produce the energy functionals , while the quartic space–time integrals can be estimated as claimed.

The vector-field norm can also be controlled in a similar way, by proving a similar quartic energy inequality of the form

3.29

for a suitable functional satisfying .

Quartic energy inequalities such as (3.25) were proved and played a key role in all the (almost) global regularity results for water waves in two dimensions. As explained above, the main ingredient for such an inequality to hold is the absence of bilinear time resonances. However, the implementation is somewhat delicate in certain quasi-linear problems, like water waves models, due to the potential loss of derivatives. See the longer discussion in the introduction of [91] or [78] for more details and references.

The calculation we present above, using integration by parts in time in Fourier variables, has similarities with the I-method of Colliander et al. [92,93], which is used extensively in semilinear problems. One should also compare this with the more involved calculation used in energy estimates in the three-dimensional gravity–capillary model described below.

(ii). Modified scattering and decay

One can start again, as in the two-dimensional case, from identities on the profile similar to (3.12)–(3.13). The phases do not vanish (except when one of the frequencies vanishes), so one can use again a normal form. As in (3.16), we have an identity of the type

3.30

where , are regular multipliers, u′ is a suitable quadratic modification of u, and is a quartic and higher-order remainder.

The situation is different in dimension 1, compared to the dimension d=2 analysed earlier, because of the slow rate of decay of solutions. In fact, it turns out that some of the terms in the right-hand side, which correspond to the cubic space–time resonances, cannot be integrated in time. These cubic space–time resonances appear only in the phases Φ₊₊₋, Φ₊₋₊, Φ₋₊₊, and correspond to the frequencies (ξ,ξ,−ξ), (ξ,ξ,−ξ) and (ξ,−ξ,ξ), respectively. To remove the non-integrable contribution, one can define the nonlinear profiles u*(t) by

where C is a suitable real constant. Using (3.30), one can now show that the renormalized profile u*(t) stays bounded and converges (quantitatively) in the Z norm as ,

if m≥0 and t₁≤t₂∈[2^m−1,2^m+1]. This leads to global control of the solution and modified scattering, as claimed.

The idea of using nonlinear profiles and modified scattering to prove global regularity was introduced in the context of water waves by Ionescu & Pusateri [76] and Alazard & Delort in [63,64]. Just like the quartic energy inequality described earlier, this idea played a key role in all the global regularity results for water waves in two dimensions.

(d). Gravity–capillary water waves in three dimensions

Finally, we consider the system (1.7) in two dimensions with (g,σ)=(1,1), which was analysed in [62]. Let Ω:=x₁∂₂−x₂∂₁ denote the rotation vector field on and let denote the space of functions defined by the norm

The main result in [62] is the following global regularity theorem.

Theorem 3.3 —

Assume that δ is sufficiently small, N₀,N₁,N₃ are sufficiently large and that the datum (h₀,ϕ₀) satisfies

3.31

where is a sufficiently small constant and the Z norm is explained below in §3d(ii); see (3.50). Then, there is a unique global solution of the system (1.7), with (h(0),ϕ(0))=(h₀,ϕ₀). In addition

3.32

for any , where .

As before, we explain some of the main ideas, including the subtle construction of the Z norm, using a simplified model. The problem is substantially more difficult in this case, and we consider the more specialized model

3.33

Note that V is real-valued, such that solutions of (3.33) satisfy the L² conservation law

3.34

This conservation is a good substitute for the Hamiltonian structure of the original water-wave systems. As before, we use a bootstrap argument, with the bootstrap hypothesis

3.35

(i). Energy estimates

Let W:=〈∇〉^NU, , and calculate

3.36

where

3.37

This is similar to (3.8)–(3.10). We note that m(ξ,η) satisfies

3.38

The depletion factor is important in establishing energy estimates, due to its correlation with the modulation function Φ (see (3.41) and (3.46) below). The presence of this factor is related to the exact conservation law (3.34).

There is a key difference between the full gravity–capillary model and the three-dimensional gravity model discussed earlier: the dispersion relation in (3.33) has stationary points when (figure 2). As a result, linear solutions can only have |t|^−5/6 pointwise decay, i.e.

even for Schwartz functions ϕ whose Fourier transforms do not vanish on the sphere {|ξ|=γ₀}. As a result, the identities (3.36) cannot be used directly to prove energy estimates, as in the three-dimensional gravity case. Moreover, quartic energy inequalities like (3.25) also fail because there are large, codimension 1, sets of quadratic resonances, with no matching null structures (see figure 3). New ideas, which we describe below, are needed to prove the energy bounds for this problem.

Figure 2.

The curves represent the dispersion relation and the group velocity λ′, for g=1=σ. The frequency γ₁ corresponds to the space–time resonant sphere. Note that while the slower decay at γ₀ is due to some degeneracy in the linear problem, γ₁ is unremarkable from the point of view of the linear dispersion. (Online version in colour.)

Figure 3.

The first picture illustrates the resonant set {η:0=Φ(ξ,η)=Λ(ξ)−Λ(η)− Λ(ξ−η)} for a fixed large frequency ξ (in the picture ξ=(100,0)). The second picture illustrates the intersection of a neighbourhood of this resonant set with the set where |ξ−η| is close to γ₀. Note in particular that, near the resonant set, ξ−η is almost perpendicular to ξ (see (3.38), (3.46)).

Step 1. We would like to estimate now the increment of . We use (3.36) and consider only the main case, when |ξ|,|η|≈2^k≫1, and |ξ−η| is close to the slowly decaying frequency γ₀. So we need to bound space–time integrals of the form

where χ_γ0 is a smooth cut-off function supported in the set {ξ:||ξ|−γ₀|≪1}, and we replaced ℜu by U (replacing ℜU by leads to a similar calculation). As before, define the linear profiles

3.39

Then, decompose the integral in dyadic pieces over the size of the modulation (3.41) and over the size of the time variable. In terms of the profiles u,w, we need to consider the space–time integrals

3.40

where

3.41

is the associated modulation (or phase), q_m is smooth and supported in the set s≈2^m, and φ_p is supported in the set {x:|x|≈2^p}.

Step 2. To estimate the integrals I_k,m,p, we consider several cases depending on the relative size of k,m,p. Assume that k,m are large, i.e. 2^k≫1, 2^m≫1, which is the harder case. To deal with the case of small modulation, when one cannot integrate by parts in time, we need an L² bound on the Fourier integral operator

where s≈2^m is fixed. The critical bound proved in [62] (the main L² lemma) is

3.42

provided that p−k/2∈[−0.99m,−0.01m]. The main gain here is the factor in 2^{(3/2)(p−k/2)} in the right-hand side (Schur's test would only give a factor of 1).

The proof of (3.42) uses a TT* argument, which is a standard tool to prove L² bounds for Fourier integral operators. This argument depends on a key non-degeneracy property of the function Φ, more precisely on what we call the restricted non-degeneracy condition

3.43

This condition, which appears to be new, can be verified explicitly in our case, when | |ξ−η|−γ₀|≪1. The function Υ does in fact vanish at two points on the resonant set {η:Φ(ξ,η)=0} (where | |ξ−η|−γ₀|≈2^−k), but our argument can tolerate vanishing up to order 1.

The non-degeneracy condition (3.43) can be interpreted geometrically: the non-degeneracy of the mixed Hessian of Φ is a standard condition that leads to optimal L² bounds on Fourier integral operators. In our case, however, we have the additional cut-off function φ_≤p(Φ(ξ,η)), so we can only integrate by parts in the directions tangent to the level sets of Φ. This explains the additional restriction to these directions in the definition of Υ in (3.43).

Given the bound (3.42), one can control the contribution of small modulations, i.e.

3.44

Step 3. In the high modulation case, we integrate by parts in time in the formula (3.40). The main contribution is when the time derivative hits the high-frequency terms, and the resulting integral is

3.45

Note that ∂_tw is a quadratic expression, as in (3.12), so that we gain a unit of decay (which is |t|^−5/6+), but lose a derivative.

In the harder case when the modulation is small we can use the depletion factor in the multiplier m (see (3.38)) and the following key algebraic correlation:

3.46

see figure 3. As a result, we gain one derivative in the integral I′_k,m,p, which compensates for the derivative loss. On the other hand, when the modulation is not small, 2^p≥1, then the denominator Φ(ξ,η) becomes a favourable factor, and one can reiterate the symmetrization procedure implicit in the energy estimates. This avoids the loss of one derivative and gives sufficient decay to estimate |I′_k,m,p| and close the energy estimate.

(ii). Dispersive analysis

The first main issue is to define an effective Z norm that can be used in the bootstrap argument. As in (3.13), we use the Duhamel formula, written in terms of the profile u=u₊=e^itΛU, ,

3.47

where the sum is taken over choices of the signs +,− and m_±± are suitable smooth multipliers.

The idea is to estimate the function using the Duhamel formula (3.13), by integrating by parts either in s or in η. As in (3.14)–(3.15), the main contribution is expected to come from the set of quadratic space–time resonances

3.48

where m=m_±± and Φ(ξ,η)=Λ(ξ)∓Λ(ξ−η)∓Λ(η). In the gravity–capillary problem, space–time resonances are present only for the phase Φ(ξ,η)=Λ(ξ)−Λ(ξ−η)−Λ(η) and the space–time resonant set is

3.49

Moreover, the space–time resonant points are non-degenerate (according to the terminology of [87]), in the sense that the Hessian of the matrix is non-singular at these points.

To gain intuition, consider the first iteration of the formula (3.13), i.e. assume that the functions u_± in the right-hand side are Schwartz functions supported at frequency ≈1, independent of s. Assume that s≈2^m. Integration by parts in η and s shows that the main contribution comes from a small neighbourhood of the stationary points where |∇_ηΦ(ξ,η)|≤2^−m/2+δm and |Φ(ξ,η)|≤2^−m+δm, up to negligible errors. Thus, the main contribution comes from space–time resonant points as in (3.14). A simple calculation shows that the main contribution to the second iteration is of the type

up to smaller contributions, where we have also ignored factors of 2^δm, and c is smooth.

We are now ready to describe more precisely the crucial choice of the Z space. The idea is to decompose the profile as a superposition of atoms, localized in both space and frequency,

The Z norm is then defined by measuring suitably every atom.

In our case, the Z space should include all Schwartz functions. It also has to include functions like , due to the considerations above, for any m large. It should measure localization in both space and frequency, and be strong enough, at least, to recover the t^−5/6+ pointwise decay. We define

3.50

up to small (but important) δ-corrections. Then we define the Z norm by applying a suitable number of vector fields D and Ω.

We emphasize that the dispersive analysis in the Z norm in the gravity–capillary problem is a lot more subtle than in the earlier papers on water waves. To illustrate how this analysis works in our problem, we consider the contribution of the integral over s≈2^m≫1 in (3.13), and assume that the frequencies are ≈1.

Step 1. Start with the contribution of small modulations,

3.51

where l=−m+δm (δ is a small constant) and q_m(s) restricts the time integral to s≈2^m, and, for simplicity, we consider only the phase Φ(ξ,η)=Λ(ξ)−Λ(ξ−η)−Λ(η). In this case, the considerations above, leading to the definition of the Z norm, are still relevant: one can integrate by parts in η, identify the main contributions as coming from small 2^−m/2+δm neighbourhoods of the stationary points and estimate these contributions in the Z norm.

Step 2. Consider now the contributions of the modulations of size 2^l, l≥−m+δm. We start from a formula similar to (3.51) and integrate by parts in s. The main case is when d/ds hits one of the profiles u. Using again the equation (see (3.13)), we have to estimate cubic expressions of the form

3.52

where Φ′(η,σ)=Λ(η)+Λ(η−σ)−Λ(σ). We combine Φ and Φ′ into the cubic phase

The most difficult case in the dispersive analysis is when l is small, say l≤−m/14, and the denominator Φ(ξ,η) in (3.52) is dangerous. We first restrict to suitably small neighbourhoods of the stationary points of in η and σ, thus to the cubic space–time resonances. Eventually, we need to rely on one more algebraic property of the form

3.53

The point of (3.53) is that in the resonant region for the cubic integral we have , so the resulting function is essentially supported when |x|≪2^m, using an approximate finite speed of propagation argument. This gain is reflected in the factor 2^j in (3.50).

In proving control of the Z norm, there are, of course, many cases to consider. The type of arguments presented above are typical in the proof: we decompose our profiles in space and frequency, localize to small sets in the frequency space, keeping track in particular of the special frequencies of size γ₀,γ₁,γ₁/2,2γ₀, use integration by parts in ξ to control the location of the output and use multilinear Hölder-type estimates to bound L² norms. An important aspect of this analysis is that we can essentially assume that all profiles are almost radial and located at frequencies , owing to the strong complementary control on Sobolev and weighted norms in the bootstrap hypothesis (3.35).

Step 3. The identity (3.47) can also be used to justify the approximate formula

3.54

as , where η_j(ξ) denote the stationary points where ∇_ηΦ(ξ,η_j(ξ))=0. This approximate formula is consistent with the asymptotic behaviour of solutions, more precisely scattering in the Z norm. Qualitatively, at space–time resonances one has Φ(ξ,η_j(ξ))=0, which leads to logarithmic growth for , while away from these space–time resonances the oscillation of e^{itΦ(ξ,η_j(ξ))} leads to convergence.

(e). Conclusions and additional references

To summarize, there is a small number of cases when one can construct global solutions of water waves systems, by perturbing around the trivial solutions.⁸ The mechanism that leads to global solutions in all these cases is based on establishing dispersion and decay.

Sometimes it is possible to prove results going beyond the local theory, but not reach full global regularity. For example, starting with data of size ε in a standard Sobolev space, one can sometimes get ≈ε⁻² time of existence by proving a quartic energy inequality like (3.25) in cases when there are no significant quadratic resonances [39,94]. See also the recent work of Berti–Delort [95], where a combination of paradifferential analysis and ideas from KAM and normal forms theory was used to prove a significant long-time (≈ε^−N) existence result for periodic two-dimensional gravity–capillary waves (one-dimensional interface), for almost all choices of (g,σ). Note that these extended lifespan results do not rely on dispersion but mainly on the absence of resonances.

In this context, a natural question to ask is if there are global or long-time regularity results for solutions with nontrivial vorticity. We emphasize that all the global regularity results so far assume irrotationality.

4. Formation of singularities and other topics

In this section, we briefly present a few additional questions concerning the evolution of water waves, and provide more references to other topics.

(a). Singularity formation

A set of fundamental questions in pure and applied fluid dynamics concerns the study of singularities. While some major open problems, such as the loss of regularity and blow-up in the (rotational) Euler flow, remain widely open, some types of ‘geometric singularities’ have been studied in the context of water waves.

In [96], the authors proved that a wave that is initially given as the graph of a function h can overturn at a later time. More importantly, Castro et al. [69] showed the existence of ‘splash’ singularities. A ‘splash’ (respectively, a ‘splat’) occurs when the surface of the fluid self-intersects at a point (respectively, on an arc) while retaining its smoothness (figure 4). This changes the topology of the domain and leads to a breakdown of the chord-arc condition, that is, the assumption (to some extent necessary for well-posedness)

where a parametrization of the interface S_t. Notice that a ‘splash’ singularity occurs while the parametrization of the interface and the velocity of the fluid retain their initial regularity. The two-dimensional result of [69] was extended to three dimensions and to some other related models by Coutand–Shkoller [70]. See also the related work [97] on the formation of singularities.

Figure 4.

Formation of a ‘splash’ singularity in two dimensions (taken from [69]). (Online version in colour.)

In [98], Fefferman et al. showed that a ‘splash’ singularity cannot happen in the case of an interface separating two fluids: the presence of a second fluid, with positive constant density, prevents the interface from self-intersecting. In other words, one fluid cannot squeeze the other one if the interface and the solution are to remain smooth. A similar result was also proved by Coutand & Shkoller in [99]. We refer the reader to §4b below for some references about the well-posedness and instability for interfaces between two fluids.

As discussed above, a self-intersection of an interface through a fluid cannot happen for sufficiently regular solutions of the water waves equations. However, it is plausible that a self-intersection could happen with the surface losing regularity, for example, pinching out and creating a cusp (figure 5).

Figure 5.

Possible scenario for a corner-like singularity: (i) a locally strong velocity field pushes two points to come close together; (ii) the fluid in the middle does not have enough time to escape; (iii) a self-intersection of the interface cannot happen with a smooth boundary, and the symmetries of the equation force the formation of a corner.

(b). Fluid interfaces

In §2, we focused our attention on the evolution problem for one fluid in vacuum. When considering the motion of waves on the surface of the ocean, one can think of the one-fluid model as a good first approximation for a water–air interface; rigorous results in this direction are provided in [100–102]. However, the motion of a free boundary between two immiscible fluids (or gases) is a more complex problem than (1.1).

The two-fluid model is a more unstable scenario than the one-fluid model and is subject to instabilities/ill-posedness in the absence of surface tension. Early works on this model and the study of its instability include [103–106]. We also mention some classical numerical works by Hou et al. [107,108]. More recent contributions can be found in [28,46,48,100–102,109–112] to cite a few. We refer the reader to the survey of Bardos & Lannes [21] for more on the instability of fluid interfaces.

We remark that there are no global regularity results for any two-fluid models.

(c). Other questions and further references

The enormous complexity of the free boundary Euler flow and the water waves equations has motivated a large amount of research, beyond the local and global well-posedness discussed above. This includes the construction of particular classes of solutions, and extensive numerical activities. We provide here just a few additional references, including books and reviews, and other research papers on various topics of interest, and refer the reader to the cited works for more references on these topics.

• — Steady, solitary, extreme waves. For the construction of steady water waves we refer to Groves’ survey [113] and the paper of Constantin & Strauss [114]; the excellent review of Strauss [115] contains an account on both the history and more modern achievements on this topic. See also the recent paper of Constantin et al. [116] for the latest developments and more references.

   The existence theory of solitary waves was developed in Friedrichs & Hyers [7], Beale [117], Amick & Toland [118] and Amick & Kirchgässner [119]; see also Rousset & Tzvetkov [120] for their transverse instability.

   The conjecture made by Stokes [3,121] that the crest of a steady wave of maximal amplitude forms a 120° angle has been extensively investigated. Classical references on asymptotics are Longuet-Higgins & Fox [122,123]; a proof of the conjecture was given by Amick et al. [124]; recent numerical works dealing with the behaviour of near Stokes’ waves are in [35]. A counterpart of Stokes’ conjecture for large standing waves was proposed by Penney–Price in the 1950s and its validity was investigated in [125]. Recent numerical computations of three-dimensional standing waves can be found in [126]; see also [127–130] for more properties of the profile of travelling gravity waves.

• — Standing waves and Hamiltonian structure. An interesting aspect of the water waves equations concerns its Hamiltonian nature, which motivates numerous questions with a strong dynamical system flavour. An informative survey paper is [131]. Works in this direction, related to small divisors and Nash–Moser techniques, are those by Plotnikov & Toland [132], Iooss et al. [133] and Iooss & Plotnikov [134] on the existence of standing waves that are periodic in space and time. Quasi-periodic standing waves have been constructed in [135] using KAM techniques for the first time in a quasi-linear setting. See also the already mentioned work [95] where a key role is played by the reversible, rather than Hamiltonian, structure. Aspects of the theory of normal forms and connections to integrable Hamiltonian systems are discussed in many works (e.g. [136–141]).

• — Approximate and asymptotics models. Because of the physical relevance of the water waves system and the aim of better describing its complex dynamics, many simplified models have been derived and studied in special regimes. Two important examples include the approximation of waves in the form of wave packets by the nonlinear Schrödinger (NLS) equation, and the approximation of long waves in shallow water by the Korteweg–de Vries (KdV) equation. We refer the reader to [43,142–145], the book [24] and the surveys [146,147] and references therein for more about reduced models and their mathematical justification.

• — Books and reviews. An interesting historical account on the early developments of the theory can be found in [2]. A classical introduction to the theory of water waves is Stoker [148]; an introduction to water waves and related models, such as KdV and NLS, can be found in [149], and a more thorough account in [22]. The recent book of Lannes [24] contains all major results on the modern well-posedness theory and on approximate models, and the book of Constantin [150] discusses many applications to oceanography.

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Authors' contributions

Both authors contributed equally to all parts of the paper.

Competing interests

We declare we have no competing interests.

Funding

A.D.I. was supported in part by NSF grant no. DMS-1600028 and by NSF-FRG grant no. DMS-1463753. F.P. was supported in part by NSF grant no. DMS-1265875.