Propagation of singularities for gravity-capillary water waves

We obtain two results of propagation for solutions to the gravity-capillary water wave system. First we show how oscillations and the spatial decay propagate at infinity; then we show a microlocal smoothing effect under the non-trapping condition of the initial free surface. These results extends the works of Craig, Kappeler and Strauss, Wunsch and Nakamura to quasilinear dispersive equations. We also prove the existence of gravity-capillary water waves in weighted Sobolev spaces. Such solutions have asymptotically Euclidean free surfaces. To obtain these results, we generalize the paradifferential calculus of Bony to weighted Sobolev spaces and develop a semiclassical paradifferential calculus. We also introduce a new family of wavefront sets -- the quasi-homogeneous wavefront sets, which is a generalization, at least in the Euclidean geometry, the wavefront sets of H\"{o}rmander, the scattering wavefront sets of Melrose, the quadratic scattering wavefront sets of Wunsch and the homogeneous wavefront sets of Nakamura.


Introduction
We are interested in the propagation of singularities for the gravity-capillary water wave equation, which is a quasilinear dispersive equation to be defined later. We shall first revisit some classical results about propagation of singularities for simpler linear dispersive equations, and see how they lead to a more generalized definition of singularities that is adaptive for gravity-capillary water waves.

Half Wave Equation.
By the classical definition, x 0 ∈ R d is called a singularity of u ∈ D ′ (R d ), if u is not C ∞ in any neighborhood of x 0 ; the singular support of u, denoted by sing supp u, is the set of all singularities of u. To study the propagation of sing supp u when u solves some PDEs, the information given by sing supp u alone is usually insufficient, as the direction of propagation for a singularity is not determined by its position, but rather by its "direction of oscillation".
A classical result says that for solutions to the half wave equation singularities travel at finite speeds along geodesics.
The purpose of this paper is to present the propagation of singularities for solutions to (1.7), including a microlocal smoothing effect. To the best of our knowledge, these results are the first of this type for quasilinear dispersive equations.
1.3.4. Existence in Weighted Sobolev Spaces. Instead of the linearization at (η, ψ) = (0, 0), if we paralinearize and symmetrize (1.7) as Alazard-Burq-Zuily [1], we obtain a fractional Schrödinger equation (with lower order terms) on Σ. The geometry of Σ is time dependent and is given by the solution itself, as (1.7) is quasilinear. We need this geometry to be asymptotically Euclidean to avoid the mess caused by the infinite speed of propagation, but the existence of such geometry is not cheap. We shall prove it by establishing the existence of asymptotically "flat" gravitycapillary water waves.
For the Cauchy problem of water waves, we refer to the initial works of Kano-Nishida [19] and Yosihara [35,36], the breakthroughs of Wu [32,33] and Beyer-Günther [7] for the local well-posedness in Sobolev spaces with general initial data. Using the paradifferential calculus, Alazard-Burq-Zuily [1,2,3] proved the local well-posedness with low Sobolev regularity. We shall prove Theorem 1.6 by combining the analysis of [1] and a spatial dyadic decomposition. More precisely, we define the dyadic paradifferential operators are supported in the dyadic annulus C −1 2 j ≤ |x| ≤ C2 j , (ii) ψ j ψ j = ψ j , (iii) {ψ j } j∈N is a partition of unity of R d , and (iv) T ψ j a is the usual paradifferential operator of Bony [8], which we shall review in §4.1. We show that the dyadic paradifferential calculus naturally extends Bony's paralinearization to weighted Sobolev spaces.
We do not attempt to lower µ to > 2+d/2, as it was the case in [1]. The range of m is so chosen that µ − m/2 > 3 + d/2, enabling us to paralinearize (1.7) in H µ m . We should mention that the well-posedness of gravity water waves, i.e., without surface tension, in uniformly local weighted Sobolev spaces was obtain by Nguyen [28] using a periodic spatial decomposition from [3].
1.3.5. Propagation at Spatial Infinity. Our first main result concerns about the propagation of (1/2, 1)-singularities at the spatial infinity, corresponding to (M.1) of Theorem 1.4.
We will see that, by Lemma 2.16, as (η, . By Alazard-Métivier [5], we expect σ to be at most µ − α − d/2 for some α > 0, corresponding to the extra gain of regularity by the remainder of the paralinearization procedure. Although Theorem 1.7 does not give the optimal upper bound for σ, as it is not our priority, but when m = 2µ − 6 − d, σ can still be as big as µ − 9/2 − d/2.
One may wonder whether the non-trapping assumption in Theorem 1.8 is necessary. We are tempted to believe that the co-geodesic flow on Σ 0 is everywhere non-trapping, both forwardly and backwardly, because Σ 0 is the graph of a function from R d to R. However, there are only two cases that are known to us be true: either when d = 1, or when ∇η 0 ∈ L ∞ and x ∇ 2 η 0 L ∞ is sufficiently small, see §6.4. In both cases we obtain the following local smoothing effect. Corollary 1.9. Under the hypothesis of Theorem 1.8, suppose that the following two conditions are satisfied, We refer to Christianson-Hur-Staffilani [11] and Alazard-Burq-Zuily [1] for local smoothing effects of 2D capillary-gravity water waves, and Alazard-Ifrim-Tataru [4] for a Morawetz inequality of 2D gravity water waves.
1.4. Outline of Paper. In §2, we present basic properties of weighted Sobolev spaces and the quasi-homogeneous wavefront set. In §3, we prove Theorem 1.4 by extending the idea of Nakamura [27]. In §4, we review the paradifferential calculus of Bony, and extend it to weighted Sobolev spaces by a spatial dyadic decomposition. We also develop a quasi-homogeneous semiclassical paradifferential calculus, and study its relations with the quasi-homogeneous wavefront set. In §5, we study the Dirichlet-Neumann operator in weighted Sobolev spaces and prove the existence of asymptotically flat gravity-capillary water waves, i.e., Theorem 1.6. In §6, we prove our main results, Theorem 1.7, Theorem 1.8 and Corollary 1.9, by extending the proof of Theorem 1.4 to the quasilinear equation using the paradifferential calculus.

Acknowledgment
The author would like to thank Thomas Alazard, Nicolas Burq and Claude Zuily for their help and encouragement. He would like to thank Shu Nakamura for his kindness and some useful discussions during the early stage of this project.
We say that a h ∈ S µ k is (µ, k)-elliptic if ∃R > 0, C > 0 such that for |x| + |ξ| ≥ R, inf We say that a h ∈ S −∞ −∞ is elliptic at (x 0 , ξ 0 ) if for some neighborhood Ω of (x 0 , ξ 0 ), inf is an isometry of L 2 (R d ), we induce the following results from the usual semiclassical calculus for which we refer to the book [40] of Zworski.
then Op δ,ρ h (a) L 2 →L 2 ≤ KM . Proposition 2.4 (Sharp Gårding Inequality). If δ + ρ > 0 and a h ∈ S −∞ −∞ such that Re a h ≥ 0, then for some C > 0, all u ∈ L 2 (R d ), and 0 < h < 1, There exists a bilinear operator Proposition 2.6. Then there exists a linear operator If a h ∈ S µ k , then ζ δ,ρ h a h ∈ S µ k . For γ > 0, define 2.2. Weighted Sobolev Spaces. Let us recall that S (R d ) is the Schwartz space, and S ′ (R d ) is the space of tempered distributions.
Definition 2.7. We say that a linear operator A : S → S ′ is of order (ν, ℓ) ∈ R 2 , and denote By Proposition 2.3 and Proposition 2.5, we obtain The converse follows by Sobolev injection. As for the second statement, clearly Proof. By the proof of Proposition 2.10, ∃M, N > 0, such that ∀ϕ ∈ S , Next, we characterize weighted Sobolev spaces by a dyadic decomposition.

Quasi-Homogeneous Wavefront Sets.
The following characterization is easy to prove by a routine construction of parametrix.
Proposition 2.14. Let u ∈ S ′ , then (x 0 , ξ 0 ) ∈ WF µ δ,ρ (u) if and only if for some By a partition of unity, we may assume that K ⊂ Ω := Ω i 0 for some Proof. (1) and (2) are easy. To prove (3), we use To prove the case where δ > 0, ρ > 0 of (5), we simply observe that, The other two cases are similar.

Model Equation
We prove Theorem 1.4 by combining the ideas of Nakamura [27] and simple methods of scaling. There is no harm in assuming that u 0 ∈ L 2 (R d ).
We deduce by Gronwall's inequality that

3.2.
Proof of (M.2). Let β = ργ − (δ + ρ) > 0, introduce the semiclassical time variable s = h −β t, and rewrite (1.9) as Proof. Each time we differentiate χ with respect to x, we get a multiplicative factor (1 + s) −1 , which is of size To estimate L s χ, we perform an explicit computation, Therefore, Similarly as above, we prove that L s χ ∈ W ∞,∞ (R ≥0 , S −∞ −1 ). We assume t 0 > 0 and µ = ∞ as the other cases are similar. Let ǫ > 0 be sufficiently small and let {λ j } j∈N ⊂ [1, 1 + ǫ[ be strictly increasing. Let φ be as in Lemma 3.2, and set ∀j ∈ N, . If such a h is found, and assume that ). By (i) and (3.3), if we replace φ with φ(λ·) for some λ > 1 sufficiently large, then for some compact set K, and sufficiently small h > 0, Therefore, by (ii), . We shall construct a h in the following form of asymptotic expansion All functions belonging to P j is smooth and non-negative for s ≥ 0. Moreover, if ψ ∈ P j , then The above asymptotic expansion is in the weak sense that, for some ǫ ′ > 0, and all N ∈ N, We begin by setting ϕ 0 ≡ 1 and choosing a 0 h satisfying By the definition of β and Proposition 2.5, Proposition 2.6, with ψ ℓ = ψ ℓ−1 +ϕ ℓ ∈ P ℓ . Summing up (3.5) and h ℓ(δ+ρ) × (3.6), we close the induction procedure.
We prove the asymptotic expansion by observing that, for all ǫ ′ > 0,

Paradifferential Calculus
We develop a paradifferential calculus on weighted Sobolev spaces, and a semiclassical paradifferential calculus.
4.1. Classical Paradifferential Calculus. We recall some classical results of the paradifferential calculus. We refer to the original work of Bony [8], and the books [17,25,6]. The proofs below are mainly based on [25], so we shall only sketch them. In the meanwhile, we shall also make some refinements regarding the estimates of the remainder terms, for the sake of the semiclassical paradifferential calculus that will be developed later.

4.2.
Dyadic Paradifferential Calculus. We develop the theory of paradifferential calculus on weighted Sobolev spaces.
Next observe that where the remainder can be estimated similarly as above, We conclude by Lemma 4.16.
More precisely, this means that, where by (4.5), Using j∈N ψ j ≡ 1, we induce that j∈N ∂ α x ψ j ≡ 0, ∀α ∈ N d \0. Therefore, Then we write where the symbols w αβ are independent of j. Set By (4.6), we prove similarly as in Proposition 4.19 that Setting ψ ′ j = |j ′ −j|≤100 ψ j ′ . We again conclude by Lemma 4.16, and the identity Proof. Decompose the product ab as follows, By Proposition 4.8 and Corollary 4.7, We conclude by Proposition 2.13.
Proof. By (4.7), ψ j a h = 0 and ψ j b h = 0 implies that j log 2 (1 + R h ). We claim that And then we can conclude by the identity Indeed, we use b h ∈ σ 0 and (4.1) to induce that F(T θ 1,0 h, * (ψ j ′ b h ) u) vanishes in a neighborhood of ξ = 0. By (4.3), for some π ∈ C ∞ (R d ) which vanishes near ξ = 0 and equals to 1 outside a neighborhood of ξ = 0, and for all m + m ′ ≤ N ∈ N, Then we use Proposition 4.8 and Remark 4.9, To estimate the remainders, we see that for each α ∈ N d with |α| = r, where we use 0 ≤ δ ≤ 1. Therefore, the first term in the remainder is Similar methods apply to the other two terms.
Combining the analysis of Proposition 4.26, Proposition 4.20, using Proposition 4.8, we obtain a similar result for the adjoint, whose proof we shall omit, as it is similar as above.
We conclude by Proposition 4.26.
Then for h > 0 sufficiently small, Proof. It suffices to use the identity (θ ǫ,0 h, * a h )♯ h (θ ǫ,0 h, * b h ) = θ ǫ,0 h, * (a h ♯ ǫ h b h ). The results above only concerned about the high frequency regime. The next lemma studies the interaction of high frequencies and low frequencies.
The admissibility of χ implies that Therefore, for any |j ′ − j| ≤ 20, We conclude by Lemma 4.16.
Proof. By a direct verification using (4.2), the homogeneity of a and the admissible function χ,

4.4.
Link with Quasi-Homogeneous Wavefront Sets. (P e f ) • ⊂ WF s+σ ǫ,1 (f ) • . If in addition e is elliptic, i.e., for some C > 0 and |ξ| sufficiently large, |e(x, ξ)| ≥ C|ξ| m , then So we may assume that proving the first statement. The second statement follows by a construction of parametrix.

Asymptotically Flat Water Waves
In this section we prove Theorem 1.6. The idea is to combine the analysis in [1] with the dyadic paradifferential calculus on weighted Sobolev spaces. We shall use the following notations for simplicity. Let w ∈ L ∞ (R d ) which is nowhere vanishing, then for A : S ′ → S ′ and f ∈ S ′ , we denote A (w) = wAw −1 , f (w) = wf . Particularly, (Af ) (w) = A (w) f (w) . For k ∈ R, we also denote, by an abuse of notation, , when there is no ambiguity. Observe that L 2 k = H 0 k is an Hilbert space with the inner product 5.1. Dirichlet-Neumann Operator. We study the Dirichlet-Neumann operator on weighted Sobolev spaces and its paralinearization. The time variable will be temporarily omitted for simplicity.
Proof. We only prove the case with n = 0. The general case follows with similar arguments and the identity Observe that, .
Proof. Let H 1,0 ̺ be the completion of the space {f ∈ C ∞ (Ω) : f vanishes in a neighborhood ofΣ} with respect to the norm where (X, Y ) L 2 ̺ :=´Ω ̺(X, Y ) dx dz. As b < ∞, by Poincaré inequality, Let 0 < ζ ∈ C ∞ (R) be such that ζ(z) = 1 for |z| ≤ 1, and ζ(z) = z for |z| ≥ 2. For some R > 0 sufficiently large to be determined later, set where for u, ϕ ∈ H 1,0 ̺ , Observe that ∇ We easily verify that L and B are continuous linear and bilinear forms on H 1,0 ̺ . Moreover B is coercive when R is sufficiently large, indeed, Therefore, by Lax-Milgram's Theorem and Lemma 5.1, Proof. By Lemma 5.1 and Lemma 5.

By Kato-Ponce commutator estimates, we verify that
Then observe that x k P x −k − P = α · ∇ xz with α ∈ L ∞ , we obtain the estimate Suppose that we have already proven that with the norm denoted by N σ . Then, Combing the estimates above, by Lemma 5.1, we have ∀ε > 0 χΛṽ 2 By choosing ε sufficiently small, this a priori estimate implies that

5.2.
Paralinearization. Now we paralinearize the system of water waves. The following results can be proven directly by combining the analysis in [1] and our dyadic paradifferential calculus, mainly Proposition 4.19, Proposition 4.21, and Proposition 4.22. We shall work in the Sobolev spaces H µ m , recalling Definition 1.5.
We shall denote ω = ψ − P B η, which is called the good unknown of Alinhac.
The proofs of following results are in the same spirit and simpler, and we shall omit them.
Define the symmetrizer where the equivalence relation is applied separately to each component of the matrices.
Finally we define L ε = LQ −1 PpJ ε P p 0 0 P 1/q J ε P q Q, and the approximate system A key identity of the operator L ε is that . To do this, let Λ µ k = P m µ−k/2 k , and set Φ = Λ µ k S η ψ . Then By Proposition 5.8, Proposition 5.5, Proposition 5.6, and Lemma 5.7, . Unfortunately, the operator to L 2 because of the sub-principal symbols do not vanish in the symbolic calculus, due to the existence of the commutators with Λ µ k . However, by Proposition 4.19, . Finally by an exact same energy estimate as in [1], we deduce that 5.6. Existence. Lemma 5.11. For all (η 0 , ψ 0 ) ∈ H µ+1/2 m × H µ m and all ε > 0, the Cauchy problem of the approximate system (5.4) has a unique maximal solution Proof. Following [1], the existence follows from the existence theory of ODEs by writing (5.4) in the compact form where F ε is a Lipschitz map on H µ+1/2 m × H µ m . The only nontrivial term is the Dirichlet-Neumann operator, whose regularity follows by combining Proposition 5.4 and the shape derivative formula (which goes back to Zakharov [38]), dG(η)ψ, ϕ := lim A standard abstract argument then shows that T ε has a strictly positive lower bound, we refer to [1] for more details.
Because k ≥ 1, by the same spirit of estimating R in Proposition 5.10, we obtain the following energy estimate d dt .
Combining the weak continuity, we induce by functional analysis that Φ ∈ C([0, T ], L 2 ). By (5.5), the paradifferential calculus, and the definition of Φ, we easily induce that . Thus we finish the proof of Theorem 1.6.

Finer Paralinearization and Symmetrization.
To study the propagation of singularities, we need much finer results of paralinearization and symmetrization than Proposition 5.5 and Proposition 5.8 so as to gain regularities in the remainder terms.
Proof. This theorem follows by replacing the usual paradifferential calculus with the dyadic paradifferential calculus in the analysis of [5]. In [5], the explicit expression for λ is given. We write it down for the sake of later applications.
Suppose that a (j) ± are defined for m ≤ j ≤ 1, then we define The principal and sub-principal symbols of λ clearly coincide with the ones given by Proposition 5.5.
Proof. Combining Proposition 6.1 and Proposition 5.8, moving the term gη to the left hand side, Given two time-dependent operators A, B : By the ellipticity of γ (3/2) , p (1/2) and q (0) , we can find paradifferential operatorsΛ µ andS by a routine construction of parametrix such that Observe that S is a lower triangular matrix, we can also chooseS to be lower triangular. Therefore, we can find ζ ∈ Σ −1/2,µ−5/2 by the symbolic calculus, such that Moreover, the principal symbols of S is We verify that the principal symbol of ζ is Then by (5.3) and the fact that the Poisson bracket between the symbol of Λ µ and γ vanishes, we find by the symbolic calculus two symbols A, B ∈ M 2×2 (Σ 0,µ−5/2 ) such that Therefore, let Φ = Λ µ S η ψ , and write we obtain by the analysis above that Finally, observe that Conversely, as WF µ+σ ǫ,1 (P B η) 6.2. Proof of Theorem 1.7. By Lemma 6.5, it is equivalent to prove the following theorem.
We use s to denote the time variable in accordance to the semiclassical time variable in the following section. Observe that where G is the geodesic flow defined in §1.3.6, and Proof. We have G ϕ 0 (x,ξ) (x, ξ) = G 0 (x, ξ) = (x, ξ) = Φ 0 (x, ξ). Then observe that ξ)). We conclude by the uniqueness of solutions to Hamiltonian ODEs.

So for any bounded set
Therefore, for any 0 < s − < s + with s − → ∞, implying that (x s , ξ s ) is a Cauchy sequence as s → ∞.

Construction of Symbol.
For h ≥ 0, and h 1/2 s ≤ T , set For h > 0, the semiclassical time variable s = h −1/2 t was inspired by Lebeau [24], see also Zhu [39] for an application in theory of control for water waves.

6.3.3.
Propagation. Now we prove Theorem 1.8. By Lemma 6.5 and Lemma 6.7, it suffices to prove the following theorem.