On the Cauchy problem of spherical capillary water waves

1.1 Equation of Motion for a Water Drop

We are interested in the initial value problem for the motion of a water droplet under zero gravity, which is the starting point of a program proposed by the author in [28]. Let us first describe the physical scenario. We pose the following assumptions on the fluid motion we aim to describe:

1. The perfect, incompressible, irrigational fluid of constant density ρ 0 occupies a smooth, compact region in ℝ 3 .

2. There is no gravity or any other external force in presence.

3. The air-fluid interface is governed by the Young–Laplace law, and the effect of air flow is neglected.

Hydrodynamics of water droplet governed by (A1)–(A3) is a long-standing interest for astrophysicists and astronautical engineers. To mention a few, astrophysicists Tsamopoulos and Brown [34], Natarajan and Brown [22] and Lundgren and Mansour [19] all carried out initial “weakly nonlinear analysis” towards the fluid motion satisfying assumptions (A1)–(A3). There are also numbers of visual materials on such experiments conducted in spacecrafts by astronauts.^[1] In this paper, we aim to set the stage for mathematical study of this problem.

We assume that the boundary of the fluid region has the topological type of a smooth compact orientable surface M, and is described by a time-dependent embedding ι ⁢ ( t , ⋅ ) : M → ℝ 3 . We will denote a point on M by x, the image of M under ι ⁢ ( t , ⋅ ) by M t , and the region enclosed by M t by Ω t . The unit outer normal will be denoted by N ⁢ ( ι ) . We also write ∇ ¯ for the flat connection on ℝ 3 .

Adopting assumption (A3), we have the Young–Laplace equation

where H ι is the (scalar) mean curvature of the embedding, σ 0 is the surface tension coefficient (which is assumed to be a constant), and p i , p e are respectively the inner and exterior air pressure at the boundary; they are scalar functions on the boundary and we assume that p e is a constant. Under assumptions (A1) and (A2), we obtain Bernoulli’s equation, sometimes referred as the pressure balance condition, on the evolving surface:

where Φ is the velocity potential of the velocity field of the air. Note that Φ is determined up to a function in t, so we shall leave the external (constant) pressure p e around for convenience reason. According to assumption (A1), the function Φ is a harmonic function within the region Ω t , so it is uniquely determined by its boundary value, and the velocity field within Ω t is ∇ ¯ ⁢ Φ . The kinetic equation on the free boundary M t is naturally obtained as

We would like to discuss the conservation laws for (1.1)–(1.2). The conservation of volume Vol ⁢ ( Ω t ) = Vol ⁢ ( Ω 0 ) is a consequence of incompressibility. Since the flow is Eulerian without any external force, the center of mass must move at a uniform speed along a fixed direction, i.e.

with Vol being the Lebesgue measure, P marking points in ℝ 3 , V 0 and C 0 being the velocity and starting position of center of mass respectively. Furthermore, the total momentum is conserved, and since the flow is a potential incompressible one, the conservation of total momentum is expressed as

Most importantly, as Zakharov pointed out in [38], (1.1)–(1.2) is a Hamilton system, with Hamiltonian

i.e. potential energy proportional to surface area plus kinetic energy of the fluid.

We explain why the scenario of oscillating almost spherical water drop is of particular interest. If the system is static, then the kinetic equation (1.2) implies that the outer normal derivative of the velocity potential Φ is zero, so Φ ≡ const . The Bernoulli equation on boundary (1.1) then implies that the embedded surface ι ⁢ ( M ) has constant mean curvature. Although the topology of M is not prescribed, the famous rigidity theorem of Alexandrov asserts that ι ⁢ ( M ) must be an Euclidean sphere (see for example [20]). Thus, Euclidean ball is the only static configuration for the physical system, and oscillating almost spherical water drop is the only possible perturbative oscillation near a static configuration. We emphasize that surface tension plays an important role: since there is no gravity in presence, it is the only constraining force preventing the fluid from spreading in the space. This is in contrast to the familiar gravity or gravity-capillary waves oscillating near the horizontal level.

1.2 Spherical Capillary Water Waves Equation

It is not hard to verify that the system (1.1)–(1.2) is invariant if ι is composed with a diffeomorphism of M. We may thus regard it as a geometric flow. If we are only interested in perturbation near a given configuration, we may reduce system (1.1)–(1.2) to a non-degenerate dispersive differential system concerning two scalar functions defined on M, just as Beyer and Günther did in [6]. In fact, during a short time of evolution, the interface can be represented as the graph of a function defined on the initial surface: if ι 0 : M → ℝ 3 is a fixed embedding close to the initial embedding ι ⁢ ( 0 , x ) , we may assume that ι ⁢ ( t , x ) = ι 0 ⁢ ( x ) + ζ ⁢ ( t , x ) ⁢ N 0 ⁢ ( x ) , where ζ is a scalar “height” function defined on M 0 , and N 0 is the outer normal vector field of M 0 . See Figure 1.

With this observation, we shall transform the system (1.1)–(1.2) into a non-local system of two real scalar functions ( ζ , ϕ ) defined on M, where ζ is the “height” function described as above, and ϕ ⁢ ( t , x ) = Φ ⁢ ( t , ι ⁢ ( t , x ) ) is the boundary value of the velocity potential, pulled back to the underlying manifold M.

Set

to be the operator mapping the pulled-back Dirichlet boundary value ϕ to the boundary value of the gradient of Φ. Define the Dirichlet–Neumann operator D ⁢ [ ζ ] ⁢ ϕ corresponding to the region enclosed by the image of ι = ι 0 + ζ ⁢ N 0 as the weighted outer normal derivative:

Thus the kinetic equation (1.2) becomes

We also need to calculate the restriction of ∂ t ⁡ Φ on M t in terms of ϕ and ι. By the chain rule,

We thus arrive at the following nonlinear system:

where H ⁢ ( ζ ) is the (scalar) mean curvature of the surface given by the height function ζ.

For M = 𝕊 2 , the case that we shall discuss in detail, we name the system as spherical capillary water waves equation. To simplify our discussion, we will be working under the center of mass frame, and require the mean of ϕ vanishes for all t. This could be easily accomplished by absorbing the mean into ϕ since the equation is invariant by a shift of ϕ. The quadratic terms in the right-hand side of (EQ(M)) are also computed explicitly using the Riemann connection ∇ 0 on 𝕊 2 corresponding to the standard spherical metric g 0 . In a word, from now on, we will be focusing on the non-dimensional capillary spherical water waves equation

By suitable spatial scaling, we may assume that σ 0 / ρ 0 = 1 , the total volume of the fluid is 4 ⁢ π 3 , and p e = - 2 , so that the conservation of volume is expressed as

where μ 0 is the standard measure on 𝕊 2 . The inertial movement of center of mass (1.3) and conservation of total momentum (1.4) under our center of mass frame are expressed respectively as

where μ ⁢ ( ι ) is the induced surface measure. Further, the Hamiltonian of the system is

and for a solution ( ζ , ϕ ) there holds 𝐇 ⁢ [ ζ , ϕ ] ≡ const .

The system (EQ) resembles the well-known Zakharov–Craig–Schanz–Sulem formulation of gravity-capillary water waves equation [8]. Formal linearization of (EQ) around the (unique) static solution ( ζ , ϕ ) = ( 0 , 0 ) indicates that the system is a dispersive one of order 3 2 . In fact, if we denote by ℰ n the space of degree n spherical harmonics, then the Dirichlet–Neumann operator D ⁢ [ 0 ] acts as the multiplier n on ℰ n , and the linearization H ′ ⁢ ( 0 ) acts as the multiplier - ( n - 2 ) ⁢ ( n + 1 ) on ℰ n . Consequently, with Π ( n ) being the projection to ℰ n , we can introduce the diagonal unknown

so that the linearization (EQ) around the static solution is a linear dispersive equation

where the 3 2 -order elliptic operator Λ is given by a multiplier

The system (EQ) for small amplitude oscillation can be formally re-written as

with 𝔑 ⁢ ( u ) = O ⁢ ( u ⊗ 2 ) vanishing quadratically as u ≃ 0 . In [28], the author provided formal analysis regarding small amplitude solutions and pointed out obstructions in understanding the long time behavior.

From a hydrophysical point of view, the linearized equation of (EQ) and the dispersive relation (1.11) dates back to Lord Rayleigh [24] (see also the well-known textbook [17, Section 275]). It has been studied by several widely cited papers on hydrodynamics [34, 22, 19], regarding the following topics: Poincaré–Lindstedt series of periodic solutions (assuming its existence, which is not guaranteed), possible “chaotic” behavior, and numerical simulation.

From a mathematical point of view, it is already known that the general free-boundary problem for Euler equation is locally well-posed, due to the work of Coutand and Shkoller [7] and Shatah and Zeng [31]. The curl equation ensures that if the flow is curl free at the beginning, then it remains so during the evolution. This justifies the local well-posedness of Cauchy problem of (EQ), at least for sufficiently regular initial data. On the other hand, Beyer and Günther [6] also showed that the Cauchy problem of (EQ) is locally well-posed, without referring to general free-boundary problem. They transformed the problem into an ODE problem defined on graded Hilbert spaces. See also [18] and [21] for proof of local well-posedness for water waves near horizontal level using Nash–Moser-type theorems. The potential-theoretic approach of Wu [36, 37] may also be transplanted to this case.

In summary, the Cauchy problem for (EQ) is locally well-posed, at least for sufficiently regular initial data. But this is all we can assert for the motion of a water droplet under zero gravity. None of the mathematical results mentioned above goes beyond the time regime guaranteed by energy estimate.

By a formal analysis conducted in [28], the author observed that the geometry of the sphere plays a fundamental role in the long-time behavior of (EQ), and leads to very different oscillatory behavior compared to gravity or gravity-capillary water waves near horizontal level. To analyze such oscillations in detail, we need to take into account the spectral properties of the sphere. This calls for the development of new analytic tools that explicitly reflects the geometry of the underlying space.

1.3 Main result and discussion

The main result of this paper is a sequence of para-differential formulas based on the toolbox developed by the author in [30]. The toolbox is briefed in Section 2, collecting precise definition of symbols, para-differential operators and a symbolic calculus. The first main result of this paper is the para-linearization formula for the Dirichlet–Neumann operator D ⁢ [ ζ ] ⁢ ϕ :

Section 4 is devoted to the proof of Theorem 1.1. The idea of the proof is quite straightforward: one factorizes the Laplacian as the product of two first order elliptic operators near the boundary, and extract the boundary value. Using this para-linearization formula, the system (EQ) is converted to a quasi-linear para-differential system of order 3 2 resembling the formal linearization (1.9)–(1.12):

The proof of Theorem 1.2 occupies Section 5. An initial application of Theorem 1.2 is the following local well-posedness theorem for (EQ):

The proof of Theorem 1.3 is only sketched in Section 6, as the key computations for this are quite parallel to the counterparts in [1]; we confine ourselves elaborating what is different. Theorem 1.3 shares the same regime of regularity as the main result in [1] does for nearly horizontal water waves. The requirement on regularity is obviously milder than that in [6], which is ( ζ , ϕ ) ∈ H 5.5 × H 5 ; or that in [7], which is ( ζ , ϕ ) ∈ H 6.5 × H 6 ; or that in [31], which is ( ζ , ϕ ) ∈ H 7 × H 6.5 . However, this should not be surprising, as the results in [31] and [7] apply to general motion of perfect fluid, without assuming irrotationality, while the argument in [6] deals with ODE on graded spaces so that some information will certainly be lost.

Let us briefly describe previous attempts towards para-differential calculus on curved manifolds. In [16], Klainerman and Rodnianski extended the Littlewood–Paley theory to compact surfaces via smooth cut-off of the Laplacian spectrum. In [9], Delort introduced a para-differential calculus on general compact Riemann manifold characterized by commutators. In [12], Bonthonneau, Guillarmou and de Poyferré developed a para-differential calculus for any smooth manifold via local coordinate chart.

However, we proceed within a formalism different from all above. In this paper, we extensively utilize the symmetry of the 2-sphere, on which (EQ) is defined. The 2-sphere is considered as a SU ⁢ ( 2 ) -homogeneous space via Hopf fibration, and the Cauchy problem of (EQ) is lifted to SU ⁢ ( 2 ) , a compact Lie group. We then employ the para-differential calculus for rough symbols on compact Lie groups proposed in [30]. The advantage of this approach is that the lower order symbols, being non-negligible for quasi-linear problems of order > 1 , are manipulated in an explicit, global, coordinate-independent way. We present this toolbox in Section 2 and Section 3. The theory turns out to be parallel to para-differential calculus on ℝ n or 𝕋 n .

With para-differential calculus constructed, the para-linearization and symmetrization of (EQ) becomes parallel to the gravity-capillary water waves problem for the flat water level. This procedure is accomplished in Sections 4–5, with method very similar to [1]. The system is then reduced to a quasi-linear, non-local, dispersive para-differential equation of order 3 2 , with precise form given in formula (1.13). The local existence result, sketched in Section 6, then follows from a standard fixed point argument.

The complexity of this para-differential reduction raises the natural question of whether it is necessary for the study of the water drop problem at all. Given the previous local well-posedness results already mentioned, such reduction does not seem unavoidable, if one is concerned with the local Cauchy theory. For example, if one employs Delort’s coordinate-free definition [9] of para-differential operators on manifolds, it is expected that the same regime of regularity remains valid for the local theory for initial fluid-air interface with higher genus. Anyway, it would be hard to imagine that the regime of short-time regularity depends on the topology of the interface.

However, previous approaches of para-differential calculus may fail to incorporate spectral properties of the sphere, and thus may cause extra difficulty in understanding long-time solutions. For example, no symbolic calculus is developed in [16] at all, since no additional structure beyond a compact surface is assumed. Both constructions in [9] and [12] result in calculus involving principal symbol alone, similar to Hörmander calculus on curved manifolds. These approaches thus may not capture the lower order symbols in quasi-linear problems that are crucial for long-time evolution. Furthermore, much of the spectral information related to dispersive properties become implicit, although not lost, under the frameworks of them.

It then seems feasible to utilize the representation-theory-based approach to para-differential calculus, since it clearly reflects the spectral properties of the dispersive system (EQ), while giving all the lower-order terms in symbolic calculus very explicit, manipulable form. Evidence of this convenience is obvious, if one takes into account the extensive volume of literature on gravity-capillary water waves with periodic boundary condition. People usually reduce the gravity-capillary water waves equation to a para-differential form, and use it to obtain global or almost global results. See for example [10, 4, 15], or the review [14]. Given that (1.13) is quite similar to the para-differential form of gravity-capillary water waves equation, we regard it as the starting point for the study of long-time behavior of small amplitude solutions of (EQ).

2.1 Representation and spectral properties

Let 𝔾 be a simply connected, compact, semisimple Lie group with dimension n and rank ϱ, i.e. the dimension of any maximal torus in 𝔾 . We write e for the identity element of 𝔾 . Let 𝔤 be the Lie algebra of 𝔾 , identified with the space of left-invariant vector fields on 𝔾 . If a basis X 1 , … , X n for 𝔤 is given, then for any multi-index α ∈ ℕ 0 n , write

for the left-invariant differential operator composed by these basis vectors, in the “normal” ordering.

Since 𝔾 is semisimple and compact, the Killing form

is non-degenerate and negative definite, where Ad is the adjoint representation. A Riemann metric on 𝔾 can thus be introduced as the opposite of the Killing form. The Laplace operator corresponding to this metric, usually called Casimir element in representation theory, is given by

where { X i } is any orthonormal basis of 𝔤 ; the operator is independent from the choice of basis.

Let 𝔾 ^ be the dual of 𝔾 , i.e. the set of equivalence classes of irreducible unitary representations of 𝔾 . For simplicity, we do not distinguish between an equivalence class ξ ∈ 𝔾 ^ and a certain unitary representation that realizes it. The ambient space of ξ ∈ 𝔾 ^ is an Hermite space ℋ [ ξ ] , with (complex) dimension d ξ . If we equip the space ℋ [ ξ ] with an orthonormal basis containing d ξ elements, then ξ : 𝔾 → U ( ℋ [ ξ ] ) can be equivalently viewed as a unitary-matrix-valued function ( ξ j ⁢ k ) j , k = 1 d ξ .

We shall equip the Lie group 𝔾 with the normalized Haar measure. For simplicity, we will denote the integration with respect to this measure by d ⁢ x . The Fourier transform of f ∈ 𝒟 ′ ⁢ ( 𝔾 ) is defined by

Conversely, for every field a ⁢ ( ξ ) on 𝔾 ^ such that a ( ξ ) ∈ End ( ℋ [ ξ ] ) for all ξ ∈ 𝔾 ^ , the Fourier inversion of a is defined by

as long as the right-hand side converges at least in the sense of distribution.

Two different norms for a ( ξ ) ∈ End ( ℋ [ ξ ] ) will be used in this paper, one being the operator norm

one being the Hilbert–Schmidt norm

To simplify notation, when there is no risk of confusion, we shall omit the dependence of these norms on the representation ξ.

We fix the convolution on 𝔾 to be right convolution:

The only property not inherited from the usual convolution on Euclidean spaces is commutativity. Every other property, including the convolution-product duality, Young’s inequality, is still valid. The only modification is that

which in general does not coincide with ( f ^ ⋅ g ^ ) ⁢ ( ξ ) since the Fourier transform is now a matrix.

The Peter–Weyl theorem is fundamental for harmonic analysis on Lie groups:

In the following, standard constructions and results in highest weight theory will be directly cited. They are already collected in Berti-Procesi [5].

Let 𝔱 ⊂ 𝔤 be the Lie algebra of a maximal torus of 𝔾 , and write 𝔱 * for its dual. Let { 𝜶 i } i = 1 ϱ ⊂ 𝔱 * be the set of positive simple roots of 𝔤 with respect to 𝔱 . They form a basis for 𝔱 * . The fundamental weights ϖ 1 , … , ϖ ϱ ∈ 𝔱 * is the unique set of vectors { ϖ i } i = 1 ϱ satisfying

Here the inner product on the dual is inherited from the Killing form on 𝔤 .

The weight lattice completely describes the eigenvalues of Δ.

We write λ ξ = | 𝑱 ⁢ ( ξ ) + ϖ | 2 - | ϖ | 2 , so that { λ ξ } is the spectrum of the positive self-adjoint operator - Δ . Imitating commutative harmonic analysis, we set the size functions | ξ | := | λ ξ | , 〈 [ 〉 ξ ] := ( 1 + λ ξ ) 1 2 . The operator | ∇ | , which is the square-root of - Δ defined via spectral analysis, will be frequently used. The values of 〈 [ 〉 ξ ] are exactly the eigenvalues of ( 1 - Δ ) 1 2 . For s ∈ ℝ , the Sobolev space H s ⁢ ( 𝔾 ) is defined to be the subspace of f ∈ 𝒟 ′ ⁢ ( 𝔾 ) such that ∥ ( 1 - Δ ) s / 2 ⁢ f ∥ L 2 ⁢ ( 𝔾 ) < ∞ . By the Peter–Weyl theorem and the above characterization of Laplacian, the Sobolev norm is computed by

To proceed further, we need to introduce a notion of partial ordering on ℒ + . Define the cone

We then introduce the dominance order on ℒ + as follows: for 𝒋 , 𝒌 ∈ ℒ + , we set 𝒋 ≺ 𝒌 if 𝒋 ≠ 𝒌 and 𝒌 - 𝒋 ∈ ℛ + , and set 𝒋 ⪯ 𝒌 to include the possibility 𝒋 = 𝒌 . The relation ≺ defines a partial ordering on ℒ + . With the aid of this root system, we obtain the following characterization of product of Laplace eigenfunctions (see, for example, [23, Proposition 3, p. 345]):

An important spectral localization property for products can be deduced from this characterization, which will play a fundamental role in dyadic analysis:

2.2 Symbol and symbolic calculus

A global notion of symbol can be defined on Lie groups due to their high symmetry. An explicit symbolic calculus was formally constructed by a series of works of Ruzhansky, Turunen, Wirth. Fischer’s work [11] provided a complete study of the object. We will closely follow [11].

We fix our compact Lie group 𝔾 as in the previous subsection. The starting point will be a Taylor’s formula on 𝔾 .

Note that the differential operators X q ( α ) are defined independently for every multi-index, so the equality X q ( α + β ) = X q ( α ) ⁢ X q ( β ) is, in general, not valid.

A symbol a on 𝔾 is simply a field a defined on 𝔾 × 𝔾 ^ , such that a ⁢ ( x , ξ ) is a distribution of value in End ( ℋ [ ξ ] ) for each ξ ∈ 𝔾 ^ . If a basis for ℋ [ ξ ] is chosen, the value a ⁢ ( x , ξ ) can be simply understood as a matrix function of size d ξ × d ξ . We define the quantization of a symbol a formally by

Conversely, if A : C ∞ ⁢ ( 𝔾 ) → 𝒟 ′ ⁢ ( 𝔾 ) is a continuous operator, then it is the quantization of the symbol

Here A ⁢ ξ is understood as entry-wise action. In this case the series (2.2) then converges in 𝒟 ′ ⁢ ( 𝔾 ) . The associated right convolution kernel for a ⁢ ( x , ξ ) is defined by

where the Fourier inversion is taken with respect to ξ. Formally, the action 𝐎𝐩 ⁢ ( a ) ⁢ f can be written as a convolution

An intrinsic notion of difference operators acting on symbols was introduced by Fischer [11], generalizing the differential operator with respect to ξ in harmonic analysis on ℝ n . For any (continuous) unitary representation ( τ , ℋ τ ) , Maschke’s theorem ensures that τ = ⊕ j ξ j for finitely many ξ j ∈ 𝔾 ^ . A symbol a ⁢ ( x , ξ ) can be naturally extended to any τ by a ⁢ ( x , τ ) = ⊕ j a ⁢ ( x , ξ j ) , up to equivalence of representations. The definition of difference operators is then given by

For a tuple of representations 𝝉 = ( τ 1 , … , τ p ) , write 𝐃 𝝉 = 𝐃 τ 1 ⁢ ⋯ ⁢ 𝐃 τ p , and 𝐃 𝝉 ⁢ a ⁢ ( x , ξ ) is then an endomorphism of ℋ τ 1 ⊗ ⋯ ⊗ ℋ τ p ⊗ ℋ [ ξ ] .

Corresponding to the functions q = { q i } i = 1 M as in Proposition 2.2, Ruzhansky et al. [25, 26, 27] introduced the so-called admissible difference operators:

We write 𝐃 q , ξ for the action of a difference operator on the ξ variable. Formally, we have

so 𝐃 q , ξ commutes with any differential operator acting on x. To compare with ℝ n , we simply notice that given a symbol a ⁢ ( x , ξ ) on ℝ n , the Fourier inversion of ∂ ξ ⁡ a ⁢ ( x , ξ ) with respect to ξ is i ⁢ y ⁢ a ∨ ⁢ ( x , y ) , i.e. multiplication by a polynomial. The functions q on 𝔾 then play the role of monomials on ℝ n .

With the aid of difference operators, Fischer [11] introduced symbol classes of interest.

In [11], Fischer proved that if q = ( q i ) i = 1 M is a strongly RT-admissible tuple, then the symbol class in Definition 2.5 does not depend on the choice of q, and in fact gives rise to the usual Hörmander class of pseudo-differential operators.

A convenient choice of strongly RT-admissible tuple is necessary for calculations. From now on, we shall just define the fundamental tuple of 𝔾 as

It is not hard to verify that the tuple Q is indeed a strongly RT-admissible tuple.

A particular property of the fundamental tuple Q deserves a specific mention: if τ is a fundamental representation of 𝔾 , then for q i ⁢ j ⁢ ( x ) = τ i ⁢ j ⁢ ( x ) - δ i ⁢ j , there holds

Thus we have the Leibniz-type property