Longtime Dynamics of Irrotational Spherical Water Drops: Initial Notes

Comparing to gravity water waves problems, the governing equation for a spherical droplet of water under zero gravity takes a very different form. At a first glance it looks similar to the water waves systems as mentioned above, but some crucial differences do arise after careful analysis. To the author’s knowledge, besides those dealing with generic free-boundary Euler equation ([CS07], [SZ08]), the only reference on this problem is Beyer-Günther [BG98], in which the local well-posedness of the equation is proved using a Nash-Moser type implicit function theorem. We will briefly discribe known results for gravity water waves problems in the next subsection.

To start with, let us pose the following assumptions on the fluid motion that we try to describe:

• •

  (A1) The perfect, irrotational fluid of constant density $\rho_{0}$ occupies a smooth, compact region in $\mathbb{R}^{3}$.

• •

  (A2) There is no gravity or any other external force in presence.

• •

  (A3) The air-fluid interface is governed by the Young-Laplace law, and the effect of air flow is neglected.

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 $\iota(t,\cdot):M\to\mathbb{R}^{3}$. We will denote a point on $M$ by $x$, the image of $M$ under $\iota(t,\cdot)$ by $M_{t}$, and the region enclosed by $M_{t}$ by $\Omega_{t}$. The outer normal will be denoted by $N(\iota)$. We also write $\bar{\nabla}$ for the flat connection on $\mathbb{R}^{3}$.

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

$$\sigma_{0}H(\iota)=p_{i}-p_{e},$$

where $H(\iota)$ is the (scalar) mean curvature of the embedding, $\sigma_{0}$ 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:

(1.1)

$$\left.\frac{\partial\Phi}{\partial t}\right|_{M_{t}}+\frac{1}{2}|\bar{\nabla}\Phi|_{M_{t}}|^{2}-p_{e}=-\frac{\sigma_{0}}{\rho_{0}}H(\iota),$$

where $\Phi$ is the velocity potential of the velocity field of the air. Note that $\Phi$ is determined up to a function in $t$, so we shall leave the constant $p_{e}$ around for convenience reason that will be explained shortly. According to assumption (A1), the function $\Phi$ is a harmonic function within the region $\Omega_{t}$, so it is uniquely determined by its boundary value, and the velocity field within $\Omega_{t}$ is $\bar{\nabla}\Phi$. The kinematic equation on the free boundary $M_{t}$ is naturally obtained as

(1.2)

$$\frac{\partial\iota}{\partial t}\cdot N(\iota)=\bar{\nabla}\Phi|_{M_{t}}\cdot N(\iota).$$

Finally, we would like to discuss the conservation laws for (1.1)-(1.2). The preservation of volume $\text{Vol}(\Omega_{t})=\text{Vol}(\Omega_{0})$ is a consequence of incompressibility. The system describes an Eulerian flow without any external force, so the center of mass moves at a uniform speed along a fixed direction, i.e.

(1.3)

$$\frac{1}{\mathrm{Vol}(\Omega_{0})}\int_{\Omega_{t}}Pd\mathrm{Vol}(P)=V_{0}t+C_{0},$$

with Vol being the Lebesgue measure, $P$ marking points in $\mathbb{R}^{3}$, $V_{0}$ 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

(1.4)

$$\int_{M_{t}}\rho_{0}\Phi N(\iota)d\mathrm{Area}(M_{t})\equiv\rho_{0}\mathrm{Vol}(\Omega_{0})V_{0}.$$

Most importantly, it is not surprising that (1.1)-(1.2) is a Hamilton system (the Zakharov formulation for water waves; see Zakharov [Zak68]), with Hamiltonian

(1.5)

$$\sigma_{0}\mathrm{Area}(\iota)+\frac{1}{2}\int_{\Omega_{t}}\rho_{0}|\bar{\nabla}\Phi|^{2}d\mathrm{Vol}=\sigma_{0}\mathrm{Area}(M_{t})+\frac{1}{2}\int_{M_{t}}\rho_{0}\Phi|_{M_{t}}\left(\bar{\nabla}\Phi|_{M_{t}}\cdot N(\iota)\right)d\mathrm{Area},$$

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

It is not hard to verify that the system (1.1)-(1.2) is invariant if $\iota$ 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 [BG98]. 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 $\iota_{0}:M\to\mathbb{R}^{3}$ is a fixed embedding close to the initial embedding $\iota(0,x)$, we may assume that $\iota(t,x)=\iota_{0}(x)+\zeta(t,x)N_{0}(x)$, where $\zeta$ is a scalar “height” function defined on $M_{0}$ and $N_{0}$ is the outer normal vector field of $M_{0}$.

With this observation, we shall transform the system (1.1)&(1.2) into a non-local system of two real scalar functions $(\zeta,\phi)$ defined on $M$, where $\zeta$ is the “height” function described as above, and $\phi(t,x)=\Phi(t,\iota(t,x))$ is the boundary value of the velocity potential, pulled back to the underlying manifold $M$.

The operator

$$B_{\zeta}:\phi\to\bar{\nabla}\Phi|_{M_{t}}$$

maps the pulled-back Dirichlet boundary value $\phi$ to the boundary value of the gradient of $\Phi$. We shall write

$$(D[\zeta]\phi)N(\iota)$$

for its normal part, where $D[\zeta]$ is the Dirichlet-Neumann operator corresponding to the region enclosed by the image of $\iota_{0}+\zeta N_{0}$. Thus

$$\frac{\partial\zeta}{\partial t}N_{0}\cdot N(\iota)=D[\zeta]\phi.$$

We also need to calculate the restriction of $\partial_{t}\Phi$ on $M_{t}$ in terms of $\phi$ and $\iota$. By the chain rule,

	$\displaystyle\left.\frac{\partial\Phi}{\partial t}\right|_{M_{t}}$	$\displaystyle=\frac{\partial\phi}{\partial t}-\bar{\nabla}\Phi|_{M_{t}}\cdot\frac{\partial\iota}{\partial t}$
		$\displaystyle=\frac{\partial\phi}{\partial t}-\left(\bar{\nabla}\Phi|_{M_{t}}\cdot N_{0}\right)\frac{\partial\zeta}{\partial t}$
		$\displaystyle=\frac{\partial\phi}{\partial t}-\frac{1}{N_{0}\cdot N(\iota)}\left(\bar{\nabla}\Phi|_{M_{t}}\cdot N_{0}\right)\cdot D[\zeta]\phi.$

We thus arrive at the following nonlinear system:

(EQ(M))

$$\left\{\begin{aligned} \frac{\partial\zeta}{\partial t}&=\frac{1}{N_{0}\cdot N(\iota)}D[\zeta]\phi,\\ \frac{\partial\phi}{\partial t}&=\frac{1}{N_{0}\cdot N(\iota)}\left(B_{\zeta}\phi\cdot N_{0}\right)\cdot D[\zeta]\phi-\frac{1}{2}|B_{\zeta}\phi|^{2}-\frac{\sigma_{0}}{\rho_{0}}H(\iota)+p_{e},\end{aligned}\right.$$

where $\iota=\iota_{0}+\zeta N_{0}$.

For $M=S^{2}$, the case that we shall discuss in detail, we refer to the system as the capillary spherical water waves equation. To simplify our discussion, we shall be working under the center of mass frame, and require the eigenmode $\Pi^{(0)}\phi$ to vanish for all $t$. This could be easily accomplished by absorbing the eigenmode into $\phi$ since the equation is invariant by a shift of $\phi$. In a word, from now on, we will be focusing on the non-dimensional capillary spherical water waves equation

(EQ)

$$\left\{\begin{aligned} \frac{\partial\zeta}{\partial t}&=\frac{1}{N_{0}\cdot N(\iota)}D[\zeta]\phi,\\ \frac{\partial\phi}{\partial t}&=\frac{1}{N_{0}\cdot N(\iota)}\left(B_{\zeta}\phi\cdot N_{0}\right)\cdot D[\zeta]\phi-\frac{1}{2}|B_{\zeta}\phi|^{2}-H(\iota),\end{aligned}\right.$$

where $\iota=(1+\zeta)\iota_{0}$, and $\Pi^{(0)}\phi\equiv 0$. We assume that the total volume of the fluid is $4\pi/3$, so that the preservation of volume is expressed as

(1.6)

$$\frac{1}{3}\int_{S^{2}}(1+\zeta)^{3}d\mu_{0}\equiv\frac{4\pi}{3},$$

where $\mu_{0}$ 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

(1.7)

$$\int_{S^{2}}(1+\zeta)^{4}N_{0}d\mu_{0}=0,\quad\int_{S^{2}}\phi N(\iota)d\mu(\iota)=0,$$

where $\mu(\iota)$ is the induced surface measure. Further, the Hamiltonian of the system is

(1.8)

$$\mathbf{H}[\zeta,\phi]=\mathrm{Area}(\iota)+\frac{1}{2}\int_{S^{2}}\phi\cdot D[\zeta]\phi\cdot d\mu(\iota),$$

and for a solution $(\zeta,\phi)$ there holds $\mathbf{H}[\zeta,\phi]\equiv 4\pi$.

Up to this point, we are still working within the realm of well-established frameworks. We already know that the general free-boundary Euler equation is locally well-posed due to the work of [CS07] or [SZ08], and due to the curl equation the curl free condition persists during the evolution. On the other hand, the Cauchy problem of system (1.1) and (1.2) is known to be locally well-posed, due to Beyer and Günther in [BG98]. They used an iteration argument very similar to a Nash-Moser type argument in the sense that it involves multiple scales of Banach spaces and “tame” maps in a certain sense. Finally, it is not hard to transplant the potential-theoretic argument of Wu [Wu99] to prove the local well-posedness.

To sum up, we already know that the system (EQ(M)) for a compact orientable surface $M$ (hence (EQ) specifically) is locally well-posed. But this is all we can assert for the motion of a water droplet under zero gravity. In the following part of this note, we will propose several questions and conjectures concerning the long-time behaviour of water droplets under zero gravity.

There has already been several different approaches to describe the motion of perfect fluid with free boundary. One is to consider the motion of perfect fluid occupying an arbitrary domain in $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$, either with or without surface tension. The motion is described by a free boundary value problem of Euler equation. This generic approach was employed by Countand-Shkoller [CS07] and Shatah-Zeng [SZ08]. Both groups proved the local-wellposedness of the problem. This approach has the advantage of being very general, applicable to all geometric shapes of the fluid.

On the other hand, when coming to potential flows of a perfect fluid, the curl free property results in dispersive nature of the problem. The motion of a curl free perfect fluid under gravity and a free boundary value condition is usually referred to as the gravity water waves problem. The first breakthroughs in understanding local well-posedness were works of Wu [Wu97] [Wu99], who proved local well-posedness of the gravity water waves equation without any smallness assumption. Lannes [Lan05] extended this to more generic bottom shapes. Taking surface tension into account, the problem becomes gravity-capillary water waves. Schweizer [Sch05] proved local well-posedness with small Cauchy data of the gravity-capillary water waves problem, and Ming-Zhang [MZ09] proved local well-posedness without smallness assumption. Alazard-Metevier [AM09] and Alazard-Burq-Zuily [ABZ11] used para-differential calculus to obtain the optimal regularity for local well-posedness of the water waves equation, either with or without surface tension.

For discussion of long time behavior, it is important to take into account the dispersive nature of the problem. For linear dispersive properties, there has been work of Christianson-Hur-Staffilani [CHS10]. For gravity water waves living in $\mathbb{R}^{2}$, works on lifespan estimate include Wu [Wu09] and Hunter-Ifrim-Tataru [HIT16] (almost global result), Ionescu-Pusateri [IP15] and Alazard-Delort [AD13] and Ifrim-Tataru [IT14] (global result). For gravity water waves living in $\mathbb{R}^{3}$, there are works of Germain-Masmoudi-Shatah [GMS12] and Wu [Wu11] (no surface tension), Germain-Masmoudi-Shatah [GMS15] (no gravity), Deng-Ionescu-Pausader-Pusateri [DIPP17] (gravity-capillary water waves) and Wang [Wan20] (gravity-capillary water waves with finite depth). These results all employed different forms of decay estimates derived from dispersive properties.

As for long time behavior of periodic water waves, Berti-Delort [BD⁺18] considered gravity-capillary water waves defined on $\mathbb{T}^{1}$, Berti-Feola-Pusateri [BFP22] considered gravity water waves defined on $\mathbb{T}^{1}$, Ionescu-Pusateri [IP19] considered gravity-capillary water waves defined on $\mathbb{T}^{2}$, and obtained an estimate on the lifespan beyond standard energy method. All of the three groups used para-differential calculus and suitable normal form reduction; the results of Berti-Delort and Ionescu-Pusateri were proved for physical data of full Lebesgue measure.

To sum up, all results on the gravity water waves problem listed above are concerned with an equation for two scalar functions defined on a fixed flat manifold, being one of the following: $\mathbb{R}^{1}$, $\mathbb{R}^{2}$, $\mathbb{T}^{1}$, $\mathbb{T}^{2}$, sometimes called the “bottom” of the fluid. These two functions represents the geometry of the liquid-gas interface and the boundary value of the velocity potential, respectively. The manifold itself is considered as the bottom of the container in which all dynamics are performed. We observe that the differential equation (EQ(M)) is, mathematically, fundamentally different from the water waves equations that have been well-studied.

We are mostly interested in the stability of the static solution of (EQ(M)). A static solution should be a fluid region whose shape stays still, with motion being a mere shift within the space. In this case, we have $p_{e}=0$ since the reference is relatively static with respect to the air. Moreover, the velocity field $\bar{\nabla}\Phi$ and “potential of acceleration” $\partial\Phi/\partial t$ must both be spatially uniform, so that the left-hand-side of the pressure balance condition (1.1) is a function of $t$ alone. It follows that $\iota(M)$ is always a compact embedded surface of constant mean curvature, hence in fact always an Euclidean sphere by the Alexandrov sphere theorem (see [MP19]), and we may just take $M=S^{2}$. Moreover, since $\iota(S^{2})$ should enclose a constant volume by incompressibility, the radius of that sphere does not change. After suitable scaling, we may assume that the radius is always 1, and $\rho_{0}=1$, $\sigma_{0}=1$, to make (1.1)-(1.2) non-dimensional. Finally, by choosing the center of mass frame, we may simply assume that the spatial shift is always zero, so that the velocity potential $\Phi\equiv\Phi_{0}$, a real constant. It is harmless to fix it to be zero.

Thus, under our convention, a static solution of (1.1)-(1.2) takes the form

(2.1)

$$\left(\begin{array}[]{c}\iota(t,x)\\ \Phi(t,x)\end{array}\right)=\left(\begin{array}[]{c}\iota_{0}(x)\\ \displaystyle{0}\end{array}\right),$$

where $a\in\mathbb{R}^{3}$ is a constant vector, and $\iota_{0}$ is the standard embedding of $S^{2}$ as $\partial B(0,1)\subset\mathbb{R}^{3}$. Equivalently, this means that a static solution of (EQ(M)) under our convention must be $(\zeta,\phi)=(0,0)$. Note here that the Gauss map of $\iota_{0}$ coincides with itself.

We can now start our perturbation analysis around a static solution at the linear level. Let $\mathcal{E}^{(n)}$ be the space of spherical harmonics of order $n$, normalized according to the standard surface measure on $S^{2}$. In particular, $\mathcal{E}^{(1)}$ is spanned by three components of $N_{0}$. Let $\Pi^{(n)}$ be the orthogonal projection on $L^{2}(S^{2})$ onto $\mathcal{E}^{(n)}$, $\Pi_{\leq n}$ be the orthogonal projection on $L^{2}(S^{2})$ onto $\bigoplus_{k\leq n}\mathcal{E}^{(k)}$, $\Pi_{\geq n}$ be the orthogonal projection on $L^{2}(S^{2})$ onto $\bigoplus_{k\geq n}\mathcal{E}^{(k)}$. For $\iota=(1+\zeta)\iota_{0}$, the linearization of $-H(\iota)$ around the sphere $\zeta\equiv 0$ is $\Delta\zeta+2\zeta$, where $\Delta$ is the Laplacian on the sphere $S^{2}$; cf. the standard formula for the second variation of area in [BC12]. Then $H^{\prime}(\iota_{0})$ acts on $\mathcal{E}^{(n)}$ as the multiplier $-(n-1)(n+2)$. Note that even if we consider the dimensional form of (EQ), there will only be an additional scaling factor $\sigma_{0}/(\rho_{0}R^{2})$, where $R$ is the radius of the sphere. On the other hand, the following solution formula for the Dirichlet problem on $B(0,1)$ is well-known: if $f\in L^{2}(S^{2})$, then the harmonic function in $B(0,1)$ with Dirichlet boundary value $f$ is determined by

$$u(r,\omega)=\sum_{n\geq 0}r^{n}(\Pi^{(n)}f)(\omega),$$

where $(r,\omega)$ is the spherical coordinate in $\mathbb{R}^{3}$. Thus the Dirichlet-Neumann operator $D[\iota_{0}]$ acts on $\mathcal{E}^{(n)}$ as the multiplier $n$. Note again that even if we consider the dimensional form (EQ), there will only be an additional scaling factor $R^{-1}$.

Thus, setting

$$u=\Pi^{(0)}\zeta+\Pi^{(1)}\zeta+\sum_{n\geq 2}\sqrt{(n-1)(n+2)}\cdot\Pi^{(n)}\zeta+i\sum_{n\geq 1}\sqrt{n}\cdot\Pi^{(n)}\phi,$$

we find that the linearization of (EQ) around the static solution (2.1) is a linear dispersive equation

(2.2)

$$\frac{\partial u}{\partial t}+i\Lambda u=0,$$

where the $3/2$-order elliptic operator $\Lambda$ is given by a multiplier

$$\Lambda=\sum_{n\geq 2}\sqrt{n(n-1)(n+2)}\cdot\Pi^{(n)}=:\sum_{n\geq 0}\Lambda(n)\Pi^{(n)}.$$

Note that $(\zeta,\phi)$ is completely determined by $u$. At the linear level, there must hold $\Pi^{(0)}u\equiv 0$ because the first variation of volume must be zero; and $\Pi^{(1)}u\equiv 0$ because of the conservation laws (1.7).

Let us also re-write the original nonlinear system (EQ) into a form that better illustrates its perturbative nature. For simplicity, we use $O(u^{\otimes k})$ to abbreviate a quantity that can be controlled by $k$-linear expressions in $u$, and disregard its continuity properties for the moment. For example, $\|u\|_{H^{1}}^{2}+\|u\|_{H^{2}}^{4}$ is an expression of order $O(u^{\otimes 2})$ when $u\to 0$.

Since the operator $\Lambda$ acts degenerately on $\mathcal{E}^{(0)}\oplus\mathcal{E}^{(1)}$, we should be more careful about the eigenmodes $\Pi^{(0)}u$ and $\Pi^{(1)}u$. The volume preservation equation (1.6) implies $\partial_{t}\Pi^{(0)}\zeta=O(u^{\otimes 2})$. Projecting (EQ) to $\mathcal{E}^{(1)}$, which is spanned by the components of $N_{0}$, we obtain $\partial_{t}\Pi^{(1)}\zeta=\Pi^{(1)}\phi+O(u^{\otimes 2})=O(u^{\otimes 2})$ since the conservation law (1.7) implies $\Pi^{(1)}\phi=O(u^{\otimes 2})$; and $\partial_{t}\Pi^{(1)}\phi=O(u^{\otimes 2})$ since $H^{\prime}(\iota_{0})=-\Delta-2$ annihilates $\mathcal{E}^{(1)}$. We can thus formally re-write the nonlinear system (EQ) as the following:

(2.3)

$$\frac{\partial u}{\partial t}+i\Lambda u=\mathfrak{N}(u),$$

with $\mathfrak{N}(u)=O(u^{\otimes 2})$ vanishing quadratically as $u\to 0$. Note that we are disregarding all regularity problems at the moment.

At the linear level, our first unanswered question is

Answer to Question 1 should be important in understanding the dispersive nature of linear capillary spherical water waves. For Schrödinger equation on a compact manifold, a widly cited result was obtained by Burq-Gérard-Tzvetkov [BGT04]:

The authors used a time-localization argument for the parametrix of $\partial_{t}-i\Delta_{g}$ to prove this result. For the sphere $S^{d}$, this inequality is not optimal. The authors further used a Bourgain space argument to obtain the optimal Strichartz inequality:

The proof is the consequence of two propositions. The first one is the “decoupling inequality on compact manifolds”, in particular the following result proved by Sogge [Sog88]:

The second one is a Bourgain space embedding result:

The key ingredient for proving this proposition is the following number-theoretic result:

$$\#\{(p,q)\in\mathbb{N}^{2}:p^{2}+q^{2}=A\}=O(A^{\varepsilon}).$$

As for optimality of the Strichartz inequality, the authors of [BGT04] implemented standard results of Gauss sums.

The parametrix and Bourgain space argument can be repeated without essential change for the linear capillary spherical water waves equation (2.2), but this time the Bourgain space argument would be more complicated: the Bourgain space norm is now

$$\|f(t,x)\|_{X^{s,b}}:=\left\|\langle\partial_{t}+i\Lambda\rangle^{b}f(t,x)\right\|_{L^{2}_{t}H^{s}_{x}},$$

and the embedding result becomes

$$\|f\|_{L^{4}(\mathbb{R}\times S^{2})}\leq C_{s,b}\|f\|_{X^{s,b}}$$

for $f\in C^{\infty}_{0}(\mathbb{R}\times S^{2})$ and all $s>3\rho/8+1/8$, $b>1/2$, where $\rho$ is the infimum of all exponents $\rho^{\prime}$ such that when $A\to\infty$, the number

$$\#\left\{(n_{1},n_{2})\in\mathbb{N}^{2}:\,\frac{1}{2}\leq\frac{n_{2}}{n_{1}}\leq 2,\,|\Lambda(n_{1})+\Lambda(n_{2})-A|\leq\frac{1}{2}\right\}\leq C_{\rho^{\prime}}A^{\rho^{\prime}}.$$

Some basic analytic number theory implies $\rho=1/3$, and thus the range of $s$ is $s>1/4$. Surprisingly this index is not better than that predicted by the parametrix method. It remains unknown whether this index could be further optimized.

To close this subsection, we note that the capillary spherical water wave lives on a compact region, so the dispersion does not take away energy from a locality to infinity. This is a crucial difference between waves on compact regions and waves in Euclidean spaces. In particular, we do not expect decay estimate for $e^{it\Lambda}f$. For the nonlinear problem (EQ), techniques like vector field method (Klainerman-Sobolev type inequalities) do not apply.

Illuminated by observations in hydrodynamical experiments under zero gravity, and suggested by the existence of standing gravity capillary water waves due to Alazard-Baldi [AB15], we propose the following conjecture:

Let us conduct the bifurcation analysis that suggests why this conjecture should be true. By time rescaling, we aim to find solution $(\zeta,\phi,\omega_{0})$ of the following system that is $2\pi$-peiodic in $t$:

(2.4)

$$\left\{\begin{aligned} \omega_{0}\frac{\partial\zeta}{\partial t}&=\frac{1}{N_{0}\cdot N(\iota)}D[\zeta]\phi,\\ \omega_{0}\frac{\partial\phi}{\partial t}&=\frac{1}{N_{0}\cdot N(\iota)}\left(B_{\zeta}\phi\cdot N_{0}\right)\cdot D[\zeta]\phi-\frac{1}{2}|B_{\zeta}\phi|^{2}-H(\iota),\end{aligned}\right.$$

together with the conservation laws (1.6)-(1.8). Here we refer $\omega_{0}>0$ as the fundamental frequency. The linearization of this system at the equilibrium $(\zeta,\phi)=(0,0)$ is

(2.5)

$$L_{\omega_{0}}\left(\begin{matrix}\zeta\\ \phi\end{matrix}\right):=\left(\begin{matrix}\omega_{0}\partial_{t}&-D[0]\\ -\Delta-2&\omega_{0}\partial_{t}\end{matrix}\right)\left(\begin{matrix}\zeta\\ \phi\end{matrix}\right)=0,\quad\Pi^{(0)}\zeta=\Pi^{(1)}\zeta=\Pi^{(1)}\phi=0.$$

We restrict to rotationally symmetric solutions of the system: that is, water droplets which are always rotationally symmetric with a fixed axis. In addition, we require $\zeta$ to be even in $t$ and $\phi$ to be odd in $t$. The solution thus should take the form

$$\zeta(t,x)=\sum_{j,n\geq 0}\zeta_{jn}\cos(jt)Y_{n}(x),\quad\phi(t,x)=\sum_{j\geq 1,n\geq 0}\phi_{jn}\sin(jt)Y_{n}(x),$$

where $Y_{n}$ is the $n$’th zonal spherical harmonic, i.e. the (unique) normalized spherical harmonic of degree $n$ that is axially symmetric. In spherical coordinates this means that $Y_{n}(\theta,\varphi)=P_{n}(\cos\theta)$, where $P_{n}$ is the $n$’th Legendre polynomial. Since $\phi_{0n}$ are irrelevant we fix them to be 0. Then

$$L_{\omega_{0}}\left(\begin{matrix}\zeta\\ \phi\end{matrix}\right)=\sum_{j,n\geq 0}\left(\begin{matrix}(-\omega_{0}j\zeta_{jn}-n\phi_{jn})\sin(jt)Y_{n}(x)\\ \left((n-1)(n+2)\zeta_{jn}+\omega_{0}j\phi_{jn}\right)\cos(jt)Y_{n}(x)\end{matrix}\right).$$

In order that $(\zeta,\phi)^{\mathrm{T}}\in\mathrm{Ker}L_{\omega_{0}}$, at the level $n=0$, we must have $\zeta_{j0}=\phi_{j0}=0$ for all $j\geq 0$. At the level $n=1$, we have $\zeta_{01}=\phi_{01}=0$, and for $j\geq 1$ there holds $\omega_{0}j\zeta_{j1}-\phi_{j1}=0$ and $\omega_{0}j\phi_{j1}=0$, so $\zeta_{j1}=\phi_{j1}=0$ for all $j\geq 0$. Hence $\zeta_{jn},\phi_{jn}$ can be nonzero only for $j\geq 1$ and $n\geq 2$.

Consequently, $L_{\omega_{0}}$ has a one-dimensional kernel if and only if the Diophantine equation

(2.6)

$$\omega_{0}^{2}j^{2}=n(n-1)(n+2),\quad j\geq 1,\,n\geq 2$$

has exactly one solution $(j_{0},n_{0})$. If $\omega_{0}$ has this property, then at the linear level, the lowest frequency of oscillation is

$$\omega_{0}j_{0}=\sqrt{n_{0}(n_{0}-1)(n_{0}+2)}.$$

We look into this Diophantine equation. Obviously $\omega_{0}^{2}$ has to be a rational number. The equation is closely related to a family of elliptic curves over $\mathbb{Q}$:

$$E_{c}:y^{2}=x(x-c)(x+2c)=x^{3}+c^{2}x^{2}-2c^{2}x,\quad c\in\mathbb{N}.$$

If we set $a/b=\omega_{0}^{2}$ (irreducible fraction), then integral solutions of (2.6) are in 1-1 correspondence with integral points with natural number coordinates on the elliptic curve $E_{ab}$, under the following map:

$$(j_{0},n_{0})\to(abn_{0},a^{2}bj_{0})\in E_{ab}.$$

Thus we just need to find natural numbers $a,b$ such that there is a unique (up to negation) integral point $(x,y)\in E_{ab}$, where $x>0$ is divided by $ab$, and $y$ is divided by $a^{2}b$. We seek for $n_{0}$ as small as possible with such property, which gives lowest frequency of oscillation as small as possible. For $ab=1,\cdots,50$, we find that if $ab=15$, then the integral points on elliptic curve $E_{15}$ (up to negation of Mordel-Weil group) are

$$(-30,0),\,(-5,50),\,(0,0),\,(15,0),\,(24,108),\,(90,900).$$

The only point $(x,y)$ with $ab|x$ and $ab^{2}|y$ is (90,900), which gives $n_{0}=6$, and the lowest frequency $\Lambda(n_{0})=\sqrt{n_{0}(n_{0}-1)(n_{0}+2)}$ of oscillation is $4\sqrt{15}\simeq 15.49\cdots$.

There are other choices of $a,b$. We list down the value of $ab$ below 50, the corresponding $n_{0}$ and the lowest frequency $\Lambda(n_{0})$:

$ab$	$n_{0}$	$\Lambda(n_{0})$
15	6	$4\sqrt{15}$
17	49	$4\sqrt{323}$
22	9	$6\sqrt{22}$
26	50	$15\sqrt{78}$
42	7	$3\sqrt{42}$
46	576	$2040\sqrt{46}$
50	25	$90\sqrt{2}$

See Appendix A for the MAGMA code used to find these values. This list suggests that $n_{0}=6$ might be the smallest order that meets the requirement, but this remains unproved. We summarize these into the following number-theoretic question:

Of course, a complete answer of Question 2 should imply very clear understanding of periodic solutions of the spherical capillary water waves equation constructed using bifurcation analysis. But at this moment we are satisfied with existence, so we may pick any $\omega_{0}=a/b$ and $(j_{0},n_{0})$ that meets the requirement, for example the simplest case $n_{0}=6$, and any of the following choices of $\omega_{0}$ and $j_{0}$:

$$\omega_{0}=\sqrt{15},\,j_{0}=4;\quad\omega_{0}=\sqrt{\frac{1}{15}},\,j_{0}=60;\quad\omega_{0}=\sqrt{\frac{3}{5}},\,j_{0}=20;\quad\omega_{0}=\sqrt{\frac{5}{3}},\,j_{0}=12.$$

We thus refine our conjecture as follows:

The counterpart of Conjecture 2.2 for gravity capillary standing water waves was proved by Alazard-Baldi [AB15] using a Nash-Moser type theorem. The key technique in their proof was to find a conjugation of the linearized operator of the gravity capillary water waves system on $\mathbb{T}^{1}$ to an operator of the form

$$\omega\partial_{t}+iT+i\lambda_{1}|D_{x}|^{1/2}+i\lambda_{-1}|D_{x}|^{-1/2}+\text{Operator of order}\leq-\frac{3}{2},$$

where $T$ is an elliptic Fourier multiplier of order $3/2$, and $\lambda_{1},\lambda_{-1}$ are real constants. The frequency $\omega$ lives in a Cantor type set that clusters around a given frequency so that the kernel of the linearized operator is 1-dimensional. With this conjugation, they were able to find periodic solutions of linearized problems required by Nash-Moser iteration.

It is expected that this technique could be transplanted to the equation (2.4), since our analysis for (2.6) suggests that the 1-dimensional kernel requirement for bifurcation analysis is met. It seems that the greatest technical issue is to find a suitable conjugation that takes the linearized operator of (2.4) to an operator of the form

$$\omega\partial_{t}+i(T_{3/2}+T_{1/2}+T_{-1/2})+\text{Operator of order}\leq-\frac{3}{2},$$

where each $T_{k}$ is a real Fourier multiplier acting on spherical harmonics. The difficulty is that, since we are working with pseudo-differential operators on $S^{2}$, the formulas of symbolic calculus are not as neat as those on flat spaces. It seems necessary to implement some global harmonic analysis for compact homogeneous spaces, e.g. extension of results collected in Ruzhansky-Turunen [RT09]. Unfortunately, those results do not include pseudo-differential operators with “rough coefficients” and para-differential operators, so it seems necessary to re-write the whole theory.

As pointed out in Section 1, the system (2.3) is locally well-posed due to a result in [BG98]. General well-posedness results [CS07], [SZ08] for free boundary value problem of Euler equation also apply. As for lifespan estimate for initial data $\varepsilon$-close to the static solution (2.1), it should not be hard to conclude that the lifespan should be bounded below by $1/\varepsilon$. The result relates to the fact that the sphere is a stable critical point of the area functional, cf. [BC12]. This is nothing new: a suitable energy inequality should imply it. However, although the clue is clear, the implementation is far from standard since we are working on a compact manifold. A rigorous proof still calls for hard technicalities.

Now we will be looking into the nonlinear equation (2.3) for its longer time behavior. Although appearing similar to the well-studied water waves equation in e.g. [IP19], [Lan05], [Wu97], [Wu99], there is a crucial difference between the dispersive relation in (2.3) and the well-studied water waves equations: the dispersive relation exhibits a strong rigidity property, i.e. the arbitrary physical constants enter into the dispersive relation $\Lambda$ only as scaling factors. For the gravity-capillary water waves, the linear dispersive relation reads

$$\sqrt{g|\nabla|+\sigma|\nabla|^{3}},$$

where $g$ is the gravitational constant and $\sigma$ is the surface tension coefficient. For the Klein-Gordon equation on a Riemannian manifold, the linear dispersive relation reads

$$\sqrt{-\Delta+m^{2}},$$

where $m$ is the mass. In [IP19], Ionescu and Pusateri referred such dispersive relations as having non-degenerate dependence on physical parameter, while in our context it is appropriate to refer to the dependence as degenerate. We will see that this crucial difference brings about severe obstructions for the long-time well-posedness of the system.

Following the idea of Delort and Szeftel [DS04], we look for a normal form reduction of (2.3) and explain why the rigidity property could cause obstructions. Not surprisingly, the obstruction is due to resonances, and strongly relates to the solvablity of a Diophantine equation. Delort and Szeftel cast a normal form reduction to the small-initial-data problem of quasilinear Klein-Gordon equation on the sphere and obtained an estimate on the lifespan longer than the one provided by standard energy method. After their work, normal form reduction has been used by mathematicians to understand water waves on flat tori, for example [BD⁺18], [BFP22] and [IP19]. The idea was inspired by the normal form reduction method introduced by Shatah [Sha85]: for a quadratic perturbation of a linear dispersive equation

$$\partial_{t}u+iLu=N(u)=O(u^{\otimes 2}),$$

using a new variable $u+B(u,\bar{u})$ with a suitably chosen quadratic addendum $B(u,\bar{u})$ can possibly eliminate the quadratic part of $N$, thus extending the lifespan estimate beyond the standard $1/\varepsilon$.

So we shall write the quadraticr part of $\mathfrak{N}(u)$ as

$$\sum_{n_{3}\geq 0}\Pi^{(n_{3})}\left[\sum_{n_{1}\geq 0}\sum_{n_{2}\geq 0}\mathcal{M}_{1}\left(\Pi^{(n_{1})}u,\Pi^{(n_{2})}u\right)+\mathcal{M}_{2}\left(\Pi^{(n_{1})}u,\Pi^{(n_{2})}\bar{u}\right)+\mathcal{M}_{3}\left(\Pi^{(n_{1})}\bar{u},\Pi^{(n_{2})}\bar{u}\right)\right],$$

where $\mathcal{M}_{1},\mathcal{M}_{2},\mathcal{M}_{3}$ are complex bi-linear operators, following the argument of Section 4 in [DS04]. They are independent of $t$ since the right-hand-side of the equation does not depend on $t$ explicitly. Let’s look for a diffeomorphism

$$u\to v:=u+\mathbf{B}[u,u]$$

in the function space $C^{\infty}(S^{2})$, where $\mathbf{B}$ is a bilinear operator, so that the equation (2.3) with quadratic nonlinearity reduces to an equation with cubic nonlinearity. The $\mathbf{B}[u,u]$ is supposed to take the form $\mathbf{B}[u,u]=\mathbf{B}_{1}[u,u]+\mathbf{B}_{2}[u,u]+\mathbf{B}_{3}[u,u]$, with

$$\mathbf{B}_{1}[u,u]=\sum_{n_{3}\geq 0}\sum_{n_{1},n_{2}\geq 0}b_{1}(n_{1},n_{2},n_{3})\Pi^{(n_{3})}\mathcal{M}_{1}\left(\Pi^{(n_{1})}u,\Pi^{(n_{2})}u\right),$$

$$\mathbf{B}_{2}[u,u]=\sum_{n_{3}\geq 0}\sum_{n_{1},n_{2}\geq 0}b_{2}(n_{1},n_{2},n_{3})\Pi^{(n_{3})}\mathcal{M}_{2}\left(\Pi^{(n_{1})}u,\Pi^{(n_{2})}\bar{u}\right),$$

$$\mathbf{B}_{3}[u,u]=\sum_{n_{3}\geq 0}\sum_{n_{1},n_{2}\geq 0}b_{3}(n_{1},n_{2},n_{3})\Pi^{(n_{3})}\mathcal{M}_{3}\left(\Pi^{(n_{1})}\bar{u},\Pi^{(n_{2})}\bar{u}\right),$$

where the $b_{j}(n_{1},n_{2},n_{3})$’s are complex numbers to be determined. Implementing (2.3), we find

	$\displaystyle(\partial_{t}$	$\displaystyle+i\Lambda)(u+\mathbf{B}[u,u])$
		$\displaystyle=\mathfrak{N}(u)+\sum_{n_{3}\geq 0}\sum_{n_{1},n_{2}\geq 0}b_{1}(n_{1},n_{2},n_{3})\Pi^{(n_{3})}\mathcal{M}_{1}\left(\Pi^{(n_{1})}\partial_{t}u,\Pi^{(n_{2})}u\right)$
		$\displaystyle\quad+\sum_{n_{3}\geq 0}\sum_{n_{1},n_{2}\geq 0}b_{1}(n_{1},n_{2},n_{3})\Pi^{(n_{3})}\mathcal{M}_{1}\left(\Pi^{(n_{1})}u,\Pi^{(n_{2})}\partial_{t}u\right)+(\text{similar terms})$
		$\displaystyle\quad+\sum_{n_{3}\geq 0}\sum_{n_{1},n_{2}\geq 0}i\Lambda(n_{3})b_{1}(n_{1},n_{2},n_{3})\Pi^{(n_{3})}\mathcal{M}_{1}\left(\Pi^{(n_{1})}u,\Pi^{(n_{2})}u\right)$
		$\displaystyle=\mathfrak{N}(u)+\sum_{n_{3}\geq 0}\sum_{\min(n_{1},n_{2})\leq 1}i\left[\Lambda(n_{3})-\Lambda(n_{1})-\Lambda(n_{2})\right]b_{1}(n_{1},n_{2},n_{3})\Pi^{(n_{3})}\mathcal{M}_{1}\left(\Pi^{(n_{1})}u,\Pi^{(n_{2})}u\right)$
		$\displaystyle\quad+\sum_{n_{3}\geq 0}\sum_{n_{1},n_{2}\geq 2}i\left[\Lambda(n_{3})-\Lambda(n_{1})-\Lambda(n_{2})\right]b_{1}(n_{1},n_{2},n_{3})\Pi^{(n_{3})}\mathcal{M}_{1}\left(\Pi^{(n_{1})}u,\Pi^{(n_{2})}u\right)$
		$\displaystyle\quad+(\text{similar terms})+O(u^{\otimes 3}).$

We aim to eliminate most of the second order portions of $\mathfrak{N}(u)$. The coefficients $b_{j}(n_{1},n_{2},n_{3})$ are fixed as follows:

(2.7)		$\displaystyle b_{1}(n_{1},n_{2},n_{3})$	$\displaystyle=i\left[\Lambda(n_{3})-\Lambda(n_{1})-\Lambda(n_{2})\right]^{-1},\quad n_{1},n_{2},n_{3}\geq 2$
		$\displaystyle b_{2}(n_{1},n_{2},n_{3})$	$\displaystyle=i\left[\Lambda(n_{3})-\Lambda(n_{1})+\Lambda(n_{2})\right]^{-1},\quad n_{1},n_{2},n_{3}\geq 2$
		$\displaystyle b_{3}(n_{1},n_{2},n_{3})$	$\displaystyle=i\left[\Lambda(n_{3})+\Lambda(n_{1})+\Lambda(n_{2})\right]^{-1},\quad n_{1},n_{2},n_{3}\geq 2$
		$\displaystyle b_{1,2,3}(n_{1},n_{2},n_{3})$	$\displaystyle=0,\quad\text{if }\Lambda(n_{3})\pm\Lambda(n_{1})\pm\Lambda(n_{2})=0\text{ or }\min(n_{1},n_{2},n_{3})\leq 1,$

then a large portion of the second order part of $\Pi_{\geq 2}\mathfrak{N}\left(u\right)$ will be eliminated. In fact, for $n_{1},n_{2},n_{3}\geq 2$, if $\Lambda(n_{3})\pm\Lambda(n_{1})\pm\Lambda(n_{2})\neq 0$, then the term

$$\Pi^{(n_{3})}\left[\sum_{n_{1},n_{2}\geq 2}\mathcal{M}_{1}\left(\Pi^{(n_{1})}u,\Pi^{(n_{2})}u\right)+\mathcal{M}_{2}\left(\Pi^{(n_{1})}u,\Pi^{(n_{2})}\bar{u}\right)+\mathcal{M}_{3}\left(\Pi^{(n_{1})}\bar{u},\Pi^{(n_{2})}\bar{u}\right)\right]$$

is cancelled out. On the other hand, by the volume preservation equality (1.6) and conservation law (1.7), there holds $\Pi^{(0)}u=O(u^{\otimes 2})$, $\Pi^{(1)}u=O(u^{\otimes 2})$, so the low-low interaction

$$\Pi_{\geq 2}\left[\sum_{\min(n_{1},n_{2})\leq 1}\mathcal{M}_{1}\left(\Pi^{(n_{1})}u,\Pi^{(n_{2})}u\right)+\mathcal{M}_{2}\left(\Pi^{(n_{1})}u,\Pi^{(n_{2})}\bar{u}\right)+\mathcal{M}_{3}\left(\Pi^{(n_{1})}\bar{u},\Pi^{(n_{2})}\bar{u}\right)\right]$$

is automatically $O(u^{\otimes 3})$.