No pure capillary solitary waves exist in 2D finite depth

Solitary water waves are localized disturbances of a fluid surface which travel at constant speed and with a fixed profile. Such waves were first observed by Russell in the mid-19th century [41], and are fundamental features of many water wave models. The objective of this paper is to settle the existence/non-existence problem for the full irrotational water wave system in 2D, with the physical parameters being gravity, surface tension, and the fluid depth. Five of the six combinations have already been dealt with, and the results are summarized in Table 1 - it is our intent to fill in the missing case.

In a nutshell, our result can be loosely formulated as follows:

A more precise formulation of the result is given later, in Theorem 4.1.

We consider the incompressible, finite depth water wave equations in two space dimensions. The motion of the water is governed by the incompressible Euler equations, with boundary conditions on the water surface and the flat, finite bottom. We emphasize that this section is purely for motivational purposes, and is not the formulation we will use to prove our non-existence result. In particular, for simplicity, this subsection assumes that $\Gamma(t)$ is a graph, but we will not assume this when working with the holomorphic formulation of our problem.

To describe the equations, denote the water domain at time $t$ by $\Omega(t)\subseteq\mathbb{R}^{2}$; we assume that $\Omega(t)$ has a flat finite bottom $\{y=-h\}$, and let $\eta(x,t)$ denote the height of the free surface as a function of the horizontal coordinate:

The free surface of the water at time $t$ will be denoted by $\Gamma(t)$. As we are interested in solitary waves, we think of $\Gamma(t)$ as being asymptotically flat at infinity to $y\approx 0$. Since the 2D finite depth capillary water wave equations do permit periodic travelling waves, this decay at infinity will factor heavily into our proof, even though we do not impose any specific rate of decay.

We denote by $u$ the fluid velocity and by $p$ the pressure. The vector field $u$ solves Euler’s equations inside $\Omega(t),$

and the bottom boundary is impenetrable:

On the upper boundary the atmospheric pressure is normalized to zero and we have the dynamic boundary condition

and the kinematic boundary condition

Here $g\geq 0$ represents the gravity,

is the mean curvature of the free boundary, and $\sigma>0$ represents the surface tension coefficient.

We adhere to the classical assumption that the flow is irrotational, so we can write $u$ in terms of a velocity potential $\phi$ as $u=\nabla\phi$. It is easy to see that $\phi$ is a harmonic function whose normal derivative is zero on the bottom. Thus, $\phi$ is determined by its trace $\psi=\phi|_{\Gamma(t)}$ on the free boundary $\Gamma(t).$ Under these assumptions, it is well-known that the fluid dynamics can be expressed in terms of a one-dimensional evolution of the pair of variables $(\eta,\psi)$ via:

Here $G$ denotes the Dirichlet to Neumann map associated to the fluid domain. This operator is one of the main analytical obstacles in this formulation of the problem, and in the next subsection we briefly discuss a change of coordinates that somewhat simplifies the analysis.

We now write down the solitary wave equations. We begin with the equations (2.1)-(2.6) as well as the irrotationality condition, and assume that the profile is uniformly translating in the horizontal direction with velocity $c$, i.e., $\phi(x,y,t)=\phi_{0}(x-ct,y)$, $\eta(x,y,t)=\eta_{0}(x-ct,y),$ and $p(x,y,t)=p_{0}(x-ct,y).$ This gives the steady water wave equations. To get to solitary waves (as opposed to, say, periodic waves), we impose some averaged decay on $\eta_{0}$ and $u_{0}$, so that in the far-field the water levels out and is essentially still. Contrary to many works which use a frame of reference travelling with the localized disturbance, we choose a frame so that the fluid is at rest near infinity. This allows us to set to zero the integration constant in the Bernoulli equation; the price to pay is that there are terms with $c$ in the equations below.

We are thus interested in states $(\eta,\phi)$ satisfying the following equations:

We prove that the above equations admit no non-trivial solutions, with appropriate (averaged) decay at infinity. Such a claim, of course, presupposes certain regularity requirements on the solutions, but this will not play a major role due to ellipticity. Indeed, the above system can be shown to be locally elliptic whenever $(\eta,\phi)$ is above critical regularity, which corresponds to $\eta\in~{}H^{\frac{3}{2}+}_{loc}$.

As mentioned, one of the main difficulties of (2.7) is the presence of the Dirichlet to Neumann operator $G(\eta)$, which depends on the free boundary. For this reason, we will reformulate the equations in holomorphic coordinates, which, in some sense, diagonalizes $G(\eta)$. We will only highlight briefly the procedure of changing coordinates; full details can be found in [23]. Moreover, although (2.7) assumes that $\Gamma(t)$ is a graph, the formulation below does not require this, which is another advantage of this approach.

The conditions we require on $\Gamma(t)$ are the same (or weaker, see the discussion below) as those listed in Section 2.3 of [23]; namely, that $\Gamma(t)$ can be parametrized to have sufficient Sobolev regularity, has no degeneracies or self-intersections, and never touches the bottom boundary. These assumptions are used in [23, Theorem 3] to justify the existence of the conformal map we refer to below.

In the holomorphic setting, the coordinates are denoted by $\alpha+i\beta\in S:=\mathbb{R}\times(-h,0)$, and the fluid domain is parameterized by the conformal map

which takes the bottom $\mathbb{R}-ih$ into the bottom, and the top $\mathbb{R}$ into the top $\Gamma(t)$. The restriction of this map to the real line is denoted by $Z$, i.e., $Z(\alpha):=z(\alpha-i0)$, and can be viewed as a parametrization of the free boundary $\Gamma(t)$. We will work with the variables $W(\alpha)=Z(\alpha)-\alpha$, and the trace $Q(\alpha)$ of the holomorphic velocity potential on the free surface. $W$ and $Q$ are traditionally called holomorphic functions, which in this terminology means that they can be realized as the trace on the upper boundary $\beta=0$ of holomorphic functions in the strip $S$ which are purely real on the lower boundary $\beta=-h$. The space of holomorphic functions is a real algebra, but is not a complex algebra.

In terms of regularity, we note that the existence of the conformal map is guaranteed by the Riemann Mapping Theorem for any simply connected fluid domain. In order to have an equivalence between Sobolev norms, it suffices to assume that the free surface $\Gamma$ has critical Besov regularity $B^{\frac{3}{2}}_{2,1}$. This, in particular, guarantees that $\Gamma$ is a graph outside of a compact set. The conformal map, then, has the matching property $\Im(W)\in B^{\frac{3}{2}}_{2,1}$, and in particular $\Im(W)$ and $W_{\alpha}$ are bounded. For more details we refer the reader to both [23, Section 2] and the stronger results in [2], as well as the more general local results of [37].

The two-dimensional finite depth gravity-capillary water wave equations in holomorphic coordinates can be written as follows:

where

and

As before, $g$ and $\sigma$ are non-negative parameters, at least one of which is non-zero. $\mathcal{T}_{h}$ denotes the Tilbert transform, which is the Fourier multiplier with symbol $-i\tanh(h\xi)$, and arises in order to characterize what it means to be a holomorphic function. Precisely, holomorphic functions are described by the relation

It is important to note that the Tilbert transform takes real-valued functions to real-valued functions, and satisfies the following product rule:

Finally, $\textbf{P}_{h}$ is the projection onto the space of holomorphic functions. In terms of $\mathcal{T}_{h}$ it can be written as

In the case of no surface tension, equations (3.1) were derived in [23]. We begin with a brief outline of how the surface tension term arises, as we are particularly interested in the case when $g=0$ and $\sigma>0$.

Following [23], we arrive at the Bernoulli equation

We then evaluate this equation on the top boundary and apply the dynamic boundary condition to replace $p$ by $-\sigma\textbf{H}.$ We then pass to the strip $S$ - so the equations are now defined on $\{\beta=0\}$ - rewrite the equations in terms of the holomorphic variables, clear common factors of $2$, and project. Running this procedure explicitly for the term with $\sigma$, we begin by parameterizing $\Gamma(t)$ by, say, $s\mapsto(\gamma_{1}(s),\gamma_{2}(s))$ and write $-\sigma\textbf{H}$ in the standard parametric way. We then use the relations

and formal calculations to write the capillary expression in terms of the holomorphic variables as:

which after projecting gives us the capillary term in (3.1).

We are now able to state our main theorem. The result is stated in the low regularity function space $B^{\frac{1}{2}}_{2,1}$ defined above. However, part of the proof involves upgrading potential solutions to sufficient regularity to justify basic computations. Comparing with the infinite depth results in [26], our function space requires more regularity for $W_{\alpha}$ at low frequency, but this is to be expected, as the same happens in the dynamic problem [23]. From a technical standpoint, the issue is that $\mathcal{T}_{h}^{-1}$ does not have good mapping properties (it is not even bounded on $L^{2}$) compared to the Hilbert transform, which satisfies $H^{-1}=-H.$ For justification of the other assumption - and conclusion - of 4.1, recall 3.1 and 3.2.