Chapter 9: Wave turbulence theory for capillary water waves

This post is based on my joint paper with Toan Nguyen.

Although many studies have been carried on to understand the Hasselmann-Zakharov weak turbulence equation for capillary waves since its derivation in the 60’s, the question about the existence and uniqueness of solutions to the equation still remains unanswered, due to the complexity of the equation. Our work provides a solution to the problem.

146886-004-29A7BC32 — Figure 1: Capillary Waves

First of all, let us review the derivation of the weak (wave) turbulence kinetic equation of capillary waves. We first recall that the law of wave dispersion on the surface of an infinite deep liquid is of the form $\omega_k=(\alpha |k|^3+g|k|)^\frac12$ , in which $k$ is the wave number, $\alpha$ is the coefficient of the surface tension and $g$ is the gravity constant. If we neglect the effect of the gravity, $\omega_k=(\alpha |k|^3)^\frac12$ . The water wave system that governs the vibrations of the surface of a liquid, in which the viscosity is omitted, reads

$\Delta \Phi(t,x,y,z) \ = \ 0, \mbox{ for } z<\zeta(t,x,y), (t,x,y,z)\in\mathbb{R}_+\times\mathbb{R}^3,$

$\zeta_t(t,x,y)-\Phi_z(t,x,y,z) \ = \ -\zeta_x(t,x,y)\Phi_x(t,x,y,z) -\zeta_y(t,x,y)\Phi_y(t,x,y,z)\Big{|}_{z=\zeta},$

$\Phi_t(t,x,y,z) -\alpha (\zeta_{xx}(t,x,y)+\zeta_{yy}(t,x,y)) \ = \ \frac{|\nabla\Phi(t,x,y,z)|^2}{2}\Big{|}_{z=\zeta},$

$\Phi(t,x,y,z)\Big{|}_{z=-\infty}\ = \ 0,$

where $\Phi(t,x,y,z)$ is the velocity potential, $\zeta(t,x,y)$ is the deviation of the surface from equilibrium. The $z$ axis is directed away from the liquid and we set the pressure to be $0$ .

Denote by $\hat\zeta(t,k)$ and $\hat\Phi(t,k)$ (or $\hat\zeta_k$ and $\hat\Phi_k$ ) the Fourier transform of $\zeta(t,x,y)$ and $\Phi(t,x,y,\zeta(t,x,y))$ in $(x,y)$ and make use the smallness of the nonlinearity, keeping in the Fourier series terms up to second order of smallness of the amplitude of the vibrations

$\partial_t\hat\zeta(t,k) - k \hat\Phi(t,k) \ = \ \iint_{\mathbb{R}^2\times\mathbb{R}^2}\delta(k-k_1-k_2)[k\cdot k_1-|k||k_1|]\hat\Phi(t,k_1)\hat\zeta(t,k_2)dk_1dk_2,$

$\partial_t\hat\Phi(t,k)+\alpha|k|^2\zeta(t,k) \ = \ \iint_{\mathbb{R}^2\times\mathbb{R}^2}\delta(k-k_1-k_2)[k\cdot k_1+|k||k_1|]\hat\Phi(t,k_1)\hat\Phi(t,k_2)dk_1dk_2.$

Now, let us define the new variables $a_k$ and $a^*_k$ , which describe the complex amplitudes of waves

$\hat{\zeta}_k = \left(\frac{4}{\alpha |k|}\right)^\frac14 (a_k+a_{-k}^*), \ \ \ \hat\Phi_k=-i(4\alpha|k|)^\frac14(a_k-a_k^*).$

The equation for capillary waves in terms of the new variables then follows

$\partial_t a_k -i\omega_ka_k \ = \ i \iint_{\mathbb{R}^2\times\mathbb{R}^2}\delta(k-k_1-k_2)V_{k,k_1,k_2}a_{k_1}a_{k_2}dk_1dk_2$

$+ 2i \iint_{\mathbb{R}^2\times\mathbb{R}^2}\delta(k-k_1-k_2)V_{k_1,k,k_2}a_{k_1}a_{k_2}^*dk_1dk_2$

$+ i \iint_{\mathbb{R}^2\times\mathbb{R}^2}\delta(k+k_1+k_2)U_{k,k_1,k_2}a_{k_1}a_{k_2}^*dk_1dk_2$

where

$V_{k,k_1,k_2} \ = \frac{1}{8\pi \sqrt{2\alpha}} \sqrt{\omega_k \omega_{k_1} \omega_{k_2}} ( \frac{L_{k_1,k_2}}{ |k| \sqrt{|k_1| |k_2|}} - \frac{L_{k,-k_1}}{ |k_2| \sqrt{|k| |k_1|}} - \frac{L_{k,-k_2}}{ |k_1| \sqrt{|k| |k_2|}})$

with $L_{k_1,k_2} = k_1 \cdot k_2 +|k_1| |k_2|.$

We omit the form of $U_{k,k_1,k_2}$ .

Setting $f(t,k)=|a_k|^2$ , we obtain the weak (wave) turbulence equation for capillary waves (Hasselmann’62, Zakharov’65)

$\partial_tf + 2 \nu |k|^2 f \ = \ Q[f]$

where $\nu$ denotes the positive coefficient of fluid viscosity, and $Q[f]$ is the collision term, describing pure resonant three-wave interactions. The equation is a three-wave kinetic one, in which the collision operator is of the form

$Q[f](k) \ = \ \iint_{\mathbb{R}^{2d}} \Big[ R_{k,k_1,k_2}[f] - R_{k_1,k,k_2}[f] - R_{k_2,k,k_1}[f] \Big] dk_1dk_2$

with

$R_{k,k_1,k_2} [f]:= 4\pi |V_{k,k_1,k_2}|^2\delta(k-k_1-k_2)\delta(\mathcal{E}_k -\mathcal{E}_{k_1}-\mathcal{E}_{k_2})(f_1f_2-ff_1-ff_2).$

We have used the short-hand notation $f = f(t,k)$ and $f_j = f(t,k_j)$ .

According to the weak turbulence theory, the equation admits nontrivial equilibria $Q[f_\infty] =0$ , called the Kolmogorov-Zakharov’s spectra: $f_\infty(k) \approx C|k|^{-\frac{17}{4}}.$ Moreover, such a solution can be interpreted as a universal spectrum in the region of transparency. These solutions are the analogs of the familiar Kolmogorov energy spectrum prediction $C|k|^{-\frac{5}{3}}$ of hydrodynamic turbulence.

Let us mention that this kinetic wave equation has a very similar structure with the quantum Boltzmann equation that describes the evolution of the excitations in a trapped Bose gas system, in which the temperature of the gas is below the Bose-Einstein condensate transition temperature. The collision operator that describes the interaction between excitations and condensates in the quantum Boltzmann equation reads

$C[f](k) \ = \ \iint_{\mathbb{R}^{2d}} \Big[ \bar{R}_{k,k_1,k_2}[f] - \bar{R}_{k_1,k,k_2}[f] - \bar{R}_{k_2,k,k_1}[f] \Big] dk_1dk_2$

with

$\bar{R}_{k,k_1,k_2} [f]:= |\bar{V}_{k,k_1,k_2}|^2\delta(k-k_1-k_2)\delta(\bar{\mathcal{E}}_k -\bar{\mathcal{E}}_{k_1}-\bar{\mathcal{E}}_{k_2})(f_1f_2-ff_1-ff_2-f)$

and $|\bar{V}_{k,k_1,k_2}|^2=\mathcal{C}^*|k||k_1||k_2|$ , $\bar{\mathcal{E}}_k=\sqrt{\kappa_1 |k|^2 + \kappa_2 |k|^4}$ for some positive constants $\mathcal{C}^*,$ $\kappa_1,$ $\kappa_2$ .

In this paper, we develop new techniques, inspired by our recent works on quantum kinetic theory, to give an answer to the question on the global existence and uniqueness of solutions to the equation.

References for the picture:

[1] https://www.britannica.com/science/capillary-wave#ref49830

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