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On a wave kinetic equation with resonance broadening in oceanography and atmospheric sciences

Young Ho Kim, Yuri V. Lvov, Leslie M. Smith, Minh-Binh Tran

TL;DR

The paper develops a resonance-broadened three-wave kinetic equation for stratified oceanic flows and proves global existence and uniqueness of strong solutions in $L^1_m(\mathbb{R}^d)$. It introduces a near-resonant broadening operator with a Lorentzian (Lorentzian) kernel, derives a weak formulation, and establishes key estimates: propagation of moments, bounds on the gain and loss terms, Hölder continuity of the collision operator, and subtangent and one-sided-Lipschitz properties. The analysis combines an abstract Banach-space ODE framework with detailed collision-operator estimates to obtain a global well-posedness result, providing a rigorous foundation for ocean-relevant near-resonant wave turbulence models. This approach moves beyond diffusion-limit approximations and aims to capture energy transfer in stratified fluids more accurately for oceanographic applications.

Abstract

In this work, we study a three-wave kinetic equation with resonance broadening arising from the theory of stratified ocean flows. Unlike Gamba-Smith-Tran(On the wave turbulence theory for stratified flows in the ocean, Math. Models Methods Appl. Sci. 30 (2020), no.1, 105--137), we employ a different formulation of the resonance broadening, which makes the present model more suitable for ocean applications. We establish the global existence and uniqueness of strong solutions to the new resonance broadening kinetic equation.

On a wave kinetic equation with resonance broadening in oceanography and atmospheric sciences

TL;DR

The paper develops a resonance-broadened three-wave kinetic equation for stratified oceanic flows and proves global existence and uniqueness of strong solutions in . It introduces a near-resonant broadening operator with a Lorentzian (Lorentzian) kernel, derives a weak formulation, and establishes key estimates: propagation of moments, bounds on the gain and loss terms, Hölder continuity of the collision operator, and subtangent and one-sided-Lipschitz properties. The analysis combines an abstract Banach-space ODE framework with detailed collision-operator estimates to obtain a global well-posedness result, providing a rigorous foundation for ocean-relevant near-resonant wave turbulence models. This approach moves beyond diffusion-limit approximations and aims to capture energy transfer in stratified fluids more accurately for oceanographic applications.

Abstract

In this work, we study a three-wave kinetic equation with resonance broadening arising from the theory of stratified ocean flows. Unlike Gamba-Smith-Tran(On the wave turbulence theory for stratified flows in the ocean, Math. Models Methods Appl. Sci. 30 (2020), no.1, 105--137), we employ a different formulation of the resonance broadening, which makes the present model more suitable for ocean applications. We establish the global existence and uniqueness of strong solutions to the new resonance broadening kinetic equation.

Paper Structure

This paper contains 10 sections, 8 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. A Preliminary estimate and estimates of $\mathbb{C}_{\mathrm{gain}}$
  3. $L^1_m$ $(m\geq 0)$ estimates
  4. Bounds of the solution
  5. Estimates for $\mathbb{C}[f]$
  6. Proof of Theorem \ref{['Theorem:Main']}
  7. Condition $(\mathscr{B})$: Subtangent condition
  8. Condition $(\mathscr{C})$: One-side Lipschitz condition
  9. Proof of Theorem \ref{['Theorem:ODE']}
  10. Conclusion

Key Result

Theorem 1

Let $N>0$, $\gamma > 2$, $T>0$, and let for some $\varsigma > 1$, where $B_*(O,\varsigma)$ denotes the ball in $L^1_{m+3}(\mathbb{R}^d)$ centered at $O$ with radii $\varsigma$. Then the weak turbulence equation WeakTurbulenceInitial admits a unique strong solution $f(t,{\bf{p}})$ such that Moreover, $f(t,{\bf{p}})\in \Omega_T$ for all $t\in [0,T)$. Since $T$ can be chosen arbitrarily large, the

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Proposition 5
  • proof : Proof of Proposition \ref{['Propo:MomentsPropa']}
  • Proposition 6
  • proof
  • ...and 4 more