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<p>We analyse a 3-wave kinetic equation, derived from the elastic beam wave equation on the lattice. The ergodicity condition states that two distinct wavevectors are supposed to be connected by a finite number of collisions. In this work, we prove that the ergodicity condition is violated and the equation domain is broken into disconnected domains, called no-collision and collisional invariant regions. If one starts with a general initial condition, whose energy is finite, then in the long-time limit, the solutions of the 3-wave kinetic equation remain unchanged on the no-collision region and relax to local equilibria on the disjoint collisional invariant regions. To our best knowledge, this is the first time that the violation of the ergodicity condition is observed and proved for a kinetic equation.</p>

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On the wave turbulence theory: ergodicity for the elastic beam wave equation

  • 期刊论文

  • 发表日期：

    2025年4月5日

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  • Benno Rumpf,
  • Avy Soffer,
  • Minh-Binh Tran

摘要

We analyse a 3-wave kinetic equation, derived from the elastic beam wave equation on the lattice. The ergodicity condition states that two distinct wavevectors are supposed to be connected by a finite number of collisions. In this work, we prove that the ergodicity condition is violated and the equation domain is broken into disconnected domains, called no-collision and collisional invariant regions. If one starts with a general initial condition, whose energy is finite, then in the long-time limit, the solutions of the 3-wave kinetic equation remain unchanged on the no-collision region and relax to local equilibria on the disjoint collisional invariant regions. To our best knowledge, this is the first time that the violation of the ergodicity condition is observed and proved for a kinetic equation.