On the Energy Cascade of 3-Wave Kinetic Equations: Beyond Kolmogorov–Zakharov Solutions

Avy Soffer, Minh Binh Tran

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


In weak turbulence theory, the Kolmogorov–Zakharov spectra is a class of time-independent solutions to the kinetic wave equations. In this paper, we construct a new class of time-dependent isotropic solutions to the decaying turbulence problems (whose solutions are energy conserved), with general initial conditions. These solutions exhibit the interesting property that the energy is cascaded from small wavenumbers to large wavenumbers. We can prove that starting with a regular initial condition whose energy at the infinity wave number | p| = ∞ is 0, as time evolves, the energy is gradually accumulated at { | p| = ∞}. Finally, all the energy of the system is concentrated at { | p| = ∞} and the energy function becomes a Dirac function at infinity Eδ{ | p | = }, where E is the total energy. The existence of this class of solutions is, in some sense, the first complete rigorous mathematical proof based on the kinetic description for the energy cascade phenomenon for waves with quadratic nonlinearities. We only represent in this paper the analysis of the statistical description of acoustic waves (and equivalently capillary waves). However, our analysis works for other cases as well.

Original languageEnglish (US)
Pages (from-to)2229-2276
Number of pages48
JournalCommunications In Mathematical Physics
Issue number3
StatePublished - Jun 1 2020

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


Dive into the research topics of 'On the Energy Cascade of 3-Wave Kinetic Equations: Beyond Kolmogorov–Zakharov Solutions'. Together they form a unique fingerprint.

Cite this