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

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Abstract

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 \(|p|=\infty \) is 0, as time evolves, the energy is gradually accumulated at \(\{|p|=\infty \}\). Finally, all the energy of the system is concentrated at \(\{|p|=\infty \}\) and the energy function becomes a Dirac function at infinity \(E\delta _{\{|p|=\infty \}}\), 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.

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Acknowledgements

A. Soffer is partially supported by NSF grant DMS 1600749 and NSFC 11671163. M.-B. Tran is partially supported by NSF Grant DMS-1814149 and NSF Grant DMS-1854453. The authors would like to thank Prof. A. Newell, Prof. A. Aceves, Prof R. Pego, Prof. B. Rumpf, Prof H. Spohn for fruitful discussions on the topic.

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  1. Mathematics Department, Rutgers University, New Brunswick, NJ, 08903, USA

    Avy Soffer

  2. Department of Mathematics, Southern Methodist University, Dallas, TX, 75275, USA

    Minh-Binh Tran

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Soffer, A., Tran, MB. On the Energy Cascade of 3-Wave Kinetic Equations: Beyond Kolmogorov–Zakharov Solutions. Commun. Math. Phys. 376, 2229–2276 (2020). https://doi.org/10.1007/s00220-019-03651-w

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