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

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| = ∞ 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.

Onset of the wave turbulence description of the longtime behavior of the nonlinear Schrödinger equation

“…The analysis of (WKE) is full of very interesting questions, see [16,22,34] for recent developments, but we will focus here on the problem of its rigorous derivation. Several partial or heuristic derivations have been put forward for (WKE), or the closely related quantum Boltzmann equations [1][2][3]10,13,17,28,30,36].…”

Spectral method for the Boltzmann equation for gases with viscosity

“…hương trình Boltzmann mô tả chuyển động của các phân tử của một chất khí trong đó sự tương tác của các phân tử là va chạm đàn hồi nhị phân (xem [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]). Việc xây dựng các phương pháp tính cho phương trình Boltzmann có tầm quan trọng trong nhiều ứng dụng, từ động lực học khí hiếm (RGD) [27], vật lý plasma [28], dòng chảy dạng hạt [17,18], chất bán dẫn [32], và lý thuyết động lượng tử [29].…”

A Numerical Scheme for Wave Turbulence: 3-Wave Kinetic Equations

“…A numerical method for solving isotropic 3-wave kinetic equation under the presence of forcing and dissipation was also introduced. In [47], Soffer and Tran show that, the solutions of (3), with γ " 2, exhibit the property that the energy is cascaded from small wavenumbers to large wavenumbers. They show that starting with a regular initial condition whose energy at the infinity wave number ω " 8 is 0, as time evolves, the energy is gradually accumulated at tω " 8u.…”

“…To be more precise, let us define the energy of the solution (3) as gpt, ωq " ωf pt, ωq. It has been proved in [47] that g can be decomposed into two parts gpt, ωq " ḡpt, ωq `gptqδ ω"8 ,…”