Full derivation of the wave kinetic equation

• 成果类型: Article
• 署名作者: Deng, Yu; Hani, Zaher
• 署名单位: University of Southern California; University of Michigan System; University of Michigan
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-023-01189-2
• 发表日期: 2023
• 页码: 543-724
• 关键词: nonlinear schrodinger-equation; linear energy transfer; sobolev norms; boltzmann-equation; GROWTH; nls; diffusion; spectrum; bounds; forms

摘要

We provide the rigorous derivation of the wave kinetic equation from the cubic nonlinear Schrodinger (NLS) equation at the kinetic timescale, under a particular scaling law that describes the limiting process. This solves a main conjecture in the theory of wave turbulence, i.e. the kinetic theory of nonlinear wave systems. Our result is the wave analog of Lanford's theorem on the derivation of the Boltzmann kinetic equation from particle systems, where in both cases one takes the thermodynamic limit as the size of the system diverges to infinity, and as the interaction strength of waves/radius of particles vanishes to 0, according to a particular scaling law (Boltzmann-Grad in the particle case). More precisely, in dimensions d >= 3, we consider the (NLS) equation in a large box of size L with a weak nonlinearity of strength alpha. In the limit L -> infinity and alpha -> 0, under the scaling law alpha similar to L-1, we show that the long-time behavior of (NLS) is statistically described by the wave kinetic equation, with well justified approximation, up to times that are O(1) (i.e. independent of L and alpha) multiples of the kinetic timescale T-kin similar to alpha(-2). This is the first result of its kind for any nonlinear dispersive system.