On the Energy Transfer toward Large Values of Wavenumbers for Solutions of 4-Wave Kinetic Equations

Abstract.

Inspired by the fundamental work of Escobedo and Velázquez [ Invent. Math., 200 (2015), pp. 761–847; Mem. Amer. Math. Soc., 238 (2015), 1124],   we prove that solutions of 4-wave kinetic equations, under very general forms of the dispersion relations, exhibit energy transfer toward large wavenumbers as time evolves. We also establish a global existence result for mild solutions of the equation under general dispersion relations. To the best of our knowledge, this is the first study on the long-term asymptotic behavior of solutions to 4-wave kinetic equations in such a general setting.