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Abstract Inspired by the pioneering work of Escobedo and Velazquez [30, 31], we develop a new framework to prove that solutions of 4-wave kinetic equations, under very general forms of dispersion relations, develop a condensation at the origin in finite time, under weaker conditions on the initial data than the ones considered in [30, 31]. We also provide some estimates on the non-condensation times of the solutions. 

ABSTRACT In this work, we study a three‐wave kinetic equation with resonance broadening arising from the theory of stratified ocean flows. Unlike Gamba et al. [Mathematical Models and Methods in Applied Sciences30, no. 1 (2020): 105–137], we employ a different formulation of the resonance broadening, which makes the present model more suitable for ocean applications. We establish the global existence and uniqueness of strong solutions to the new resonance broadening kinetic equation. 

This article introduces a novel numerical approach, based on finite-volume techniques, for studying fully nonlinear coagulation–fragmentation models, where both the coagulation and fragmentation components of the collision operator are nonlinear. The models come from three-wave kinetic equations, a pivotal framework in wave turbulence theory. Despite the importance of wave turbulence theory in physics and mechanics, there have been very few numerical schemes for three-wave kinetic equations, in which no additional assumptions are manually imposed on the evolution of the solutions, and the current manuscript provides one of the first of such schemes. To the best of our knowledge, this also is the first numerical scheme capable of accurately capturing the long-term asymptotic behaviour of solutions to a fully nonlinear coagulation–fragmentation model. The scheme is implemented on some test problems, demonstrating strong alignment with theoretical predictions of energy cascade rates, rigorously obtained in the work (Soffer & Tran. 2020Commun. Math. Phys.376, 2229–2276. (doi:10.1007/BF01419532)). We further introduce a weighted finite-volume variant to ensure energy conservation across varying degrees of kernel homogeneity. Convergence and first-order consistency are established through theoretical analysis and verified by experimental convergence orders in test cases.