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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. 

In the second part of our work, we aim to establish the global-in-time well-posedness of classical solution of the master equations associated with general mean field games studied in Part I, which is beyond the specific linear-quadratic setting, provided the mean field sensitivity effect is not too large. We characterize the gradient of the value function by the backward process of the forward-backward stochastic differential equations (FBSDEs) introduced in Part I. Then we study the higher regularity of Jacobian flows of the FBSDEs in the state and measure variables so as to establish classical well-posedness of the master equation on Rd. As far as we know, it is the first work to investigate the master equations, with general cost functions having quadratic growth and allowing non-convexity in the state variable, under the small mean field effect. Our current approach directly imposes the structural assumptions (most notably, the small mean field sensitivity effect) on the cost functions, which provides the following advantages: (i) the structural conditions imposed in this work are easily verified and less demanding on the assumptions of the cost functions; (ii) we illustrate how the displacement monotonicity should be formulated when the assumptions are imposed on the cost functions instead of the Hamiltonian; and (iii) we provide an accurate lifespan, which may not be that small in many circumstances, for the local-in-time existence when the mean field sensitivity effect is relatively large, the cost functions are not convex in the state variable or we do not have the monotonicity of cost functions. 

The primary objective of this article is to present a general framework for users and applications of the master equation approach in extended mean field type control, for- mulated with a McKean-Vlasov stochastic differential equation that depends on the law of both the control and state variables. This control problem has recently gained significant attention and has been extensively studied at the level of the Bellman equa- tion. Here, we extend the analysis to the master equation and derive the corresponding Hamilton-Jacobi-Bellman equation. A key novelty of our approach is that we do not directly rely on the Fokker-Planck equation, which surprisingly leads to a significant simplification. We provide a concise theoretical presentation with proofs, as the stan- dard theory of stochastic control is not directly applicable. In the current work, the solution is constructed using an ansatz-based approach to dynamic programming via the master equation.We illustrate this method with a practical example. All proofs are presented in a self-contained manner. This paper offers a structured presentation of the extended mean field type control problem, serving as a valuable toolbox for users who are less focused on mathematical intricacies but seek a general framework for application.