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This article presents the variant of the approach introduced in the recent work of Bensoussan, Wong, Yam and Yuan [13] to the generic first-order mean field game problem. A major contribution here is the provision of new crucial a priori estimates, whose establishment is fundamentally different from the mentioned work since the associated forward-backward ordinary differential equation (FBODE) system is notably different. In addition, we require monotonicity conditions intimately on the coefficient functions but not on the Hamiltonians to handle their non-separable nature and nonlinear dynamics; as tackling Hamiltonians directly, it potentially dissolves much useful information. Compared with the assumptions used in [13], we introduce an additional requirement that the first-order derivative of the drift function in the measure variable cannot be too large relative to the convexity of the running cost function; this requirement only arises when the Hamiltonian is non-separable, and this phenomenon can also be seen in the existing literature. On the other hand, we require less here for the second-order differentiability of the coefficient functions in comparison to that in [13]. Our approach involves first demonstrating the local existence of a solution over small time interval, followed by the provision of new crucial a priori estimates for the sensitivity of the backward equation with respect to the initial condition of forward dynamics; and finally, smoothly gluing the local solutions together to form a global solution. In addition, we establish the local and global existence and uniqueness of classical solutions for the mean field game and its master equation., Comment: Both this work and arXiv:2305.11848 apply a similar approach to investigate the global-in-time well-posedness of two significant problems in mean field theory, namely mean field games in this work, and the mean field type control problems in arXiv:2305.11848

In this article, by using several new crucial {\it a priori} estimates which are still absent in the literature, we provide a comprehensive resolution of the first order generic mean field type control problems and also establish the global-in-time classical solutions of their Bellman and master equations. Rather than developing the analytical approach via tackling the Bellman and master equation directly, we apply the maximum principle approach by considering the induced forward-backward ordinary differential equation (FBODE) system; indeed, we first show the local-in-time unique existence of the solution of the FBODE system for a variety of terminal data by Banach fixed point argument, and then provide crucial a priori estimates of bounding the sensitivity of the terminal data for the backward equation by utilizing a monotonicity condition that can be deduced from the positive definiteness of the Schur complement of the Hessian matrix of the Lagrangian in the lifted version and manipulating first order condition appropriately; this uniform bound over the whole planning horizon $[0,T]$ allows us to partition $[0,T]$ into a number of sub-intervals with a common small length and then glue the consecutive local-in-time solutions together to form the unique global-in-time solution of the FBODE system. The regularity of the global-in-time solution follows from that of the local ones due to the regularity assumptions on the coefficient functions. Moreover, the regularity of the value function will also be shown with the aid of the regularity of the solution couple of the FBODE system and the regularity assumptions on the coefficient functions, with which we can further deduce that this value function and its linear functional derivative satisfy the Bellman and master equations, respectively., Comment: This new version corrected some typos with further clarifications after comments received

In this paper, we investigate steady inviscid compressible flows with radial symmetry in an annulus. The major concerns are transonic flows with or without shocks. One of the main motivations is to elucidate the role played by the angular velocity in the structure of steady inviscid compressible flows. We give a complete classification of flow patterns in terms of boundary conditions at the inner and outer circle. Due to the nonzero angular velocity, many new flow patterns will appear. There exists accelerating or decelerating smooth transonic flows in an annulus satisfying one side boundary conditions at the inner or outer circle with all sonic points being nonexceptional and noncharacteristically degenerate. More importantly, it is found that besides the well-known supersonic-subsonic shock in a divergent nozzle as in the case without angular velocity, there exists a supersonic-supersonic shock solution, where the downstream state may change smoothly from supersonic to subsonic. Furthermore, there exists a supersonic-sonic shock solution where the shock circle and the sonic circle coincide, which is new and interesting., Comment: 22 pages

This paper concerns the structural stability of smooth cylindrically symmetric transonic flows in a concentric cylinder. Both cylindrical and axi-symmetric perturbations are considered. The governing system here is of mixed elliptic-hyperbolic and changes type and the suitable formulation of boundary conditions at the boundaries is of great importance. First, we establish the existence and uniqueness of smooth cylindrical transonic spiral solutions with nonzero angular velocity and vorticity which are close to the background transonic flow with small perturbations of the Bernoulli's function and the entropy at the outer cylinder and the flow angles at both the inner and outer cylinders independent of the symmetric axis, and it is shown that in this case, the sonic points of the flow are nonexceptional and noncharacteristically degenerate, and form a cylindrical surface. Second, we also prove the existence and uniqueness of axi-symmetric smooth transonic rotational flows which are adjacent to the background transonic flow, whose sonic points form an axi-symmetric surface. The key elements in our analysis are to utilize the deformation-curl decomposition for the steady Euler system introduced in \cite{WengXin19} to deal with the hyperbolicity in subsonic regions and to find an appropriate multiplier for the linearized second order mixed type equations which are crucial to identify the suitable boundary conditions and to yield the important basic energy estimates., Comment: 39 pages