1,720,975 research outputs found
Maximal Lq -Regularity for Parabolic Hamilton–Jacobi Equations and Applications to Mean Field Games
Full text linkIn this paper we investigate maximal Lq-regularity for time-dependent viscous Hamilton–Jacobi equations with unbounded right-hand side and superlinear growth in the gradient. Our approach is based on the interplay between new integral and Hölder estimates, interpolation inequalities, and parabolic regularity for linear equations. These estimates are obtained via a duality method à la Evans. This sheds new light on the parabolic counterpart of a conjecture by P.-L. Lions on maximal regularity for Hamilton–Jacobi equations, recently addressed in the stationary framework by the authors. Finally, applications to the existence problem of classical solutions to Mean Field Games systems with unbounded local couplings are provided
Lipschitz regularity for viscous Hamilton-Jacobi equations with Lp terms
Full text linkWe provide Lipschitz regularity for solutions to viscous time-dependent Hamilton-Jacobi equations with right-hand side belonging to Lebesgue spaces. Our approach is based on a duality method, and relies on the analysis of the regularity of the gradient of solutions to a dual (Fokker-Planck) equation. Here, the regularizing effect is due to the non-degenerate diffusion and coercivity of the Hamiltonian in the gradient variable
Long time behavior and turnpike solutions in mildly non-monotone mean field games
Full text linkWe consider mean field game systems in time-horizon (0, T), where the individual cost functional depends locally on the density distribution of the agents, and the Hamiltonian is locally uniformly convex. We show that, even if the coupling cost functions are mildly non-monotone, then the system is still well posed due to the effect of individual noise. The rate of anti-monotonicity (i.e. the aggregation rate of the cost functions) which can be afforded depends on the intensity of the diffusion and on global bounds of solutions. We give applications to either the case of globally Lipschitz Hamiltonians or the case of quadratic Hamiltonians and couplings having mild growth. Under similar conditions, we investigate the long time behavior of solutions and we give a complete description of the ergodic and long term properties of the system. In particular we prove: (i) the turnpike property of solutions in the finite (long) horizon (0, T), (ii) the convergence of the system from (0, T) towards (0, ∞), (iii) the vanishing discount limit of the infinite horizon problem and the long time convergence towards the ergodic stationary solution. This way we extend previous results which were known only for the case of monotone and smoothing couplings; our approach is self-contained and does not need the use of the linearized system or of the master equation
Existence and non-existence for time-dependent mean field games with strong aggregation
Full text linkWe investigate the existence of classical solutions to second-order quadratic Mean-Field Games systems with local and strongly decreasing couplings of the form - σmα,α≥ 2 / N, where m is the population density and N is the dimension of the state space. We prove the existence of solutions under the assumption that σ is small enough. For large σ, we show that existence may fail whenever the time horizon T is large
On the existence and uniqueness of solutions to time-dependent fractional MFG
Full text linkWe establish existence and uniqueness of solutions to evolutive fractional mean field game systems with regularizing coupling for any order of the fractional Laplacian s ∈ (0,1). The existence is addressed via the vanishing viscosity method. In particular, we prove that in the subcritical regime s > 1/2 the solution of the system is classical, while if s ≤ 1/2, we find a distributional energy solution. To this aim, we develop an appropriate functional setting based on parabolic Bessel potential spaces. We show uniqueness of solutions both under monotonicity conditions and for short time horizons
On the Problem of Maximal Lq -regularity for Viscous Hamilton–Jacobi Equations
Full text linkIn this paper we prove a conjecture by P.-L. Lions on maximal regularity of Lq-type for periodic solutions to - Δ u+ | Du| γ= f in Rd, under the (sharp) assumption that q>dγ-1γ
One-dimensional multi-agent optimal control with aggregation and distance constraints: Qualitative properties and mean-field limit
Full text linkIn this paper we consider an optimal control problem for a large population of interacting agents with deterministic dynamics, aggregating potential and constraints on reciprocal distances, in dimension 1. We study existence and qualitative properties of periodic in time optimal trajectories of the finite agents optimal control problem, with particular interest on the compactness of the solutions' support and on the saturation of the distance constraint. Moreover, we prove, through a Γ-convergence result, the consistency of the mean-field optimal control problemwith density constraintswith the corresponding underlying finite agent one and we deduce some qualitative results for the time periodic equilibria of the limit problem
On the existence of oscillating solutions in non-monotone Mean-Field Games
Full text linkFor non-monotone single and two-populations time-dependent Mean-Field Game systems we obtain the existence of an infinite number of branches of non-trivial solutions. These non-trivial solutions are in particular shown to exhibit an oscillatory behaviour when they are close to the trivial (constant) one. The existence of such branches is derived using local and global bifurcation methods, that rely on the analysis of eigenfunction expansions of solutions to the associated linearized problem. Numerical analysis is performed on two different models to observe the oscillatory behaviour of solutions predicted by bifurcation theory, and to study further properties of branches far away from bifurcation points
A priori estimates and large population limits for some nonsymmetric Nash systems with semimonotonicity
Full text linkWe address the problem of regularity of solutions (Formula presented.) to a family of semilinear parabolic systems of (Formula presented.) equations, which describe closed-loop equilibria of some (Formula presented.) -player differential games with Lagrangian having quadratic behaviour in the velocity variable, running costs (Formula presented.) and final costs (Formula presented.). By global (semi)monotonicity assumptions on the data (Formula presented.) and (Formula presented.), and assuming that derivatives of (Formula presented.) in directions (Formula presented.) are of order (Formula presented.) for (Formula presented.), we prove that derivatives of (Formula presented.) enjoy the same property. The estimates are uniform in the number of players (Formula presented.). Such a behaviour of the derivatives of (Formula presented.) arise in the theory of Mean Field Games, though here we do not make any symmetry assumption on the data. Then, by the estimates obtained we address the convergence problem (Formula presented.) in a ‘heterogeneous’ Mean Field framework, where players all observe the empirical measure of the whole population, but may react differently from one another. We also discuss some results on the joint (Formula presented.) and vanishing viscosity limit
Ergodic mean field games: existence of local minimizers up to the Sobolev critical case
No full textWe investigate the existence of solutions to viscous ergodic Mean Field Games systems in bounded domains with Neumann boundary conditions and local, possibly aggregative couplings. In particular we exploit the associated variational structure and search for constrained minimizers of a suitable functional. Depending on the growth of the coupling, we detect the existence of global minimizers in the mass subcritical and critical case, and of local minimizers in the mass supercritical case, notably up to the Sobolev critical case
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