arXiv subjects

P. Jameson Graber

Publications and source records attributed to P. Jameson Graber.

24 records · Page 2Linked to original sources

arXiv2018

Sobolev regularity for first order Mean Field Games

In this paper we obtain Sobolev estimates for weak solutions of first oder variational Mean Field Game systems with coupling terms that are local function of the density variable. Under some coercivity condition on the coupling, we obtain first order Sobolev estimates for the density variable, while under similar coercivity condition on the Hamiltonian we obtain second order Sobolev estimates for the value function. These results are valid both for stationary and time-dependent problems. In the latter case the estimates are fully global in time, thus we resolve a question which was left open in [PS17]. Our methods apply to a large class of Hamiltonians and coupling functions.

math.AP↗

arXiv2017

Variational mean field games for market competition

In this paper, we explore Bertrand and Cournot Mean Field Games models for market competition with reflection boundary conditions. We prove existence, uniqueness and regularity of solutions to the system of equations, and show that this system can be written as an optimality condition of a convex minimization problem. We also provide a short proof of uniqueness to the system addressed in [Graber, P. and Ben- soussan, A., Existence and uniqueness of solutions for Bertrand and Cournot mean field games, Applied Mathematics & Optimization (2016)], where uniqueness was only proved for small parameters $ε$. Finally, we prove existence and uniqueness of weak solutions to the corresponding first order system at the deterministic limit.

math.AP↗

arXiv2017

Mixed finite elements for global tide models with nonlinear damping

We study mixed finite element methods for the rotating shallow water equations with linearized momentum terms but nonlinear drag. By means of an equivalent second-order formulation, we prove long-time stability of the system without energy accumulation. We also give rates of damping in unforced systems and various continuous dependence results on initial conditions and forcing terms. \emph{A priori} error estimates for the momentum and free surface elevation are given in $L^2$ as well as for the time derivative and divergence of the momentum. Numerical results confirm the theoretical results regarding both energy damping and convergence rates.

math.NA↗

arXiv2016

Linear Quadratic Mean Field Type Control and Mean Field Games with Common Noise, with Application to Production of an Exhaustible Resource

We study a general linear quadratic mean field type control problem and connect it to mean field games of a similar type. The solution is given both in terms of a forward/backward system of stochastic differential equations and by a pair of Riccati equations. In certain cases, the solution to the mean field type control is also the equilibrium strategy for a class of mean field games. We use this fact to study an economic model of production of exhaustible resources. Keywords: mean field type control, mean field games, linear-quadratic, optimal control, riccati equations, exhaustible resource production

math.OC↗

arXiv2015

Existence and uniqueness of solutions for Bertrand and Cournot mean field games

We study a system of partial differential equations used to describe Bertrand and Cournot competition among a continuum of producers of an exhaustible resource. By deriving new a priori estimates, we prove the existence of classical solutions under general assumptions on the data. Moreover, under an additional hypothesis we prove uniqueness. Keywords: mean field games, Hamilton-Jacobi, Fokker-Planck, coupled systems, optimal control, nonlinear partial differential equations

math.AP↗

arXiv2015

Attractors for Strongly Damped Wave Equations with Nonlinear Hyperbolic Dynamic Boundary Conditions

We establish the well-posedness of a strongly damped semilinear wave equation equipped with nonlinear hyperbolic dynamic boundary conditions. Results are carried out with the presence of a parameter distinguishing whether the underlying operator is analytic, $α>0$, or only of Gevrey class, $α=0$. We establish the existence of a global attractor for each $α\in[0,1],$ and we show that the family of global attractors is upper-semicontinuous as $α\rightarrow0.$ Furthermore, for each $α\in[0,1]$, we show the existence of a weak exponential attractor. A weak exponential attractor is a finite dimensional compact set in the weak topology of the phase space. This result insures the corresponding global attractor also possess finite fractal dimension in the weak topology; moreover, the dimension is independent of the perturbation parameter $α$. In both settings, attractors are found under minimal assumptions on the nonlinear terms.

math.AP↗