200 research outputs found
A comparison among various notions of viscosity solutions for Hamilton-Jacobi equations on networks
Three definitions of viscosity solutions for Hamilton-Jacobi equations on
networks recently appeared in literature ([1,4,6]). Being motivated by various
applications, they appear to be considerably different. Aim of this note is to
establish their equivalence
Stationary Mean Field Games systems defined on networks
We consider a stationary Mean Field Games system defined on a network. In
this framework, the transition conditions at the vertices play a crucial role:
the ones here considered are based on the optimal control interpretation of the
problem. We prove separately the well-posedness for each of the two equations
composing the system. Finally, we prove existence and uniqueness of the
solution of the Mean Field Games system
Continuous dependence estimates and homogenization of quasi-monotone systems of fully nonlinear second order parabolic equations
Aim of this paper is to extend the continuous dependence estimates proved in
\cite{JK1} to quasi-monotone systems of fully nonlinear second-order parabolic
equations. As by-product of these estimates, we get an H\"older estimate for
bounded solutions of systems and a rate of convergence estimate for the
vanishing viscosity approximation. In the second part of the paper we employ
similar techniques to study the periodic homogenization of quasi-monotone
systems of fully nonlinear second-order uniformly parabolic equations. Finally,
some examples are discussed
Memory effects in measure transport equations
Transport equations with a nonlocal velocity field have been introduced as a
continuum model for interacting particle systems arising in physics, chemistry
and biology. Fractional time derivatives, given by convolution integrals of the
time-derivative with power-law kernels, are typical for memory effects in
complex systems. In this paper we consider a nonlinear transport equation with
a fractional time-derivative. We provide a well-posedness theory for weak
measure solutions of the problem and an integral formula which generalizes the
classical push-forward representation formula to this setting
A time-fractional mean field game
We consider a Mean Field Games model where the dynamics of the agents is
subdiffusive. According to the optimal control interpretation of the problem,
we get a system involving fractional time-derivatives for the
Hamilton-Jacobi-Bellman and the Fokker-Planck equations. We discuss separately
the well-posedness for each of the two equations and then we prove existence
and uniqueness of the solution to the Mean Field Games syste
A differential model for growing sandpiles on networks
We consider a system of differential equations of Monge-Kantorovich type
which describes the equilibrium configurations of granular material poured by a
constant source on a network. Relying on the definition of viscosity solution
for Hamilton-Jacobi equations on networks, recently introduced by P.-L. Lions
and P. E. Souganidis, we prove existence and uniqueness of the solution of the
system and we discuss its numerical approximation. Some numerical experiments
are carried out
A numerical method for Mean Field Games on networks
We propose a numerical method for stationary Mean Field Games defined on a
network. In this framework a correct approximation of the transition conditions
at the vertices plays a crucial role. We prove existence, uniqueness and
convergence of the scheme and we also propose a least squares method for the
solution of the discrete system. Numerical experiments are carried out
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