arXiv subjects

Alpár Richárd Mészáros

Publications and source records attributed to Alpár Richárd Mészáros.

6 recordsLinked to original sources

arXiv2021

Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients

This paper is devoted to the study of some nonlinear parabolic equations with discontinuous diffusion intensities. Such problems appear naturally in physical and biological models. Our analysis is based on variational techniques and in particular on gradient flows in the space of probability measures equipped with the distance arising in the Monge-Kantorovich optimal transport problem. The associated internal energy functionals in general fail to be differentiable, therefore classical results do not apply directly in our setting. We study the combination of both linear and porous medium type diffusions and we show the existence and uniqueness of the solutions in the sense of distributions in suitable Sobolev spaces. Our notion of solution allows us to give a fine characterization of the emerging critical regions, observed previously in numerical experiments. A link to a three phase free boundary problem is also pointed out.

math.AP↗

arXiv2017

Uniqueness issues for evolution equations with density constraints

In this paper we present some basic uniqueness results for evolutive equations under density constraints. First, we develop a rigorous proof of a well-known result (among specialists) in the case where the spontaneous velocity field satisfies a monotonicity assumption: we prove the uniqueness of a solution for first order systems modeling crowd motion with hard congestion effects, introduced recently by \emph{Maury et al.} The monotonicity of the velocity field implies that the $2-$Wasserstein distance along two solutions is $λ$-contractive, which in particular implies uniqueness. In the case of diffusive models, we prove the uniqueness of a solution passing through the dual equation, where we use some well-known parabolic estimates to conclude an $L^1-$contraction property. In this case, by the regularization effect of the non-degenerate diffusion, the result follows even if the given velocity field is only $L^\infty$ as in the standard Fokker-Planck equation.

math.AP↗

arXiv2017

On the variational formulation of some stationary second order mean field games systems

We consider the variational approach to prove the existence of solutions of second order stationary Mean Field Games on a bounded domain $Ω\subseteq \mathbb{R}^{d}$, with Neumann boundary conditions, and with and without density constraints. We consider Hamiltonians which growth as $|\cdot|^{q'}$, where $q'=q/(q-1)$ and $q>d$. Despite this restriction, our approach allows us to prove the existence of solutions in the case of rather general coupling terms. When density constraints are taken into account, our results improve those in \cite{MesSil}. Furthermore, our approach can be used to obtain solutions of systems with multiple populations.

math.AP↗

arXiv2016

First order Mean Field Games with density constraints: Pressure equals Price

In this paper we study Mean Field Game systems under density constraints as optimality conditions of two optimization problems in duality. A weak solution of the system contains an extra term, an additional price imposed on the saturated zones. We show that this price corresponds to the pressure field from the models of incompressible Euler's equations {à} la Brenier. By this observation we manage to obtain a minimal regularity, which allows to write optimality conditions at the level of single agent trajectories and to define a weak notion of Nash equilibrium for our model.

math.OC↗

arXiv2016

A variational approach to second order mean field games with density constraints: the stationary case

In this paper we study second order stationary Mean Field Game systems under density constraints on a bounded domain $Ω\subset \mathbb{R}^d$. We show the existence of weak solutions for power-like Hamiltonians with arbitrary order of growth. Our strategy is a variational one, i.e. we obtain the Mean Field Game system as the optimality condition of a convex optimization problem, which has a solution. When the Hamiltonian has a growth of order $q' \in ]1, d/(d-1)[$, the solution of the optimization problem is continuous which implies that the problem constraints are qualified. Using this fact and the computation of the subdifferential of a convex functional introduced by Benamou-Brenier, we prove the existence of a solution of the MFG system. In the case where the Hamiltonian has a growth of order $q'\geq d/(d-1)$, the previous arguments do not apply and we prove the existence by means of an approximation argument.

math.AP↗

arXiv2016

Advection-diffusion equations with density constraints

In the spirit of the macroscopic crowd motion models with hard congestion (i.e. a strong density constraint $ρ\leq 1$) introduced by Maury {\it et al.} some years ago, we analyze a variant of the same models where diffusion of the agents is also taken into account. From the modeling point of view, this means that individuals try to follow a given spontaneous velocity, but are subject to a Brownian diffusion, and have to adapt to a density constraint which introduces a pressure term affecting the movement. From the PDE point of view, this corresponds to a modified Fokker-Planck equation, with an additional gradient of a pressure (only living in the saturated zone $\{ρ=1\}$) in the drift. The paper proves existence and some estimates, based on optimal transport techniques.

math.AP↗