Vardan Voskanyan

Short-time existence of solutions for mean-field games with congestion

We consider time-dependent mean-field games with congestion that are given by a system of a Hamilton-Jacobi equation coupled with a Fokker-Planck equation. The congestion effects make the Hamilton-Jacobi equation singular. These models are motivated by crowd dynamics where agents have difficulty moving in high-density areas. Uniqueness of classical solutions for this problem is well understood. However, existence of classical solutions, was only known in very special cases - stationary problems with quadratic Hamiltonians and some time-dependent explicit examples. Here, we prove short-time existence of

Existence of positive solutions for an approximation of stationary mean-field games

C^\infty

A Priori Bounds for Stationary Models

solutions in the case of sub-quadratic Hamiltonians.

A Priori Bounds for Time-Dependent Models

Here, we consider a regularized mean-field game model that features a low-order regularization. We prove the existence of solutions with positive density. To do so, we combine a priori estimates with the continuation method. In contrast with high-order regularizations, the low-order regularizations are easier to implement numerically. Moreover, our methods give a theoretical foundation for this approach.

The Nonlinear Adjoint Method

We draw upon our earlier results to study stationary MFGs. Here, we illustrate various techniques in three models. First, we use the Bernstein estimates given in Theorem 3.11, to obtain Sobolev estimates for the value function. Next, we consider a congestion problem and show, through a remarkable identity, that m > 0. Finally, we examine an MFG with a logarithmic nonlinearity. This model presents substantial challenges since the logarithm is not bounded from below. However, a clever integration by parts argument gives the necessary bounds for its study.

Non-local Mean-Field Games: Existence

We continue our study of the regularity of MFGs by considering the time-dependent problem

Estimates for the Transport and Fokker–Planck Equations

\displaystyle{ \left \{\begin{array}{@{}l@{\quad }l@{}} -u_{t} + \frac{1} {\gamma } \left \vert Du\right \vert ^{\gamma } = \Delta u + m^{\alpha } \quad &\;\;\;\mbox{ in}\;\;\;\mathbb{T}^{d} \times [0,T], \\ m_{t} -\mathop{\mathrm{div}}\nolimits (\left \vert Du\right \vert ^{\gamma -1}m) = \Delta m\quad &\;\;\;\mbox{ in}\;\;\;\mathbb{T}^{d} \times [0,T], \end{array} \right. }

A Priori Bounds for Models with Singularities

where 1 0. For γ 2, based on the nonlinear adjoint method. In the next chapter, we investigate two time-dependent problems with singularities—the logarithmic nonlinearity and the congestion problem—for which different methods are required.