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2015

Weak solutions for mean field games with congestion

Graber, Philip Jameson

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We study the short-time existence and uniqueness of solutions to a coupled system of partial differential equations arising in mean field game theory.

It has the generic form $$ \left\{ \begin{array}{c} -\partial_t u - \Delta u + H(t,x,m,\nabla u) = f(t,x,m) \\ \partial_t m - \Delta m - \mathrm{div} \left(m\nabla_p H(t,x,m,\nabla u)\right) = 0 \end{array}\right.
$$ plus initial-final and boundary conditions.
The novelty of the problem is that the Hamiltonian $H(t,x,m,p)$ may take such forms as $m^{-\alpha}|p|^r$ for some $\alpha \geq 0$ and $r > 1$.

Weak solutions for mean field games with congestion · Around

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O. A. Ladyzhenskai͡a, V. A. Solonnikov, and N. N. Ural’ceva
1968
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J. Simon
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1986
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A. Porretta
1999
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J.-M. Lasry and P.-L. Lions
2007
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