This paper is concerned with generalized, discontinuous solutions of initial value problems for nonlinear first order partial differential equations. “Layering” is a method of approximating an arbitrary generalized solution by dividing its domain, say a half-space $t\ge 0$, into thin layers $(i-\ell )h\le t\le ih$, $i=1,2,...(h\>0)$, and using a strict solution $u_i$ in the $i$-th layer. On the interface $t=(i-\ell )h$, $u_t$ is required to reduce to a smooth function approximating the values on that plane of $u_{i-\ell }$. The resulting stratified configuration of strict solutions...

Il s’agit de comparer les différents résultats et théorèmes concernant dans un cadre essentiellement déterministe des systèmes de particules. Cela conduit à étudier la notion de hiérarchies d’équations et à comparer les modèles non linéaires et linéaires. Dans ce dernier cas on met en évidence le rôle de l’aléatoire. Ce texte réfère à une série de travaux en collaboration avec F. Golse, A. Gottlieb, D. Levermore et N. Mauser.

We discuss some implications of linear programming for Mather theory [13, 14, 15] and its finite dimensional approximations. We find that the complementary slackness condition of duality theory formally implies that the Mather set lies in an $n$-dimensional graph and as well predicts the relevant nonlinear PDE for the “weak KAM” theory of Fathi [6, 7, 8, 5].

We discuss some implications of linear programming for Mather theory [13-15] and its finite dimensional approximations. We find that the complementary slackness condition of duality theory formally implies that the Mather set lies in an n-dimensional graph and as well predicts the relevant nonlinear PDE for the “weak KAM” theory of Fathi [5-8].