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].

We consider a 1-D tank containing an inviscid incompressible irrotational fluid. The tank is subject to the control which consists of horizontal moves. We assume that the motion of the fluid is well-described by the Saint–Venant equations (also called the shallow water equations). We prove the local controllability of this nonlinear control system around any steady state. As a corollary we get that one can move from any steady state to any other steady state.

We consider a 1-D tank containing an inviscid incompressible irrotational fluid. The tank is subject to the control which consists of horizontal moves. We assume that the motion of the fluid is well-described by the Saint–Venant equations (also called the shallow water equations). We prove the local controllability of this nonlinear control system around any steady state. As a corollary we get that one can move from any steady state to any other steady state.

We introduce a phenomenological model for anti-angiogenic therapy in the treatment of metastatic cancers. It is a structured transport equation with a nonlocal boundary condition describing the evolution of the density of metastases that we analyze first at the continuous level. We present the numerical analysis of a lagrangian scheme based on the characteristics whose convergence establishes existence of solutions. Then we prove an error estimate and use the model to perform interesting simulations...

We introduce a phenomenological model for anti-angiogenic therapy in the treatment of metastatic cancers. It is a structured transport equation with a nonlocal boundary condition describing the evolution of the density of metastases that we analyze first at the continuous level. We present the numerical analysis of a lagrangian scheme based on the characteristics whose convergence establishes existence of solutions. Then we prove an error estimate and use the model to perform interesting simulations...

We introduce a phenomenological model for anti-angiogenic therapy in the treatment of metastatic cancers. It is a structured transport equation with a nonlocal boundary condition describing the evolution of the density of metastases that we analyze first at the continuous level. We present the numerical analysis of a lagrangian scheme based on the characteristics whose convergence establishes existence of solutions. Then we prove an error estimate and use the model to perform interesting simulations...

In this article, we wish to investigate the behavior of a two-layer $k-\epsilon$ turbulence model from the mathematical point of view, as this model is useful for the near-wall treatment in numerical simulations. First, we explain the difficulties inherent in the model. Then, we present a new variable $\theta$ that enables the mathematical study. Due to a problem of definition of the turbulent viscosity on the wall boundary, we consider an alternative version of the original equation. We show that some physical aspects...

In this article, we wish to investigate the behavior of a two-layer k - ε turbulence model from the mathematical point of view, as this model is useful for the near-wall treatment in numerical simulations. First, we explain the difficulties inherent in the model. Then, we present a new variable θ that enables the mathematical study. Due to a problem of definition of the turbulent viscosity on the wall boundary, we consider an alternative version of the original equation. We show that some physical...

Treating cancer patients with metastatic disease remains an ultimate challenge in clinical oncology. Because invasive cancer precludes or limits the use of surgery, metastatic setting is often associated with (poor) survival, rather than sustained remission, in patients with common cancers like lung, digestive or breast carcinomas. Mathematical modeling may help us better identify non detectable metastatic status to in turn optimize treatment for...

We consider an Hamilton-Jacobi equation of the form$$H(x,Du)=0x\in \Omega \subset {ℝ}^N,\left(1\right)$$where $H(x,p)$ is assumed Borel measurable and quasi-convex in $p$. The notion of Monge solution, introduced by Newcomb and Su, is adapted to this setting making use of suitable metric devices. We establish the comparison principle for Monge sub and supersolution, existence and uniqueness for equation (1) coupled with Dirichlet boundary conditions, and a stability result. The relation among Monge and Lipschitz subsolutions is also discussed.

We consider an Hamilton-Jacobi equation of the form

 $$H(x,Du)=0x\in \Omega \subset {ℝ}^N,\left(1\right)$$ 
 where H(x,p) is assumed Borel measurable and quasi-convex in p. The notion of Monge solution, introduced by Newcomb and Su, is adapted to this setting making use of suitable metric devices. We establish the comparison principle for Monge sub and supersolution, existence and uniqueness for equation ([see full text]) coupled with Dirichlet boundary conditions, and a stability result. The relation among Monge and Lipschitz subsolutions is also...

The paper deals with the existence of solutions of the form $Au=b$ with operators monotone in a broader sense, including pseudomonotone operators and operators satisfying conditions $S$ and $M$. The first part of the paper which has a methodical character is concluded by the proof of an existence theorem for the equation on a reflexive separable Banach space with a bounded demicontinuous coercive operator satisfying condition ${\left(M\right)}_0$. The second part which has a character of a survey compares various types of...