75 research outputs found
An error estimate for a new scheme for mean curvature motion
International audienceIn this work, we propose a new numerical scheme for the anisotropic mean curvature equation. The solution of the scheme is not unique, but for all numerical solutions, we provide an error estimate between the continuous solution and the numerical approximation. This error estimate is not optimal, but as far as we know, this is the first one for mean curvature type equation. Our scheme is also applicable to compute the solution to dislocations dynamics equation
Existence and uniqueness of traveling wave for accelerated Frenkel-Kontorova model
In this paper, we study the existence and uniqueness of traveling wave
solution for the accelerated Frenkel-Kontorova model. This model consists in a
system of ODE that describes the motion particles in interaction. The most
important applications we have in mind is the motion of crystal defects called
dislocations. For this model, we prove the existence of traveling wave
solutions under very weak assumptions. The uniqueness of the velocity is also
studied as well as the uniqueness of the profile which used different types of
strong maximum principle. As far as we know, this is the first result
concerning traveling waves for accelerated, spatially discrete system
Comparison principle for a Generalized Fast Marching Method
International audienceIn \cite{CFFM06}, the authors have proposed a generalization of the classical Fast Marching Method of Sethian for the eikonal equation in the case where the normal velocity depends on space and time and can change sign. The goal of this paper is to propose a modified version of the Generalized Fast Marching Method proposed in \cite{CFFM06} for which we state a general comparison principle. We also prove the convergence of the new algorithm
Uniqueness and existence of spirals moving by forced mean curvature motion
In this paper, we study the motion of spirals by mean curvature type motion
in the (two dimensional) plane. Our motivation comes from dislocation dynamics;
in this context, spirals appear when a screw dislocation line reaches the
surface of a crystal. The first main result of this paper is a comparison
principle for the corresponding parabolic quasi-linear equation. As far as
motion of spirals are concerned, the novelty and originality of our setting and
results come from the fact that, first, the singularity generated by the
attached end point of spirals is taken into account for the first time, and
second, spirals are studied in the whole space. Our second main result states
that the Cauchy problem is well-posed in the class of sub-linear weak
(viscosity) solutions. We also explain how to get the existence of smooth
solutions when initial data satisfy an additional compatibility condition.Comment: This new version contains new results: we prove that the weak
(viscosity) solutions of the Cauchy problem are in fact smooth. This is a
consequence of some gradient estimates in time and spac
Singular perturbation of optimal control problems on multi-domains
International audienceThe goal of this paper is to study a singular perturbation problem in the framework of optimal control on multi-domains. We consider an optimal control problem in which the controlled system contains a fast and a slow variables. This problem is reformulated as an Hamilton-Jacobi-Bellman (HJB) equation. The main difficulty comes from the fact that the fast variable lives in a multi-domain. The geometric singularity of the multi-domains leads to the discontinuity of the Hamiltonian. Under a controllability assumption on the fast variables, the limit equation (as the velocity of the fast variable goes to infinity) is obtained via a PDE approache and by means of the tools of the control theory
From heterogeneous microscopic traffic flow models to macroscopic models
The goal of this paper is to derive rigorously macroscopic traffic flow models from microscopic models. More precisely, for the microscopic models, we consider follow-the-leader type models with different types of drivers and vehicles which are distributed randomly on the road. After a rescaling, we show that the cumulative distribution function converge to the solution of a macroscopic model. We also make the link between this macroscopic model and the so-called LWR model
A convergent scheme for a non-local coupled system modelling dislocations densities dynamics
International audienceIn this paper, we study a non-local coupled system that arises in the theory of dislocations densities dynamics. Within the framework of viscosity solutions, we prove a long time existence and uniqueness result for the solution of this model. We also propose a convergent numerical scheme and we prove a Crandall-Lions type error estimate between the continuous solution and the numerical one. As far as we know, this is the first error estimate of Crandall-Lions type for Hamilton-Jacobi systems. We also provide some numerical simulations
Homogenization of accelerated Frenkel-Kontorova models with types of particles
We consider systems of ODEs that describe the dynamics of particles. Each
particle satisfies a Newton law (including the acceleration term) where the
force is created by the interactions with the other particles and with a
periodic potential. The presence of a damping term allows the system to be
monotone. Our study takes into account the fact that the particles can be
different. After a proper hyperbolic rescaling, we show that the solutions to
this system of ODEs converge to the solution of a macroscopic homogenized
Hamilton-Jacobi equation.Comment: 37 pages, to appear in TAM
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