Related papers: Homogenization of the Peierls-Nabarro model for di…
The paper deals with the existence and almost periodic homogenization of some model of generalized Navier-Stokes equations. We first establish an existence result for non-stationary Ladyzhenskaya equations with a given non constant density.…
We consider the homogenization of monotone systems of viscous Hamilton-Jacobi equations with convex nonlinearities set in the stationary, ergodic setting. The primary focus of this paper is on collapsing systems which, as the microscopic…
Plastic deformation of crystals is a physical phenomenon, which has immensely driven the development of human civilisation since the onset of the Chalcolithic period. This process is primarily governed by the motion of line defects, called…
We study the existence and uniqueness of solutions to the vector field Peierls-Nabarro model for curved dislocations in a transversely isotropic medium. Under suitable assumptions for the misfit potential on the slip plane, we reduce the 3D…
We study the stochastic homogenization for a Cauchy problem for a first-order Hamilton-Jacobi equation whose operator is not coercive w.r.t. the gradient variable. We look at Hamiltonians like $H(x,\sigma(x)p,\omega)$ where $\sigma(x)$ is a…
We study nonlocal first-order equations arising in the theory of dislocations. We prove the existence and uniqueness of the solutions of these equations in the case of positive and negative velocities, under suitable regularity assumptions…
We study a continuum model of dislocation transport in order to investigate the formation of heterogeneous dislocation patterns. We propose a physical mechanism which relates the formation of heterogeneous patterns to the dynamics of a…
In this paper we study the convergence of monotone $P1$ finite element methods for fully nonlinear Hamilton-Jacobi-Bellman equations with degenerate, isotropic diffusions. The main result is strong convergence of the numerical solutions in…
This paper deals with the homogenization problem for a one-dimensional parabolic PDE with random stationary mixing coefficients in the presence of a large zero order term. We show that under a proper choice of the scaling factor for the…
We study the long time behavior of an Ornstein-Uhlenbeck process under the influence of a periodic drift. We prove that, under the standard diffusive rescaling, the law of the particle position converges weakly to the law of a Brownian…
We study homogenization for a class of stationnary Hamilton-Jacobi equations in which the Hamiltonian is obtained by perturbing near the origin an otherwise periodic Hamiltonian. We prove that the limiting problem consists of a…
A new formulation for the equation of motion of interacting dislocations is derived. From this solution it is shown that additional coupling forces, of kinetic and inertial origin, should be considered in Dislocation Dynamics (DD)…
We establish homogenization for nondegenerate viscous Hamilton-Jacobi equations in one space dimension when the diffusion coefficient $a(x,\omega) > 0$ and the Hamiltonian $H(p,x,\omega)$ are general stationary ergodic processes in $x$. Our…
We present a Hilbert space perspective to homogenization of standard linear evolutionary boundary value problems in mathematical physics and provide a unified treatment for (non-)periodic homogenization problems in thermodynamics,…
When an amorphous solid is deformed homogeneously, the response exhibits heterogeneous plastic instabilities with localized cooperative rearrangement of cluster of particles. The heterogeneous behavior plays an important role in deciding…
We consider periodic homogenization problems for the L{\'e}vy operators with asymmetric L{\'e}vy densities. The formal asymptotic expansion used for the $\a$-stable (symmetric) L{\'e}vy operators ($\a\in (0,2)$) is not applicable directly…
In this paper we study the connection between four models describing dislocation dynamics: a generalized 2D Frenkel-Kontorova model at the atomic level, the Peierls-Nabarro model, the discrete dislocation dynamics and a macroscopic model…
In this work, we study the effective behavior of a two-dimensional variational model within finite crystal plasticity for high-contrast bilayered composites. Precisely, we consider materials arranged into periodically alternating thin…
We consider a semi-linear integro-differential equation in dimension one associated to the half Laplacian whose solution represents the atom dislocation in a crystal. The equation comprises the evolutive version of the classical…
This work is devoted to deriving small mass limiting equation for a class of Hamiltonian systems with multiplicative L\'evy noise. Derivation of the limiting equation depends on the structure of the stochastic Hamiltonian systems, in which…