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We consider uniformly (DC) or periodically (AC) driven generalized infinite elastic chains (a generalized Frenkel-Kontorova model) with gradient dynamics. We first show that the union of supports of all the invariant measures, denoted by A, projects injectively to a dynamical system on a 2dimensional cylinder. We also prove existence of ergodic invariant measures supported on a set of rotationaly ordered configurations with an arbitrary (rational or irrational) rotation number. This shows that the Aubry-Mather structure of ground states persists if an arbitrary AC or DC force is applied. The set A attracts almost surely (in probability) configurations with bounded spacing. In the DC case, A consists entirely of equilibria and uniformly sliding solutions. The key tool is a new weak Lyapunov function on the space of translationally invariant probability measures on the state space, which counts intersections.

We prove that all the invariant measures for the monotone gradient dynamics of the (non-tilted) Frenkel-Kontorova model are supported on the set of stationary configurations, and that the ω-limit set of each configuration is either a single stationary configuration or contains an invariant circle of stationary configurations. A consequence is a new proof of the existence of Mather's shadowing orbits of twist diffeomorphisms. We compare the results with the dynamics of other dissipative spatially extended systems, such as the real Ginzburg-Landau equation, and suggest application to non-conservative diffusion on 1D lattices.

International audienceWe consider the incompressible Navier-Stokes equations in the cylinder R × T, with no exterior forcing, and we investigate the long-time behavior of solutions arising from merely bounded initial data. Although we do not prove that such solutions stay uniformly bounded for all times, we show that they converge in an appropriate sense to the family of spatially homogeneous equilibria as t → ∞. Convergence is uniform on compact subdomains, and holds for all times except on a sparse subset of the positive real axis. We also improve the known upper bound on the L ∞ norm of the solutions, although our results in this direction are not optimal. Our approach is based on a detailed study of the local energy dissipation in the system, in the spirit of a recent work devoted to a class of dissipative partial differential equations with a formal gradient structure [5]. Keywords: Navier-Stokes equations, localized energy estimates, extended dissipative system. MSC(2010): 35Q30, 35B40, 76D0

This work is concerned with computational modelling of viscoplastic fluids. The flows considered are assumed to be incompressible, while the viscoplastic laws are obtained by incorporating a yield stress below which the fluid is assumed to remain non‐deformable. The Bingham fluid is chosen as a model problem and is considered in detail in the text. The finite element formulation adopted in this work is based on a version of the stabilised finite element method, known as the Galerkin/least‐squares method, originally developed by Hughes and co‐workers. This methodology allows use of low and equal order interpolation of the pressure and velocity fields, thus providing an efficient finite element framework. The Newton‐Raphson method has been chosen for solution of the incremental non‐linear problem arising through the temporal discretisation of the evolution problem. Numerical examples are provided to illustrate the main features of the described methodology.