Benoît Merlet

Optimally swimming stokesian robots

We study self-propelled stokesian robots composed of assemblies of balls, in dimensions 2 and 3, and prove that they are able to control their position and orientation. This is a result of controllability , and its proof relies on applying Chows theorem in an analytic framework, similar to what has been done in [4] for an axisymmetric system swimming along the axis of symmetry. We generalize the analyticity result given in [4] to the situation where the swimmers can move either in a plane or in three-dimensional space, hence experiencing also rotations. We then focus our attention on energetically optimal strokes, which we are able to compute numerically. Some examples of computed optimal strokes are discussed in detail.

Phase segregation for binary mixtures of Bose-Einstein Condensates

We study the strong segregation limit for mixtures of Bose-Einstein condensates modelled by a Gross-Pitaievskii functional. Our first main result is that in presence of a trapping potential, for different intracomponent strengths, the Thomas-Fermi limit is sufficient to determine the shape of the minimizers. Our second main result is that for asymptotically equal intracomponent strengths, one needs to go to the next order. The relevant limit is a weighted isoperimetric problem. We then study the minimizers of this limit problem, proving radial symmetry or symmetry breaking for different values of the parameters. We finally show that in the absence of a confining potential, even for non-equal intracomponent strengths, one needs to study a related isoperimetric problem to gain information about the shape of the minimizers.

APPROXIMATE SHOCK CURVES FOR NON-CONSERVATIVE HYPERBOLIC SYSTEMS IN ONE SPACE DIMENSION

For non-conservative hyperbolic systems several definitions of shock waves have been introduced in the literature. In this paper, we propose a new and simple definition in the case of genuinely nonlinear fields. Relying on a vanishing viscosity process we prove the existence of shock curves for viscosity matrix commuting with the matrix of the hyperbolic system. This setting generalizes a recent result by Bianchini and Bressan. Furthermore we prove that all definitions agree to third order near a given state.

A phase-field approximation of the Steiner problem in dimension two

Abstract In this paper we consider the branched transportation problem in two dimensions associated with a cost per unit length of the form 1 + β ⁢ θ {1+\beta\,\theta} , where θ denotes the amount of transported mass and β > 0 {\beta>0} is a fixed parameter (notice that the limit case β = 0 {\beta=0} corresponds to the classical Steiner problem). Motivated by the numerical approximation of this problem, we introduce a family of functionals ( { ℱ ε } ε > 0 {\{\mathcal{F}_{\varepsilon}\}_{\varepsilon>0}} ) which approximate the above branched transport energy. We justify rigorously the approximation by establishing the equicoercivity and the Γ-convergence of { ℱ ε } {\{\mathcal{F}_{\varepsilon}\}} as ε ↓ 0 {\varepsilon\downarrow 0} . Our functionals are modeled on the Ambrosio–Tortorelli functional and are easy to optimize in practice. We present numerical evidences of the efficiency of the method.

Optimized Schwarz Waveform Relaxation for the Primitive Equations of the Ocean

In this article we are interested in the derivation of efficient domain decomposition methods for the viscous primitive equations of the ocean. We consider the rotating three-dimensional incompressible hydrostatic Navier-Stokes equations with free surface. Performing an asymptotic analysis of the system in the regime of small Rossby numbers, we compute an approximate Dirichlet to Neumann operator and build an optimized Schwarz waveform relaxation algorithm. We establish that the algorithm is well defined and provide numerical evidence of the convergence of the method.

Positive Lower Bound for the Numerical Solution of a Convection-Diffusion Equation

In this work, we apply a method due to De Giorgi [3] in order to establish a positive lower bound for the numerical solution of a stationary convection-diffusion equation.

Design and Analysis of a Finite Volume Scheme for a Concrete Carbonation Model

In this paper we introduce a finite volume scheme for a concrete carbonation model proposed by Aiki and Muntean in [1]. It consists in a Euler discretisation in time and a Scharfetter–Gummel discretisation in space. We give here some hints for the proof of the convergence of the scheme and show numerical experiments.