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Le Bourget-du-Lac, France

Dutykh D.,CNRS Mathematics Laboratory | Pelinovsky E.,Nizhny Novgorod State Technical University
Physics Letters, Section A: General, Atomic and Solid State Physics | Year: 2014

The collective behaviour of soliton ensembles (i.e. the solitonic gas) is studied using the methods of the direct numerical simulation. Traditionally this problem was addressed in the context of integrable models such as the celebrated KdV equation. We extend this analysis to non-integrable KdV-BBM type models. Some high resolution numerical results are presented in both integrable and nonintegrable cases. Moreover, the free surface elevation probability distribution is shown to be quasi-stationary. Finally, we employ the asymptotic methods along with the Monte Carlo simulations in order to study quantitatively the dependence of some important statistical characteristics (such as the kurtosis and skewness) on the Stokes-Ursell number (which measures the relative importance of nonlinear effects compared to the dispersion) and also on the magnitude of the BBM term. © 2014 Elsevier B.V. Source

Olivier J.,CNRS Mathematics Laboratory
Zeitschrift fur Angewandte Mathematik und Physik | Year: 2010

In this article, we rigourously prove several asymptotical results for the flow curves of the Hébraud-Lequeux model, a rheological model which describes the behaviour of soft glassy fluids. This model has a control parameter α which governs the behaviour of the fluid at low shear rate. More precisely, we consider τ(γ̇) the stress in a block that is sheared at a constant rate γ̇ and we prove that the system exhibits a transition in its behaviour at low shear rate when α goes through a critical value. The study is complicated by the fact that one of the parameter is only given implicitly and also we have to study two variable function in the neighbourhood of singularities. © 2009 Birkhäuser Verlag Basel/Switzerland. Source

Bucur D.,CNRS Mathematics Laboratory
Archive for Rational Mechanics and Analysis | Year: 2012

For every k ∈ ℕ, we prove the existence of a quasi-open set minimizing the k-th eigenvalue of the Dirichlet Laplacian among all sets of prescribed Lebesgue measure. Moreover, we prove that every minimizer is bounded and has a finite perimeter. The key point is the observation that such quasi-open sets are shape subsolutions for an energy minimizing free boundary problem. © 2012 Springer-Verlag. Source

Bucur D.,CNRS Mathematics Laboratory | Giacomini A.,University of Brescia
Archive for Rational Mechanics and Analysis | Year: 2010

The isoperimetric inequality for the first eigenvalue of the Laplace operator with Robin boundary conditions was recently proved by Daners in the context of Lipschitz sets. This paper introduces a new approach to the isoperimetric inequality, based on the theory of special functions of bounded variation (SBV). We extend the notion of the first eigenvalue λ1 for general domains with finite volume (possibly unbounded and with irregular boundary), and we prove that the balls are the unique minimizers of λ1 among domains with prescribed volume. © 2010 Springer-Verlag. Source

Bucur D.,CNRS Mathematics Laboratory | Feireisl E.,Academy of Sciences of the Czech Republic | Necasova S.,Academy of Sciences of the Czech Republic
Archive for Rational Mechanics and Analysis | Year: 2010

We consider a family of solutions to the evolutionary Navier-Stokes system supplemented with the complete slip boundary conditions on domains with rough boundaries. We give a complete description of the asymptotic limit by means of Γ-convergence arguments, and identify a general class of boundary conditions. © 2009 Springer-Verlag. Source

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