arXiv subjects

Benoit Desjardins

Publications and source records attributed to Benoit Desjardins.

3 recordsLinked to original sources

arXiv2019

Normal mode decomposition and dispersive and nonlinear mixing in stratified fluids

Motivated by the analysis of the propagation of internal waves in a stratified ocean, we consider in this article the incompressible Euler equations with variable density in a flat strip, and we study the evolution of perturbations of the hydrostatic equilibrium corresponding to a stable vertical strati-fication of the density. We show the local well-posedness of the equations in this configuration and provide a detailed study of their linear approximation. Performing a modal decomposition according to a Sturm-Liouville problem associated to the background stratification, we show that the linear approximation can be described by a series of dispersive perturbations of linear wave equations. When the so called Brunt-Vais{ä}l{ä} frequency is not constant, we show that these equations are coupled, hereby exhibiting a phenomenon of dispersive mixing. We then consider more specifically shallow water configurations (when the horizontal scale is much larger than the depth); under the Boussinesq approximation (i.e. neglecting the density variations in the momentum equation), we provide a well-posedness theorem for which we are able to control the existence time in terms of the relevant physical scales. We can then extend the modal decomposition to the nonlinear case and exhibit a non-linear mixing of different nature than the dispersive mixing mentioned above. Finally, we discuss some perspectives such as the sharp stratification limit that is expected to converge towards two-fluids systems.

math.AP↗

arXiv2015

Two-velocity hydrodynamics in fluid mechanics: Part II Existence of global $κ$-entropy solutions to compressible Navier-Stokes systems with degenerate viscosities

This paper addresses the issue of global existence of so-called $κ$-entropy solutions to the Navier-Stokes equations for viscous compressible and barotropic fluids with degenerate viscosities. We consider the three dimensional space domain with periodic boundary conditions. Our solutions satisfy the weak formulation of the mass and momentum conservation equations and also a generalization of the BD-entropy identity called: $κ$-entropy. This new entropy involves a mixture parameter $κ\in (0,1)$ between the two velocities $\bf{u}$ and $\bf{u} + 2\nablaφ(\varrho)$ (the latter was introduced by the first two authors in [C. R. Acad. Sci. Paris 2004]), where $\bf{u}$ is the velocity field and $φ$ is a function of the density $\varrho$ defined by $φ'(s)=μ'(s)/s$. As a byproduct of the existence proof, we show that two-velocity hydrodynamics (in the spirit of S.C. Shugrin 1994) is a possible formulation of a model of barotropic compressible flow with degenerate viscosities. It is used in construction of approximate solutions based on a conservative augmented approximate scheme.

math.AP↗

arXiv2013

On the Semi-Markov Equivalence of Causal Models

The variability of structure in a finite Markov equivalence class of causally sufficient models represented by directed acyclic graphs has been fully characterized. Without causal sufficiency, an infinite semi-Markov equivalence class of models has only been characterized by the fact that each model in the equivalence class entails the same marginal statistical dependencies. In this paper, we study the variability of structure of causal models within a semi-Markov equivalence class and propose a systematic approach to construct models entailing any specific marginal statistical dependencies.

cs.AI↗