Decay of solutions to equations modelling incompressible bipolar non-Newtonian fluids.
Density-dependent incompressible fluids with non-Newtonian viscosity
We study the system of PDEs describing unsteady flows of incompressible fluids with variable density and non-constant viscosity. Indeed, one considers a stress tensor being a nonlinear function of the symmetric velocity gradient, verifying the properties of -coercivity and -growth, for a given parameter . The existence of Dirichlet weak solutions was obtained in [2], in the cases if or if , being the dimension of the domain. In this paper, with help of some new estimates (which lead...
Derivation and mathematical analysis of a nonlocal model for large amplitude internal waves
This note is devoted to the study of a bi-fluid generalization of the nonlinear shallow-water equations. It describes the evolution of the interface between two fluids of different densities. In the case of a two-dimensional interface, this systems contains unexpected nonlocal terms (that are of course not present in the usual one-fluid shallow water equations). We show here how to derive this systems from the two-fluid Euler equations and then show that it is locally well-posed.
Derivation and well-posedness of Boussinesq/Boussinesq systems for internal waves
We consider the propagation of internal waves at the interface between two layers of immiscrible fluids of different densities, under the rigid lid assumption, with the presence of surface tension and with uneven bottoms. We are interested in the case where the flow has a Boussinesq structure in both the upper and lower fluid domains. Following the global strategy introduced recently by Bona, Lannes and Saut [J. Math. Pures Appl. 89 (2008)], we derive an asymptotic model in this regime, namely the...
Development of a mathematical theory of inner gravitational waves in deep reservoirs of variable width.
Difference schemes for pluriparabolic equations.
Diffusion with dissolution and precipitation in a porous medium: Mathematical analysis and numerical approximation of a simplified model
Modeling the kinetics of a precipitation dissolution reaction occurring in a porous medium where diffusion also takes place leads to a system of two parabolic equations and one ordinary differential equation coupled with a stiff reaction term. This system is discretized by a finite volume scheme which is suitable for the approximation of the discontinuous reaction term of unknown sign. Discrete solutions are shown to exist and converge towards a weak solution of the continuous problem. Uniqueness...
Diffusive limit for finite velocity Boltzmann kinetic models.
We investigate, in the diffusive scaling, the limit to the macroscopic description of finite-velocity Boltzmann kinetic models, where the rate coefficient in front of the collision operator is assumed to be dependent of the mass density. It is shown that in the limit the flux vanishes, while the evolution of the mass density is governed by a nonlinear parabolic equation of porous medium type. In the last part of the paper we show that our method adapts to prove the so-called Rosseland approximation...
Discontinuous Galerkin method for a 2D nonlocal flocking model
We present our work on the numerical solution of a continuum model of flocking dynamics in two spatial dimensions. The model consists of the compressible Euler equations with a nonlinear nonlocal term which requires special treatment. We use a semi-implicit discontinuous Galerkin scheme, which proves to be efficient enough to produce results in 2D in reasonable time. This work is a direct extension of the authors' previous work in 1D.
Discrete coagulation-fragmentation system with transport and diffusion
We prove the existence of solutions to two infinite systems of equations obtained by adding a transport term to the classical discrete coagulation-fragmentation system and in a second case by adding transport and spacial diffusion. In both case, the particles have the same velocity as the fluid and in the second case the diffusion coefficients are equal. First a truncated system in size is solved and after we pass to the limit by using compactness properties.
Discretization errors at free boundaries of the Grad-Schlüter-Shafranov equation.
Dispersion relations for the multivelocity acoustic Peierls equations and some properties of the scalar acoustic Peierls potential. I, II.
Dissipative Boussinesq equations on non-cylindrical domains in .
Dissipative quasi-geostrophic equations with data.
Domain sensitivity in singular limits of compressible viscous fluids
In this note, we discuss several recently developed methods for studying stability of a singular limit process with respect to the shape of the underlying physical space. As a model example, we consider a compressible viscous barotropic fluid occupying a spatial domain . In what follows, we describe two rather different problems: (i) the choice of effective boundary conditions; (ii) the fluid flow in the low Mach number regime. In the remaining part of the paper, we analyze these two issues simultaneously...