Nous considérons un modèle provenant de la biologie, composé d03équations de chimiotactisme couplées aux équations de fluide visqueux incompressible par le transport et le for04age externe. L03existence globale des solutions du problème de Cauchy est étudiée sous certaines conditions. Précisément, pour le système chimiotactisme–Navier–Stokes en deux dimensions d03espace, nous obtenons l03existence globale pour des données grandes. En trois dimensions d03espace, nous démontrons l03existence globale des solutions faibles pour le système chimiotactisme–Stokes avec une diffusion non-linéaire de la densité des cellules.

In this article the authors present a new solver for the Poisson problem on spherical shells. The method is derived by application of the Fast Fourier Transform (FFT) in all three spherical variables. Through a particular change of variables valid on spherical shells, this new approach avoids a variable coefficient on the radial differential operator, resulting in a constant-coefficient formulation that can be fast-diagonalized via FFT. The authors describe the Fourier expansion in the angular variables and the fast diagonalization of the radial derivative using the FFT, including second-, fourth- and sixth-order finite difference approximations. An algorithmic description of the sixth-order solver is described, and numerical tests that demonstrate the theoretical rate of convergence are presented.

This paper deals with the derivation and analysis of the the Hall Magneto-Hydrodynamic equations. We first provide a derivation of this system from a two-fluids Euler-Maxwell system for electrons and ions, through a set of scaling limits. We also propose a kinetic formulation for the Hall-MHD equations which contains as fluid closure different variants of the Hall-MHD model. Then, we prove the existence of global weak solutions for the incompressible viscous resistive Hall-MHD model. We use the particular structure of the Hall term which has zero contribution to the energy identity. Finally, we discuss particular solutions in the form of axisymmetric purely swirling magnetic fields and propose some regularization of the Hall equation.

The purpose of this paper is to establish a new method for proving the convergence of the particle method applied to the Camassa-Holm (CH) equation. The CH equation is a strongly nonlinear, bi-Hamiltonian, completely integrable model in the context of shallow water waves. The equation admits solutions that are nonlinear superpositions of traveling waves that have a discontinuity in the first derivative at their peaks and therefore are called peakons. This behavior admits several diverse scientific applications, but introduce difficult numerical challenges. To accurately capture these solutions, one may apply the particle method to the CH equation. Using the concept of space-time bounded variation, we show that the particle solution converges to a global weak solution of the CH equation for positive Radon measure initial data. Read More: http://www.worldscientific.com/doi/abs/10.1142/9789814417099_0033

The computation of compressible flows becomes more challenging when the Mach number has different orders of magnitude. When the Mach number is of order one, modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions. However, if the Mach number is small, the acoustic waves lead to stiffness in time and exces-sively large numerical viscosity, thus demanding much smaller time step and mesh size than normally needed for incompressible flow simulation. In this paper, we develop an all-speed asymptotic preserving (AP) numerical scheme for the compressible isentropic Euler and Navier-Stokes equations that is uni-formly stable and accurate for all Mach numbers. Our idea is to split the system into two parts: one involves a slow, nonlinear and conservative hyper-bolic system adequate for the use of modern shock capturing methods, and the other a linear hyperbolic system which contains the stiff acoustic dynam-ics, to be solved implicitly. This implicit part is reformulated into a standard pressure Poisson projection system, and thus possesses sufficient structure for efficient fast Fourier transform solution techniques. In the zero Mach number limit, the scheme automatically becomes a projection method-like incompress-ible solver. We present numerical results in one and two dimensions in both compressible and incompressible regimes.