KINETIC FUNCTIONS IN MAGNETOHYDRODYNAMICS WITH RESISTIVITY AND HALL EFFECT

We consider a nonlinear hyperbolic system of two conservation laws which arises in ideal magnetohydrodynamics and includes second-order terms accounting for magnetic resistivity and Hall effect. We show that the initial value problem for this model may lead to solutions exhibiting complex wave structures, including undercompressive nonclassical shock waves. We investigate numerically the subtle competition that takes place between the hyperbolic, diffusive, and dispersive parts of the system. Following Abeyratne, Knowles, LeFloch, and Truskinovsky, who studied similar questions arising in fluid and solid flows, we determine the associated kinetic function which characterizes the dynamics of undereompressive shocks driven by resistivity and Hall effect. To this end, we design a new class of ＂schemes with eontroled dissipation＂, following recent work by LeFloch and Mohammadian. It is now recognized that the equivalent equation associated with a scheme provides a guideline to design schemes that capture physically relevant, nonclassical shocks. We propose a new class of schemes based on high-order entropy conservative, finite differences for the hyperbolic flux, and high-order central differences for the resistivity and Hall terms. These schemes are tested for several regimes of （co-planar or not） initial data and parameter values, and allow us to analyze the properties of nonclassical shocks and establish the existence of monotone kinetic functions in magnetohydrodynamics.

A Geometry-Preserving Finite Volume Method for Compressible Fluids on Schwarzschild Spacetime

We consider the relativistic Euler equations governing spherically symmetric,perfect fluid flows on the outer domain of communication of Schwarzschild spacetime,and we introduce a version of the finite volume method which is formulated from the geometric formulation(and thus takes the geometry into account at the discretization level)and is well-balanced,in the sense that it preserves steady solutions to the Euler equations on the curved geometry under consideration.In order to formulate our method,we first derive a closed formula describing all steady and spherically symmetric solutions to the Euler equations posed on Schwarzschild spacetime.Second,we describe a geometry-preserving,finite volume method which is based from the family of steady solutions to the Euler system.Our scheme is second-order accurate and,as required,preserves the family of steady solutions at the discrete level.Numerical experiments are presented which demonstrate the efficiency and robustness of the proposed method even for solutions containing shock waves and nonlinear interacting wave patterns.As an application,we investigate the late-time asymptotics of perturbed steady solutions and demonstrate its convergence for late time toward another steady solution,taking the overall effect of the perturbation into account.

Global Null Controllability of the 1-Dimensional Nonlinear Slow Diffusion Equation

The authors prove the global null controllability for the 1-dimensional nonlinear slow diffusion equation by using both a boundary and an internal control. They assume that the internal control is only time dependent. The proof relies on the return method in combination with some local controllability results for nondegenerate equations and rescaling techniques.