Differential Equations with singular fields

This paper investigates the well posedness of ordinary differential equations and more precisely the existence (or uniqueness) of a flow through explicit compactness estimates. Instead of assuming a bounded divergence condition on the vector field, a compressibility condition on the flow (bounded jacobian) is considered. The main result provides existence under the condition that the vector field belongs to $BV$ in dimension 2 and $SBV$ in higher dimensions

Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient

The Cauchy problem for a multidimensional linear transport equation with discontinuous coefficient is investigated. Provided the coefficient satisfies a one-sided Lipschitz condition, existence, uniqueness and weak stability of solutions are obtained for either the conservative backward problem or the advective forward problem by duality. Specific uniqueness criteria are introduced for the backward conservation equation since weak solutions are not unique. A main point is the introduction of a generalized flow in the sense of partial differential equations, which is proved to have unique jacobian determinant, even though it is itself nonunique.Comment: 19-03-200

Lagrangian solutions to the 2D euler system with L^1 vorticity and infinite energy

We consider solutions to the two-dimensional incompressible Euler system with only integrable vorticity, thus with possibly locally infinite energy. With such regularity, we use the recently developed theory of Lagrangian flows associated to vector fields with gradient given by a singular integral in order to define Lagrangian solutions, for which the vorticity is transported by the flow. We prove strong stability of these solutions via strong convergence of the flow, under only the assumption of L^1 weak convergence of the initial vorticity. The existence of Lagrangian solutions to the Euler system follows for arbitrary L^1 vorticity. Relations with previously known notions of solutions are established

Classical solvability of the relativistic Vlasov-Maxwell system with bounded spatial density

In (Arch. Rational. Mech. Anal 1986, 92:59-90), Glassey and Strauss showed that if the growth in the momentum of the particles is controlled, then the relativistic Vlasov-Maxwell system has a classical solution globally in time. Later they proved that such control is achieved if the kinetic energy density of the particles remains bounded for all time (Math. Meth. Appl. Sci. 1987, 9:46-52). Here, we show that the latter assumption can be weakened to the boundedness of the spatial density.Comment: AMS-LaTeX, 9 page

On massless electron limit for a multispecies kinetic system with external magnetic field

We consider a three-dimensional kinetic model for a two species plasma consisting of electrons and ions confined by an external nonconstant magnetic field. Then we derive a kinetic-fluid model when the mass ratio $m_e/m_i$ tends to zero. Each species initially obeys a Vlasov-type equation and the electrostatic coupling follows from a Poisson equation. In our modeling, ions are assumed non-collisional while a Fokker-Planck collision operator is taken into account in the electron equation. As the mass ratio tends to zero we show convergence to a new system where the macroscopic electron density satisfies an anisotropic drift-diffusion equation. To achieve this task, we overcome some specific technical issues of our model such as the strong effect of the magnetic field on electrons and the lack of regularity at the limit. With methods usually adapted to diffusion limit of collisional kinetic equations and including renormalized solutions, relative entropy dissipation and velocity averages, we establish the rigorous derivation of the limit model

Numerical simulations of the Euler system with congestion constraint

In this paper, we study the numerical simulations for Euler system with maximal density constraint. This model is developed in [1, 3] with the constraint introduced into the system by a singular pressure law, which causes the transition of different asymptotic dynamics between different regions. To overcome these difficulties, we adapt and implement two asymptotic preserving (AP) schemes originally designed for low Mach number limit [2,4] to our model. These schemes work for the different dynamics and capture the transitions well. Several numerical tests both in one dimensional and two dimensional cases are carried out for our schemes

A new model for shallow viscoelastic fluids

International audienceWe propose a new reduced model for gravity-driven free-surface flows of shallow elastic fluids. It is obtained by an asymptotic expansion of the upper-convected Maxwell model for elastic fluids. The viscosity is assumed small (of order epsilon, the aspect ratio of the thin layer of fluid), but the relaxation time is kept finite. Additionally to the classical layer depth and velocity in shallow models, our system describes also the evolution of two scalar stresses. It has an intrinsic energy equation. The mathematical properties of the model are established, an important feature being the non-convexity of the physically relevant energy with respect to conservative variables, but the convexity with respect to the physically relevant pseudo-conservative variables. Numerical illustrations are given, based on a suitable well-balanced finite-volume discretization involving an approximate Riemann solver