In their seminal work [3], R. DiPerna and P.-L. Lions established the existence of global weak solutions to the Vlasov-Maxwell system. In the present notes we give a somewhat simplified proof of this result for the relativistic version of this system, the main purpose being to make this important result of kinetic theory more easily accessible to newcomers in the field. We show that the weak solutions preserve the total charge.

We study in this article the bifurcation and stability of the solutions of the Boussinesq equations, and the onset of the Rayleigh-Benard convection. A nonlinear theory for this problem is established in this article using a new notion of bifurcation called attractor bifurcation and its corresponding theorem developed recently by the authors in [6]. This theory includes the following three aspects. First, the problem bifurcates from the trivial solution an attractor A_R when the Rayleigh number R crosses the first critical Rayleigh number R_c for all physically sound boundary conditions, regardless of the multiplicity of the eigenvalue R_c for the linear problem. Second, the bifurcated attractor A_R is asymptotically stable. Third, when the spatial dimension is two, the bifurcated solutions are also structurally stable and are classified as well. In addition, the technical method developed provides a recipe, which can be used for many other problems related to bifurcation and pattern formation.

We continue the study of the nonconforming multiscale finite element method (Ms-FEM) introduced in [17, 14] for second order elliptic equations with highly oscillatory coefficients. The main difficulty in MsFEM, as well as other numerical upscaling methods, is the scale resonance effect. It has been show that the leading order resonance error can be effectively removed by using an over-sampling technique. Nonetheless, there is still a secondary cell resonance error of O(e²h²). Here, we introduce a Petrov-Galerkin MsFEM formulation with nonconforming multiscale trial functions and linear test functions. We show that the cell resonance error is eliminated in this formulation and hence the convergence rate is greatly improved. Moreover, we show that a similar formulation can be used to enhance the convergence of an immersed-interface finite element method for elliptic interface problems.

We give examples of divergence free vector fields. For such fields the Cauchy problem for the linear transport equation has unique bounded solutions. The first example has nonuniqueness in the Cauchy problem for the ordinary differential equation defining characteristics. In addition, there are smooth initial data so that the unique bounded solution is not continuous on any neighborhood of the origin. The second example is a field of similar regularity and intial data of bounded variation.

Several Glimm-type functionals for (piecewise smooth) approximate solutions of nonlinear hyperbolic systems have been introduced in recent years. In this paper, following a work by Baiti and Bressan on genuinely nonlinear systems we provide a framework to prove that such functionals can be extended to general functions with bounded variation and we investigate their lower semi-continuity properties with respect to the strong L1topology. In particular, our result applies to the functionals introduced by Iguchi-LeFloch and Liu-Yang for systems with general flux-functions, as well as the functional introduced by Baiti-LeFloch-Piccoli for nonclassical entropy solutions. As an illustration of the use of continuous Glimm-type functionals, we also extend a result by Bressan and Colombo for genuinely nonlinear systems, and establish an estimate on the spreading of rarefaction waves in solutions of hyperbolic systems with general flux-function.

Recently Y. Meyer derived a characterization of the minimizer of the Rudin-Osher-Fatemi functional in a functional analytical framework. In statistics the discrete version of this functional is used to analyze one dimensional data and belongs to the class of nonparametric regression models. In this work we generalize the functional analytical results of Meyer and apply them to a class of regression models, such as quantile, robust, logistic regression, for the analysis of multidimensional data. The characterization of Y. Meyer and our generalization is based on G-norm properties of the data and the minimizer. A geometric point of view of regression minimization is provided.

We introduce and study a class of model prototype hybrid systems comprised of a microscopic stochastic surface process modeling adsorption/desorption and/or surface di.usion of particles coupled to an ordinary di.erential equation (ODE) displaying bifurcations excited by a critical noise parameter. The models proposed here are caricatures of realistic systems arising in diverse applications ranging from surface processes and catalysis to atmospheric and oceanic models. We obtain deterministic mesoscopic models from the hybrid system by employing two methods: stochastic averaging principle and mean field closures. In this paper we focus on the case where phase transitions do not occur in the stochastic system. In the averaging principle case a faster stochastic mechanism is assumed compared to the ODE relaxation and a local equilibrium is induced with respect to the Gibbs measure on the lattice system. Under these circumstances remarkable agreement is observed between the hybrid system and the averaged system predictions. We exhibit several Monte Carlo simulations testing a variety of parameter regimes and displaying numerically the extent, limitations and validity of the theory. As expected fluctuation driven rare events do occur in several parameter regimes which could not possibly be captured by the deterministic averaging principle equation.

We consider the stochastic model of concentrated Liquid Crystal Polymers(LCPs) in the plane Couette flow. The dynamic equation for the liquid crystal polymers is described by a nonlinear stochastic differential equation with Maier-Saupe interaction potential. The stress tensor is obtained from an ensemble average of microscopic polymer configurations. We present the local existence and uniqueness theorem for the solution of the coupled fluid-polymer system. We also analyze the error of a fully .nite di.erence-Monte Carlo hybrid numerical scheme by investigating the asymptotic behavior of weakly interacting processes. The rate of convergence of the full discretized scheme is derived.

We prove that bounded solutions of the vanishing hyper-viscosity equation, converge to the entropy solution of the corresponding convex conservation law. The hyper-viscosity case lacks the monotonicity which underlines the Krushkov BV theory in the viscous case s = 1. Instead we show how to adapt the Tartar-Murat compensated compactness theory together with a weaker entropy dissipation bound to conclude the convergence of the vanishing hyper-viscosity.