Mathematical Models and Methods in Applied Sciences

The Vlasov–Maxwell–Fokker–Planck system is used in modeling distribution of charged particles in plasma, where particles interact via collisions and through their self-consistent electromagnetic field. We prove the existence of global in time classical solutions to the Cauchy problem near Maxwellians.

In order to study a one-dimensional analogue of the spontaneous curvature model for two-component lipid bilayer membranes, we consider planar curves that are made of a material with two phases. Each phase induces a preferred curvature to the curve, and these curvatures as well as phase boundaries may lead to the development of kinks. We introduce a family of energies for smooth curves and phase fields, and we show that these energies Γ-converge to an energy for curves with a finite number of kinks. The theoretical result is illustrated by some numerical examples.

We present a three-dimensional phenomenological model for the magneto-mechanical behavior of magnetic shape memory materials such as Ni₂MnGa. Moving from micromagnetic considerations, we advance to some thermodynamically consistent constitutive relations describing the magnetically-induced cubic-to-tetragonal martensitic transformation in a single crystal. We present an existence analysis for both the constitutive relation problem and the three-dimensional quasi-static evolution problem. Finally, we discuss the reduction of this model to some simpler one by means of a rigorous Γ-convergence analysis.

We consider the imaging of objects buried in unknown heterogeneous media. The medium is probed by using classical (e.g. acoustic or electromagnetic) waves. When heterogeneities in the medium become too strong, inversion methodologies based on a microscopic description of wave propagation (e.g. a wave equation or Maxwell's equations) become strongly dependent on the unknown details of the heterogeneous medium. In some situations, it is preferable to use a macroscopic model for a quantity that is quadratic in the wave fields. Here, such macroscopic models take the form of radiative transfer equations also referred to as transport equations. They can model either the energy density of the propagating wave fields or more generally the correlation of two wave fields propagating in possibly different media. In particular, we consider the correlation of the two fields propagating in the heterogeneous medium when the inclusion is absent and present, respectively. We present theoretical and numerical results showing that reconstructions based on this correlation are more accurate than reconstructions based on measurements of the energy density.

In this paper, we upscale the classical convection-diffusion equation in a narrow slit. We suppose that the transport parameters are such that we are in Taylor's regime, i.e. we deal with dominant Péclet numbers. In contrast to the classical work of Taylor, we undertake a rigorous derivation of the upscaled hyperbolic dispersion equation. Hyperbolic effective models were proposed by several authors and our goal is to confirm rigorously the effective equations derived by Balakotaiah et al. in recent years using a formal Lyapounov–Schmidt reduction. Our analysis uses the Laplace transform in time and an anisotropic singular perturbation technique, the small characteristic parameter ε being the ratio between the thickness and the longitudinal observation length. The Péclet number is written as Cε^-α, with α < 2. Hyperbolic effective model corresponds to a high Péclet number close to the threshold value when Taylor's regime turns to turbulent mixing and we characterize it by assuming 4/3 < α < 2. We prove that the difference between the dimensionless physical concentration and the effective concentration, calculated using the hyperbolic upscaled model, divided by ε^2-α (the local Péclet number) converges strongly to zero in L²-norm. For Péclet numbers considered in this paper, the hyperbolic dispersion equation turns out to give a better approximation than the classical parabolic Taylor model.

In this paper, we derive a simplified "quasi-1D" limit model for the propagation of low frequency acoustic waves in a laminar flow filling a 2D duct. We analyze the well-posedness of this model in function of the Mach profile of the flow. This can be reduced to the study of the spectrum of a bounded non-normal operator. As a by-product of this analysis, we establish new results for hydrodynamic instabilities of Kelvin–Helmholtz type in compressible fluids.

We consider the problem of reinforcing an elastic medium by a strong, rough, thin external layer. This model is governed by the Poisson equation with homogeneous Dirichlet boundary condition. We characterize the asymptotic behavior of the solution as the shear modulus of the layer goes to infinity. We find that there are four types of behaviors: the limiting solution satisfies Poisson equation with Dirichlet boundary condition, Robin boundary condition or Neumann boundary condition, or the limiting solution does not exist. The specific type depends on the integral of the load on the medium, the curvature of the interface and the scaling relations among the shear modulus, the thickness and the oscillation period of the layer.

In this paper we prove a homogenization theorem for interfacial discrete energies defined on an a-periodic Penrose tiling in ℝ². A general result on the homogenization of surface energies cannot be directly adapted to this case; the existence of the limit interfacial energy is therefore proved by showing some refined "quasi-periodic" properties of the tilings.