Mathematical Models and Methods in Applied Sciences

We show the existence of global-in-time weak solutions to a general class of coupled Hookean-type bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier–Stokes equations in a bounded domain in ℝ^d, d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker–Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker–Planck equation; in particular, the drag term need not be corotational. With a square-integrable and divergence-free initial velocity datum ṵ₀ for the Navier–Stokes equation and a non-negative initial probability density function ψ₀ for the Fokker–Planck equation, which has finite relative entropy with respect to the Maxwellian M, we prove, via a limiting procedure on certain regularization parameters, the existence of a global-in-time weak solution t ↦ (ṵ(t), ψ(t)) to the coupled Navier–Stokes–Fokker–Planck system, satisfying the initial condition (ṵ(0), ψ(0)) = (ṵ₀, ψ₀), such that t ↦ ṵ(t) belongs to the classical Leray space and t ↦ ψ(t) has bounded relative entropy with respect to M and t ↦ ψ(t)/M has integrable Fisher information (with respect to the measure ) over any time interval [0, T], T>0. If the density of body forces on the right-hand side of the Navier–Stokes momentum equation vanishes, then a weak solution constructed as above is such that t ↦ (ṵ(t), ψ(t)) decays exponentially in time to in the -norm, at a rate that is independent of (ṵ₀, ψ₀) and of the center-of-mass diffusion coefficient. Our arguments rely on new compact embedding theorems in Maxwellian-weighted Sobolev spaces and a new extension of the Kolmogorov–Riesz theorem to Banach-space-valued Sobolev spaces.

We study the approximation of solutions to the steady Stokes problem with a discontinuous viscosity coefficient (interface Stokes problem) in the 2D "Discrete Duality Finite Volume" (DDFV) framework. In order to take into account the discontinuities of the viscosity and to prevent consistency defect in the scheme, we propose to modify the definition of the numerical fluxes on the edges of the mesh where the discontinuity occurs. We first show how to design our modified scheme, called m-DDFV, and we analyze its well-posedness and its convergence properties. Finally, we provide numerical results which confirm that the m-DDFV scheme significantly improves the convergence rate of the usual DDFV method for Stokes problems.

We consider the inverse problem of determining an optical mask that produces a desired circuit pattern in photolithography. We set the problem as a shape design problem in which the unknown is a two-dimensional domain. The relationship between the target shape and the unknown is modeled through diffractive optics. We develop a variational formulation that is well-posed and propose an approximation that can be shown to have convergence properties. The approximate problem can serve as a foundation to numerical methods, much like the Ambrosio–Tortorelli's approximation of the Mumford–Shah functional in image processing.

The guiding center drift induced by the homogenization of the Lorentz forces is studied. It generates memory effects. The memory (or nonlocal) kernel is described by the Volterra integral equation. The memory kernel can be characterized explicitly in terms of a Radon measure. It describes the extra velocity drift. By way of velocity drift, we view the Gauss's law with polarization charges.

This paper is devoted to the derivation of an a posteriori residual-based error estimator for the A-φ magnetodynamic harmonic formulation of the Maxwell system. The weak continuous and discrete formulations are established, and the well-posedness of both of them is addressed. Some useful analytical tools are derived. Among them, an ad hoc Helmholtz decomposition is proven, which allows to pertinently split the error. Consequently, an a posteriori error estimator is obtained, which is proven to be reliable and locally efficient. Finally, numerical tests confirm the theoretical results.