Mathematical Models and Methods in Applied Sciences

We show the existence of global-in-time weak solutions to a general class of coupled FENE-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 centre-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker–Planck equation; in particular, the drag term needs 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 regularisation 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 (w.r.t. 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 centre-of-mass diffusion coefficient.

This paper proposes a numerical scheme to approximate the solution of (vectorial) limit load problems. The method makes use of a strictly convex perturbation of the problem, which corresponds to a projection of the deformation field under bounded deformation and incompressibility constraints. The discretized formulation of this perturbation converges to the solution of the original landslide problem when the amplitude of the perturbation tends to zero. The projection is computed numerically with a multi-step gradient descent on the dual formulation of the problem.

We consider homogenization problems for first-order Hamilton–Jacobi equations with u^∊/∊ periodic dependence, recently introduced by Imbert and Monneau, and also studied by Barles: this unusual dependence leads to nonstandard cell problems. We study the rate of convergence of the solution to the solution of the homogenized problem when the parameter ∊ tends to 0. We obtain the same rates as those obtained by Capuzzo Dolcetta and Ishii for the more usual homogenization problems without the dependence in u^∊/∊. In the second part, we study Eulerian schemes for the approximation of the cell problems. We prove that when the grid steps tend to zero, the approximation of the effective Hamiltonian converges to the effective Hamiltonian.

We consider a two- or three-dimensional time-dependent diffusion–convection–reaction problem and its discretization by the method of characteristics and standard finite elements. We perform the a posteriori error analysis of this discretization and prove optimal error estimates, which lead to an efficient adaptivity strategy both for the time step and the spatial mesh. The estimates are robust with respect to the ratios of the diffusion to the reaction or convection. Some numerical experiments support the theoretical results.

We prove a conjecture by De Giorgi on the elliptic regularization of semilinear wave equations in the finite-time case.

The most accurate model for real-life electrical impedance tomography is the complete electrode model, which takes into account electrode shapes and (usually unknown) contact impedances at electrode-object interfaces. When the electrodes are small, however, it is tempting to formally replace them by point sources. This simplifies the model considerably and completely eliminates the effect of contact impedance.

In this work we rigorously justify such a point electrode model for the important case of having difference measurements ("relative data") as data for the reconstruction problem. We do this by deriving the asymptotic limit of the complete model for vanishing electrode size. This is supplemented by an analogous result for the case that the distance between two adjacent electrodes also tends to zero, thus providing a physical interpretation and justification of the so-called backscattering data introduced by two of the authors.