Mathematical Models and Methods in Applied Sciences

In this paper we undertake a rigorous derivation of the interface conditions between a poroelastic medium (the pay zone) and an elastic body (the non-pay zone). We assume that the poroelastic medium contains a pore structure of the characteristic size ε and that the fluid/structure interaction regime corresponds to diphasic Biot's law. The question is challenging because the Biot's equations for the poroelastic part contain one unknown more than the Navier equations for the non-pay zone. The solid part of the pay zone (the matrix) is elastic and the pores contain a viscous fluid. The fluid is assumed viscous and slightly compressible. We study the case when the contrast of property is of order ε², i.e. the normal stress of the elastic matrix is of the same order as the fluid pressure. We assume a periodic matrix and obtain the a priori estimates. Then we let the characteristic size of the inhomogeneities tend to zero and pass to the limit in the sense of the two-scale convergence. The obtained effective equations represent a two-scale system for three displacements and two pressures, coupled through the interface conditions with the Navier equations for the elastic displacement in the non-pay zone. We prove that the appropriate interface conditions at the interface between an elastic and a poroelastic medium are: (i) the effective displacement continuity, (ii) the effective normal stress continuity and (iii) the normal Darcy velocity zero from the poroelastic side. In addition we determine the effective boundary conditions for the contact between a poroelastic body and a rigid obstacle, giving us the effective outer boundary conditions.

We derive the equivalent energy of a square lattice that either deforms into the three- dimensional Euclidean space or remains planar. Interactions are not restricted to pairs of points and take into account changes of angles. Under some relationships between the local energies associated with the four vertices of an elementary square, we show that the limit energy can be obtained by mere quasiconvexification of the elementary cell energy and that the limit process does not involve any relaxation at the atomic scale. In this case, it can be said that the Cauchy–Born rule holds true. Our results apply to classical models of mechanical trusses that include torques between adjacent bars and to atomistic models.

This report is concerned with the question of computing accurate approximations to the motion of large structures in turbulent flows. We choose a fundamental closure model used in Large Eddy Simulation and investigate how a simple finite element method can be used to compute discrete solutions. We determine that both stability and accuracy of the discretization depend strongly on how filtering is performed. A numerical example is provided that supports our theoretical findings.

We consider the linear Wigner–Fokker–Planck equation subject to confining potentials which are smooth perturbations of the harmonic oscillator potential. For a certain class of perturbations we prove that the equation admits a unique stationary solution in a weighted Sobolev space. A key ingredient of the proof is a new result on the existence of spectral gaps for Fokker–Planck type operators in certain weighted L²-spaces. In addition we show that the steady state corresponds to a positive density matrix operator with unit trace and that the solutions of the time-dependent problem converge towards the steady state with an exponential rate.

We consider a third order in time equation which arises, e.g. as a model for wave propagation in viscous thermally relaxing fluids. This equation displays, even in the linear version, a variety of dynamical behaviors for its solution that depend on the physical parameters in the equation. These range from non-existence and instability to exponential stability (in time) as was shown for the constant coefficient case in Ref. 23. In case of vanishing diffusivity of the sound, there is a lack of generation of a semigroup associated with the linear dynamics. If diffusivity of the sound is positive, the linear dynamics is described by a strongly continuous hyperbolic-like evolution. This evolution is exponentially stable provided sufficiently large viscous damping is accounted for in the model.

In this paper, we consider the full nonlinear model referred to as Jordan–Moore–Gibson–Thompson equation. This model can be seen as a "hyperbolic" version of Kuznetsov's equation, where the linearization of the latter corresponds to an analytic semigroup. This is no longer valid for the presently considered third-order model whose linearization is associated with a group structure.

In order to carry out the analysis of the nonlinear model, we first consider time and space-dependent viscosity which then leads to evolution rather than semigroup generators. Decay rates for both "natural" and "higher" level energies are derived. Relevant physical parameters that are responsible for spectral behavior (continuous and point spectrum) are identified. The theoretical estimates proved in the paper are confirmed by numerical simulations. The derived energy estimates are then used in order to establish global well-posedness and exponential decay for the solutions to the nonlinear equation.

We are interested in the uniqueness of solutions to Maxwell's equations when the magnetic permeability μ and the permittivity ε are symmetric positive definite matrix-valued functions in ℝ³. We show that a unique continuation result for globally W^{1, ∞} coefficients in a smooth, bounded domain, allows one to prove that the solution is unique in the case of coefficients which are piecewise W^{1, ∞} with respect to a suitable countable collection of subdomains with C⁰ boundaries. Such suitable collections include any bounded finite collection. The proof relies on a general argument, not specific to Maxwell's equations. This result is then extended to the case when within these subdomains the permeability and permittivity are only L^∞ in sets of small measure.

We consider stable semidiscrete approximations of parametrized curve networks for gradient flows of elastic type functionals. Here meaningful and relevant conditions at junction points, such as double and triple junctions, need to be derived and suitably discretized. Examples for double junction types are C⁰ attachment and C¹ continuity. We develop strong and weak formulations for the elastic flow for curve networks with such junction points. For junctions with three or more curves, the conditions at the junctions are derived here for the first time. Possible applications include a simplified one-dimensional model of geometric biomembranes, as well as nonlinear splines in two and higher dimensions. The numerical results presented in this paper demonstrate the practicality of the introduced finite element approximations.