An overview of recent results pertaining to the hydrodynamic description (both Newtonianand non-Newtonian) of granular gases described by the Boltzmann equation for inelasticMaxwell models is presented. The use of this mathematical model allows us to get exactresults for different problems. First, the Navier–Stokes constitutive equations withexplicit expressions for the corresponding transport coefficients are derived by applyingthe Chapman–Enskog method to inelastic gases. Second, the non-Newtonian rheologicalproperties in the uniform shear flow (USF) are obtained in the steady state as well as inthe transient unsteady regime. Next, an exact solution for a special class of Couetteflows characterized by a uniform heat flux is worked out. This solution shares the samerheological properties as the USF and, additionally, two generalized transportcoefficients associated with the heat flux vector can be identified. Finally, the problemof small spatial perturbations of the USF is analyzed with a Chapman–Enskog-like methodand generalized (tensorial) transport coefficients are obtained.

We present a dynamic geometric model of phyllotaxis based on two postulates, primordiaformation and meristem expansion. We find that Fibonacci, Lucas, bijugate and multijugateare all variations of the same unifying phenomenon and that the difference lies in thechanges in position of initial primordia. We explore the set of all initial positions andcolor-code its points depending on the phyllotactic pattern that arises.

Shape optimization of mechanical devices is investigated in the context of large, geometrically strongly nonlinear deformations and nonlinear hyperelastic constitutive laws. A weighted sum of the structure compliance, its weight, and its surface area are minimized. The resulting nonlinear elastic optimization problem differs significantly from classical shape optimization in linearized elasticity. Indeed, there exist different definitions for the compliance: the change in potential energy of the surface load, the stored elastic deformation energy, and the dissipation associated with the deformation. Furthermore, elastically optimal deformations are no longer unique so that one has to choose the minimizing elastic deformation for which the cost functional should be minimized, and this complicates the mathematical analysis. Additionally, along with the non-uniqueness, buckling instabilities can appear, and the compliance functional may jump as the global equilibrium deformation switches between different bluckling modes. This is associated with a possible non-existence of optimal shapes in a worst-case scenario. In this paper the sharp-interface description of shapes is relaxed via an Allen-Cahn or Modica-Mortola type phase-field model, and soft material instead of void is considered outside the actual elastic object. An existence result for optimal shapes in the phase field as well as in the sharp-interface model is established, and the model behavior for decreasing phase-field interface width is investigated in terms of Γ-convergence. Computational results are based on a nested optimization with a trust-region method as the inner minimization for the equilibrium deformation and a quasi-Newton method as the outer minimization of the actual objective functional. Furthermore, a multi-scale relaxation approach with respect to the spatial resolution and the phase-field parameter is applied. Various computational studies underline the theoretical observations.

In the nonconvex case, solutions of rate-independent systems may develop jumps as afunction of time. To model such jumps, we adopt the philosophy that rate-independenceshould be considered as limit of systems with smaller and smaller viscosity. For thefinite-dimensional case we study the vanishing-viscosity limit of doubly nonlinearequations given in terms of a differentiable energy functional and a dissipation potentialthat is a viscous regularization of a given rate-independent dissipation potential. Theresulting definition of “BV solutions” involves, in a nontrivial way, both therate-independent and the viscous dissipation potential, which play crucial roles in thedescription of the associated jump trajectories. We shall prove general convergenceresults for the time-continuous and for the time-discretized viscous approximations andestablish various properties of the limiting BV solutions. In particular, we shall providea careful description of the jumps and compare the new notion of solutions with therelated concepts of energetic and local solutions to rate-independent systems.

We give a first contribution to the homogenization of many-body structures that areexposed to large deformations and obey the noninterpenetration constraint. The many-bodystructures considered here resemble cord-belts like they are used to reinforce pneumatictires. We establish and analyze an idealized model for such many-body structures in whichthe subbodies are assumed to be hyperelastic with a polyconvex energy density and shallexhibit an initial brittle bond with their neighbors. Noninterpenetration of matter istaken into account by the Ciarlet-Nečas condition and we demand deformations to preservethe local orientation. By studying Γ-convergence of the correspondingtotal energies as the subbodies become smaller and smaller, we find that thehomogenization limits allow for deformations of class special functions of boundedvariation while the aforementioned kinematic constraints are conserved. Depending on themany-body structures’ geometries, the homogenization limits feature new mechanical effectsranging from anisotropy to additional kinematic constraints. Furthermore, we introduce theconcept of predeformations in order to provide approximations for special functions ofbounded variation while preserving the natural kinematic constraints of geometricallynonlinear solid mechanics.

The paper is concerned with a class of optimal blocking problems in the plane. Weconsider a time dependent set R(t) ⊂ ℝ2,described as the reachable set for a differential inclusion. To restrict its growth, abarrier Γ can be constructed, in real time. This is a one-dimensionalrectifiable set which blocks the trajectories of the differential inclusion. In this paperwe introduce a definition of “regular strategy”, based on a careful classification ofblocking arcs. Moreover, we derive local and global necessary conditions for an optimalstrategy, which minimizes the total value of the burned region plus the cost ofconstructing the barrier. We show that a Lagrange multiplier, corresponding to theconstraint on the construction speed, can be interpreted as the “instantaneous value oftime”. This value, which we compute by two separate formulas, remains constant when freearcs are constructed and is monotone decreasing otherwise.

We present qualitative and quantitative comparisons of various analytical and numerical approximation methods for calculating a position of the early exercise boundary of American put options paying zero dividends. We analyse the asymptotic behaviour of these methods close to expiration, and introduce a new numerical scheme for computing the early exercise boundary. Our local iterative numerical scheme is based on a solution to a nonlinear integral equation. We compare numerical results obtained by the new method to those of the projected successive over-relaxation method and the analytical approximation formula recently derived by Zhu [‘A new analytical approximation formula for the optimal exercise boundary of American put options’, Int. J. Theor. Appl. Finance9 (2006) 1141–1177].

The aim of this paper is to develop a finite element method which allows computingthe buckling coefficients and modes of a non-homogeneous Timoshenko beam.Studying the spectral properties of a non-compact operator,we show that the relevant buckling coefficients correspond to isolatedeigenvalues of finite multiplicity.Optimal order error estimates are proved for the eigenfunctionsas well as a double order of convergence forthe eigenvalues using classical abstract spectral approximation theory for non-compact operators.These estimates are valid independently of the thickness of the beam, whichleads to the conclusion that the method is locking-free.Numerical tests are reported in order to assess the performance of the method.

In this article, we investigate numerical schemes for solvinga three component Cahn-Hilliard model. The space discretization isperformed by usinga Galerkin formulation and the finite element method.Concerning the time discretization,the main difficulty is to write a scheme ensuring,at the discrete level, the decrease of the free energyand thus the stability of the method.We study three different schemes and proveexistence and convergence theorems. Theoretical results areillustrated by various numerical examples showing that the new semi-implicitdiscretization that we propose seems to be a good compromise between robustnessand accuracy.

We consider a two-player zero-sum-game in a bounded open domain Ωdescribed as follows: at a point x ∈ Ω, Players I and IIplay an ε-step tug-of-war game with probability α, andwith probability β (α + β = 1), arandom point in the ball of radius ε centered at x ischosen. Once the game position reaches the boundary, Player II pays Player I the amountgiven by a fixed payoff function F. We give a detailed proof of the factthat the value functions of this game satisfy the Dynamic Programming Principle

for x ∈ Ω withu(y) = F(y) wheny ∉ Ω. This principle implies the existence ofquasioptimal Markovian strategies.

A Bernoulli free boundary problem with geometrical constraints is studied. The domain Ω is constrained to lie in the half space determined by x1 ≥ 0 and its boundary to contain a segment of the hyperplane  {x1 = 0}  where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution have already been addressed and this paper is devoted first to the study of geometric and asymptotic properties of the solution and then to the numerical treatment of the problem using a shape optimization formulation. The major difficulty and originality of this paper lies in the treatment of the geometric constraints.

In this paper we study two weak notions of Jacobian determinant for Sobolev maps, namely the distributional Jacobian and the relaxed total variation, which in general could be different. We show some cases of equality and use them to give an explicit expression for the relaxation of some polyconvex functionals.

In this paper, we introduce a new method for feedback controller design for the complex distributed parameter networks governed by wave equations, which ensures the stability of the closed loop system. This method is based on the uniqueness theory of ordinary differential equations and cutting-edge approach in the graph theory, but it is not a simple extension. As a realization of this idea, we investigate a bush-type wave network. The well-posedness of the closed loop system is obtained via Lax-Milgram’s lemma and semigroup theory. The validity of cutting-edge method is proved by spectral analysis approach. In particular, we give a detailed procedure of cutting-edge for the bush-type wave networks. The results show that if we impose feedback controllers, consisting of velocity and position terms, at all the boundary vertices and at most three velocity feedback controllers on the cycle, the system is asymptotically stabilized. Finally, some examples are given.

The stabilization with time delay in observation or control represents difficultmathematical challenges in the control of distributed parameter systems. It is well-knownthat the stability of closed-loop system achieved by some stabilizing output feedback lawsmay be destroyed by whatever small time delay there exists in observation. In this paper,we are concerned with a particularly interesting case: Boundary output feedbackstabilization of a one-dimensional wave equation system for which the boundary observationsuffers from an arbitrary long time delay. We use the observer and predictor to solve theproblem: The state is estimated in the time span where the observation is available; andthe state is predicted in the time interval where the observation is not available. It isshown that the estimator/predictor based state feedback law stabilizes the delay systemasymptotically or exponentially, respectively, relying on the initial data beingnon-smooth or smooth. Numerical simulations are presented to illustrate the effect of thestabilizing controller.

In this paper we analyse an approximate controllability result for a nonlinear population dynamics model. In this model the birth term is nonlocal and describes the recruitment process in newborn individuals population, and the control acts on a small open set of the domain and corresponds to an elimination or a supply of newborn individuals. In our proof we use a unique continuation property for the solution of the heat equation and the Kakutani-Fan-Glicksberg fixed point theorem.

We study integral functionals constrained to divergence-free vector fields in Lp on a thin domain, under standard p-growth and coercivity assumptions, 1 < p < ∞. We prove that as the thickness of the domain goes to zero, the Gamma-limit with respect to weak convergence in Lp is always given by the associated functional with convexified energy density wherever it is finite. Remarkably, this happens despite the fact that relaxation of nonconvex functionals subject to the limiting constraint can give rise to a nonlocal functional as illustrated in an example.

In this paper we study a one dimensional model of ferromagnetic nano-wires of finitelength. First we justify the model by Γ-convergence arguments.Furthermore we prove the existence of wall profiles. These walls being unstable, westabilize them by the mean of an applied magnetic field.

Cell-centered and vertex-centered finite volume schemes for the Laplace equationwith homogeneous Dirichlet boundary conditionsare considered on a triangular mesh and on the Voronoi diagram associated to its vertices.A broken P 1 function is constructed from the solutions of both schemes.When the domain is two-dimensional polygonal convex,it is shown that this reconstructionconverges with second-order accuracy towards the exact solution in the L 2 norm,under the sufficient condition that the right-hand side of the Laplace equation belongs to H 1(Ω).

We consider multiscale systems for which only a fine-scalemodel describing the evolution of individuals (atoms,molecules, bacteria, agents) is given, while we are interested in theevolution of the population density on coarse space and timescales. Typically, this evolution is described by a coarseFokker-Planck equation. In this paper, we consider a numerical procedure to compute the solution ofthis Fokker-Planck equation directly on the coarse level, based on the estimation of the unknown parameters (drift and diffusion) using only appropriately chosen realizations of the fine-scale,individual-basedsystem. As these parameters might be space- and time-dependent, theestimation is performed in every spatial discretization point and atevery time step. If the fine-scale model is stochastic, the estimationprocedure introduces noise on the coarse level.We investigate stability conditions for this procedure in thepresence of this noise and present ananalysis of the propagation of the estimation error in the numericalsolution of the coarse Fokker-Planck equation.

We consider linear elliptic systems which arisein coupled elastic continuum mechanical models. In these systems, the straintensor ε P := sym (P -1∇u) is redefined to include amatrix valued inhomogeneity P(x) which cannot be described by a spacedependent fourth order elasticity tensor. Such systems arise naturally ingeometrically exact plasticity or in problems with eigenstresses.The tensor field P induces a structural change of the elasticity equations. Forsuch a model the FETI-DP method is formulated and a convergence estimateis provided for the special case that P -T = ∇ψ is a gradient.It is shown that the condition number depends only quadratic-logarithmicallyon the number of unknowns of each subdomain. Thedependence of the constants of the bound on P is highlighted. Numericalexamples confirm our theoretical findings. Promising results are also obtainedfor settings which are not covered by our theoretical estimates.