Authors:Georges Griso; Bernadette Miara Pages: 397 - 426 Abstract: Consider an elastic thin three-dimensional body made of a periodic distribution of elastic inclusions. When both the thickness of the beam and the size of the heterogeneities tend simultaneously to zero the authors obtain three different one-dimensional models of beam depending upon the limit of the ratio of these two small parameters. PubDate: 2018-05-01 DOI: 10.1007/s11401-018-0075-7 Issue No:Vol. 39, No. 3 (2018)

Authors:Philippe Destuynder; Caroline Fabre Pages: 427 - 450 Abstract: The development of foils for racing boats has changed the strategy of sailing. Recently, the America’s cup held in San Francisco, has been the theatre of a tragicomic history due to the foils. During the last round, the New-Zealand boat was winning by 8 to 1 against the defender USA. The winner is the first with 9 victories. USA team understood suddenly (may be) how to use the control of the pitching of the main foils by adjusting the rake in order to stabilize the ship. And USA won by 9 victories against 8 to the challenger NZ. The goal in this paper is to point out few aspects which could be taken into account in order to improve this mysterious control law which is known as the key of the victory of the USA team. There are certainly many reasons and in particular the cleverness of the sailors and of all the engineering team behind this project. But it appears interesting to have a mathematical discussion, even if it is a partial one, on the mechanical behaviour of these extraordinary sailing boats. The numerical examples given here are not the true ones. They have just been invented in order to explain the theoretical developments concerning three points: The possibility of tacking on the foils for sailing upwind, the nature of foiling instabilities, if there are, when the boat is flying and the control laws. PubDate: 2018-05-01 DOI: 10.1007/s11401-018-0076-6 Issue No:Vol. 39, No. 3 (2018)

Authors:Giuseppe Geymonat; Françoise Krasucki; Michele Serpilli Pages: 451 - 460 Abstract: The authors use the asymptotic expansion method by P. G. Ciarlet to obtain a Kirchhoff-Love-type plate model for a linear soft ferromagnetic material. They also give a mathematical justification of the obtained model by means of a strong convergence result. PubDate: 2018-05-01 DOI: 10.1007/s11401-018-0077-5 Issue No:Vol. 39, No. 3 (2018)

Authors:Alain Bensoussan; Miroslav Bulíček; Jens Frehse Pages: 461 - 486 Abstract: The authors deal with nonlinear elliptic and parabolic systems that are the Bellman like systems associated to stochastic differential games with mean field dependent dynamics. The key novelty of the paper is that they allow heavily mean field dependent dynamics. This in particular leads to a system of PDE’s with critical growth, for which it is rare to have an existence and/or regularity result. In the paper, they introduce a structural assumptions that cover many cases in stochastic differential games with mean field dependent dynamics for which they are able to establish the existence of a weak solution. In addition, the authors present here a completely new method for obtaining the maximum/minimum principles for systems with critical growths, which is a starting point for further existence and also qualitative analysis. PubDate: 2018-05-01 DOI: 10.1007/s11401-018-0078-4 Issue No:Vol. 39, No. 3 (2018)

Authors:Luca Dedè; Alfio Quarteroni Pages: 487 - 512 Abstract: The authors consider a phase field model for Darcy flows with discontinuous data in porous media; specifically, they adopt the Hele-Shaw-Cahn-Hillard equations of [Lee, Lowengrub, Goodman, Physics of Fluids, 2002] to model flows in the Hele-Shaw cell through a phase field formulation which incorporates discontinuities of physical data, namely density and viscosity, across interfaces. For the spatial approximation of the problem, the authors use NURBS—based isogeometric analysis in the framework of the Galerkin method, a computational framework which is particularly advantageous for the solution of high order partial differential equations and phase field problems which exhibit sharp but smooth interfaces. In this paper, the authors verify through numerical tests the sharp interface limit of the phase field model which in fact leads to an internal discontinuity interface problem; finally, they show the efficiency of isogeometric analysis for the numerical approximation of the model by solving a benchmark problem, the so-called “rising bubble” problem. PubDate: 2018-05-01 DOI: 10.1007/s11401-018-0079-3 Issue No:Vol. 39, No. 3 (2018)

Authors:Maria Malin; Cristinel Mardare Pages: 513 - 534 Abstract: The authors establish several estimates showing that the distance in W1,p, 1 < p < ∞, between two immersions from a domain of R n into Rn+1 is bounded by the distance in L p between two matrix fields defined in terms of the first two fundamental forms associated with each immersion. These estimates generalize previous estimates obtained by the authors and P. G. Ciarlet and weaken the assumptions on the fundamental forms at the expense of replacing them by two different matrix fields. PubDate: 2018-05-01 DOI: 10.1007/s11401-018-0080-x Issue No:Vol. 39, No. 3 (2018)

Authors:Shihai Zhao; Yao Yu; Tsorng-Whay Pan; Roland Glowinski Pages: 535 - 552 Abstract: In this article, a computational model and related methodologies have been tested for simulating the motion of a malaria infected red blood cell (iRBC for short) in Poiseuille flow at low Reynolds numbers. Besides the deformability of the red blood cell membrane, the migration of a neutrally buoyant particle (used to model the malaria parasite inside the membrane) is another factor to determine the iRBC motion. Typically an iRBC oscillates in a Poiseuille flow due to the competition between these two factors. The interaction of an iRBC and several RBCs in a narrow channel shows that, at lower flow speed, the iRBC can be easily pushed toward the wall and stay there to block the channel. But, at higher flow speed, RBCs and iRBC stay in the central region of the channel since their migrations are dominated by the motion of the RBC membrane. PubDate: 2018-05-01 DOI: 10.1007/s11401-018-0081-9 Issue No:Vol. 39, No. 3 (2018)

Authors:Zhi-Tao Wen; Roderick Wong; Shuai-Xia Xu Pages: 553 - 596 Abstract: In this paper, the authors consider the asymptotic behavior of the monic polynomials orthogonal with respect to the weight function w(x) = x 2αe−(x4+tx2), x ∈ R, where α is a constant larger than −1/2 and t is any real number. They consider this problem in three separate cases: (i) c > −2, (ii) c = −2, and (iii) c < −2, where c:= tN−1/2 is a constant, N = n + α and n is the degree of the polynomial. In the first two cases, the support of the associated equilibrium measure μt is a single interval, whereas in the third case the support of μt consists of two intervals. In each case, globally uniform asymptotic expansions are obtained in several regions. These regions together cover the whole complex plane. The approach is based on a modified version of the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou (1993). PubDate: 2018-05-01 DOI: 10.1007/s11401-018-0082-8 Issue No:Vol. 39, No. 3 (2018)

Authors:Michel Chipot Pages: 597 - 606 Abstract: The author presents a method allowing to obtain existence of a solution for some elliptic problems set in unbounded domains, and shows exponential rate of convergence of the approximate solution toward the solution. PubDate: 2018-05-01 DOI: 10.1007/s11401-018-0083-7 Issue No:Vol. 39, No. 3 (2018)

Authors:Hervé Le Dret; Amira Mokrane Pages: 163 - 182 Abstract: This paper deals with minimization problems in the calculus of variations set in a sequence of domains, the size of which tends to infinity in certain directions and such that the data only depend on the coordinates in the directions that remain constant. The authors study the asymptotic behavior of minimizers in various situations and show that they converge in an appropriate sense toward minimizers of a related energy functional in the constant directions. PubDate: 2018-03-01 DOI: 10.1007/s11401-018-1058-4 Issue No:Vol. 39, No. 2 (2018)

Authors:Patrizia Donato; Sorin Mardare; Bogdan Vernescu Pages: 183 - 200 Abstract: The Bingham fluid model has been successfully used in modeling a large class of non-Newtonian fluids. In this paper, the authors extend to the case of Bingham fluids the results previously obtained by Chipot and Mardare, who studied the asymptotics of the Stokes flow in a cylindrical domain that becomes unbounded in one direction, and prove the convergence of the solution to the Bingham problem in a finite periodic domain, to the solution of the Bingham problem in the infinite periodic domain, as the length of the finite domain goes to infinity. As a consequence of this convergence, the existence of a solution to a Bingham problem in the infinite periodic domain is obtained, and the uniqueness of the velocity field for this problem is also shown. Finally, they show that the error in approximating the velocity field in the infinite domain with the velocity in a periodic domain of length 2ℓ has a polynomial decay in ℓ, unlike in the Stokes case (see [Chipot, M. and Mardare, S., Asymptotic behaviour of the Stokes problem in cylinders becoming unbounded in one direction, Journal de Mathématiques Pures et Appliquées, 90(2), 2008, 133–159]) where it has an exponential decay. This is in itself an important result for the numerical simulations of non-Newtonian flows in long tubes. PubDate: 2018-03-01 DOI: 10.1007/s11401-018-1059-3 Issue No:Vol. 39, No. 2 (2018)

Authors:Jixun Chu; Jean-Michel Coron; Peipei Shang; Shu-Xia Tang Pages: 201 - 212 Abstract: In this paper, the authors consider the Gevrey class regularity of a semigroup associated with a nonlinear Korteweg-de Vries (KdV for short) equation. By estimating the resolvent of the corresponding linear operator, the authors conclude that the semigroup generated by the linear operator is not analytic but of Gevrey class δ ∈ (3/2, ∞) for t > 0. PubDate: 2018-03-01 DOI: 10.1007/s11401-018-1060-x Issue No:Vol. 39, No. 2 (2018)

Authors:Olivier Pironneau Pages: 213 - 232 Abstract: The conservation laws of continuum mechanics, written in an Eulerian frame, do not distinguish fluids and solids, except in the expression of the stress tensors, usually with Newton’s hypothesis for the fluids and Helmholtz potentials of energy for hyperelastic solids. By taking the velocities as unknown monolithic methods for fluid structure interactions (FSI for short) are built. In this paper such a formulation is analysed when the solid is compressible and the fluid is incompressible. The idea is not new but the progress of mesh generators and numerical schemes like the Characteristics-Galerkin method render this approach feasible and reasonably robust. In this paper the method and its discretisation are presented, stability is discussed through an energy estimate. A numerical section discusses implementation issues and presents a few simple tests. PubDate: 2018-03-01 DOI: 10.1007/s11401-018-1061-9 Issue No:Vol. 39, No. 2 (2018)

Authors:Tatsien Li; Xing Lu; Bopeng Rao Pages: 233 - 252 Abstract: In this paper, for a coupled system of wave equations with Neumann boundary controls, the exact boundary synchronization is taken into consideration. Results are then extended to the case of synchronization by groups. Moreover, the determination of the state of synchronization by groups is discussed with details for the synchronization and for the synchronization by 3-groups, respectively. PubDate: 2018-03-01 DOI: 10.1007/s11401-018-1062-8 Issue No:Vol. 39, No. 2 (2018)

Authors:Alaaeddine Hammoudi; Oana Iosifescu Pages: 253 - 280 Abstract: The goal of this paper is to study the mathematical properties of a new model of soil carbon dynamics which is a reaction-diffusion system with a chemotactic term, with the aim to account for the formation of soil aggregations in the bacterial and microorganism spatial organization (hot spot in soil). This is a spatial and chemotactic version of MOMOS (Modelling Organic changes by Micro-Organisms of Soil), a model recently introduced by M. Pansu and his group. The authors present here two forms of chemotactic terms, first a “classical” one and second a function which prevents the overcrowding of microorganisms. They prove in each case the existence of a nonnegative global solution, and investigate its uniqueness and the existence of a global attractor for all the solutions. PubDate: 2018-03-01 DOI: 10.1007/s11401-018-1063-7 Issue No:Vol. 39, No. 2 (2018)

Authors:Pierre Lissy; Enrique Zuazua Pages: 281 - 296 Abstract: This paper deals with the problem of internal controllability of a system of heat equations posed on a bounded domain with Dirichlet boundary conditions and perturbed with analytic non-local coupling terms. Each component of the system may be controlled in a different subdomain. Assuming that the unperturbed system is controllable—a property that has been recently characterized in terms of a Kalman-like rank condition—the authors give a necessary and sufficient condition for the controllability of the coupled system under the form of a unique continuation property for the corresponding elliptic eigenvalue system. The proof relies on a compactness-uniqueness argument, which is quite unusual in the context of parabolic systems, previously developed for scalar parabolic equations. The general result is illustrated by two simple examples. PubDate: 2018-03-01 DOI: 10.1007/s11401-018-1064-6 Issue No:Vol. 39, No. 2 (2018)

Authors:Lourenço Beirão Da Veiga; Franco Brezzi; Franco Dassi; Luisa Donatelia Marini; Alessandro Russo Pages: 315 - 334 Abstract: The authors study the use of the virtual element method (VEM for short) of order k for general second order elliptic problems with variable coefficients in three space dimensions. Moreover, they investigate numerically also the serendipity version of the VEM and the associated computational gain in terms of degrees of freedom. PubDate: 2018-03-01 DOI: 10.1007/s11401-018-1066-4 Issue No:Vol. 39, No. 2 (2018)

Authors:Alain Damlamian Pages: 335 - 344 Abstract: A recent joint paper with Doina Cioranescu and Julia Orlik was concerned with the homogenization of a linearized elasticity problem with inclusions and cracks (see [Cioranescu, D., Damlamian, A. and Orlik, J., Homogenization via unfolding in periodic elasticity with contact on closed and open cracks, Asymptotic Analysis, 82, 2013, 201–232]). It required uniform estimates with respect to the homogenization parameter. A Korn inequality was used which involves unilateral terms on the boundaries where a nopenetration condition is imposed. In this paper, the author presents a general method to obtain many diverse Korn inequalities including the unilateral inequalities used in [Cioranescu, D., Damlamian, A. and Orlik, J., Homogenization via unfolding in periodic elasticity with contact on closed and open cracks, Asymptotic Analysis, 82, 2013, 201–232]. A preliminary version was presented in [Damlamian, A., Some unilateral Korn inequalities with application to a contact problem with inclusions, C. R. Acad. Sci. Paris, Ser. I, 350, 2012, 861–865]. PubDate: 2018-03-01 DOI: 10.1007/s11401-018-1067-3 Issue No:Vol. 39, No. 2 (2018)

Authors:Yongqiang Fu; Houwang Li; Patrizia Pucci Pages: 357 - 372 Abstract: The authors study the following Dirichlet problem of a system involving fractional (p, q)-Laplacian operators: $$\left\{ {\begin{array}{*{20}{c}} {\left( { - \Delta } \right)_p^su = \lambda a\left( x \right){{\left u \right }^{p - 2}}u + \lambda b\left( x \right){{\left u \right }^{\alpha - 2}}{{\left v \right }^\beta }u + \frac{{\mu \left( x \right)}}{{\alpha \delta }}{{\left u \right }^{\gamma - 2}}{{\left v \right }^\delta }uin\Omega ,} \\ {\left( { - \Delta } \right)_q^sv = \lambda c\left( x \right){{\left v \right }^{q - 2}}v + \lambda b\left( x \right){{\left u \right }^\alpha }{{\left v \right }^{\beta - 2}}v + \frac{{\mu \left( x \right)}}{{\beta \gamma }}{{\left u \right }^\gamma }{{\left v \right }^{\delta - 2}}vin\Omega ,} \\ {u = v = 0on{\mathbb{R}^N}\backslash \Omega ,} \end{array}} \right.$$ where λ > 0 is a real parameter, Ω is a bounded domain in R N , with boundary ∂Ω Lipschitz continuous, s ∈ (0, 1), 1 < p ≤ q < ∞, sq < N, while (−Δ) p s u is the fractional p-Laplacian operator of u and, similarly, (−Δ) q s v is the fractional q-Laplacian operator of v. Since possibly p ≠ q, the classical definitions of the Nehari manifold for systems and of the Fibering mapping are not suitable. In this paper, the authors modify these definitions to solve the Dirichlet problem above. Then, by virtue of the properties of the first eigenvalue λ1 for a related system, they prove that there exists a positive solution for the problem when λ < λ1 by the modified definitions. Moreover, the authors obtain the bifurcation property when λ → λ1 -. Finally, thanks to the Picone identity, a nonexistence result is also obtained when λ ≥ λ1. PubDate: 2018-03-01 DOI: 10.1007/s11401-018-1069-1 Issue No:Vol. 39, No. 2 (2018)

Authors:Felipe Cucker Pages: 373 - 396 Abstract: In recent years, a family of numerical algorithms to solve problems in real algebraic and semialgebraic geometry has been slowly growing. Unlike their counterparts in symbolic computation they are numerically stable. But their complexity analysis, based on the condition of the data, is radically different from the usual complexity analysis in symbolic computation as these numerical algorithms may run forever on a thin set of ill-posed inputs. PubDate: 2018-03-01 DOI: 10.1007/s11401-018-1070-8 Issue No:Vol. 39, No. 2 (2018)