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:Alexandru Kristály Pages: 297 - 314 Abstract: The author proves the Poincaré lemma on some (n + 1)-dimensional corank 1 sub-Riemannian structures, formulating the \(\frac{{\left( {n - 1} \right)n\left( {{n^2} + 3n - 2} \right)}}{8}\) necessarily and sufficiently “curl-vanishing” compatibility conditions. In particular, this result solves partially an open problem formulated by Calin and Chang. The proof in this paper is based on a Poincaré lemma stated on Riemannian manifolds and a suitable Cesàro-Volterra path integral formula established in local coordinates. As a byproduct, a Saint-Venant lemma is also provided on generic Riemannian manifolds. Some examples are presented on the hyperbolic space and Carnot/Heisenberg groups. PubDate: 2018-03-01 DOI: 10.1007/s11401-018-1065-5 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:Jean Clairambault; Benoît Perthame; Andrada Quillas Maran Pages: 345 - 356 Abstract: Systems describing the dynamics of proliferative and quiescent cells are commonly used as computational models, for instance for tumor growth and hematopoiesis. Focusing on the very earliest stages of hematopoiesis, stem cells and early progenitors, the authors introduce a new method, based on an energy/Lyapunov functional to analyze the long time behavior of solutions. Compared to existing works, the method in this paper has the advantage that it can be extended to more complex situations. The authors treat a system with space variable and diffusion, and then adapt the energy functional to models with three equations. PubDate: 2018-03-01 DOI: 10.1007/s11401-018-1068-2 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)

Authors:Hao Zhao; Linan Zhong Pages: 1 - 8 Abstract: Let p be an odd prime. The authors detect a nontrivial element ã p of order p2 in the stable homotopy groups of spheres by the classical Adams spectral sequence. It is represented by \(a_0^{p - 2} h_1 \in Ext_A^{p - 1,pq + p - 2} (\mathbb{Z}/p,\mathbb{Z}/p)\) in the E2-term of the ASS and meanwhile p · ã p is the first periodic element α p . PubDate: 2018-01-01 DOI: 10.1007/s11401-018-1046-8 Issue No:Vol. 39, No. 1 (2018)

Authors:Yang Liu; Zhihua Chen; Yifei Pan Pages: 9 - 16 Abstract: The authors prove a general Schwarz lemma at the boundary for holomorphic mappings from the polydisc to the unit ball in any dimensions. For the special case of one complex variable, the obtained results give the classic boundary Schwarz lemma. PubDate: 2018-01-01 DOI: 10.1007/s11401-018-1047-7 Issue No:Vol. 39, No. 1 (2018)

Authors:Nan Gao; Xiaojing Xu Pages: 47 - 58 Abstract: The aim of this paper is two-fold. Given a recollement (T′, T, T″, i*, i*, i!, j!, j*, j*), where T′, T, T″ are triangulated categories with small coproducts and T is compactly generated. First, the authors show that the BBD-induction of compactly generated t-structures is compactly generated when i* preserves compact objects. As a con-sequence, given a ladder (T′, T, T″, T, T′) of height 2, then the certain BBD-induction of compactly generated t-structures is compactly generated. The authors apply them to the recollements induced by homological ring epimorphisms. This is the first part of their work. Given a recollement (D(B-Mod),D(A-Mod),D(C-Mod), i*, i*, i!, j!, j*, j*) induced by a homological ring epimorphism, the last aim of this work is to show that if A is Gorenstein, A B has finite projective dimension and j! restricts to D b (C-mod), then this recollement induces an unbounded ladder (B-Gproj,A-Gproj, C-Gproj) of stable categories of finitely generated Gorenstein-projective modules. Some examples are described. PubDate: 2018-01-01 DOI: 10.1007/s11401-018-1050-z Issue No:Vol. 39, No. 1 (2018)

Authors:Boyan Wei; Haipeng Qu; Yanfeng Luo Pages: 59 - 68 Abstract: A subgroup of index p k of a finite p-group G is called a k-maximal subgroup of G. Denote by d(G) the number of elements in a minimal generator-system of G and by δ k (G) the number of k-maximal subgroups which do not contain the Frattini subgroup of G. In this paper, the authors classify the finite p-groups with δd(G)(G) ≤ p2 and δd(G)−1(G) = 0, respectively. PubDate: 2018-01-01 DOI: 10.1007/s11401-018-1051-y Issue No:Vol. 39, No. 1 (2018)

Authors:Aiting Shen; Mei Yao; Benqiong Xiao Pages: 83 - 96 Abstract: In this paper, the complete convergence and the complete moment convergence for extended negatively dependent (END, in short) random variables without identical distribution are investigated. Under some suitable conditions, the equivalence between the moment of random variables and the complete convergence is established. In addition, the equivalence between the moment of random variables and the complete moment convergence is also proved. As applications, the Marcinkiewicz-Zygmund-type strong law of large numbers and the Baum-Katz-type result for END random variables are established. The results obtained in this paper extend the corresponding ones for independent random variables and some dependent random variables. PubDate: 2018-01-01 DOI: 10.1007/s11401-018-1053-9 Issue No:Vol. 39, No. 1 (2018)

Authors:Yu Wang; Tianzeng Li; Guosong Zhao Pages: 97 - 110 Abstract: The authors compute non-zero structure constants of the full flag manifold M = SO(7)/T with nine isotropy summands, then construct the Einstein equations. With the help of computer they get all the forty-eight positive solutions (up to a scale ) for SO(7)/T, up to isometry there are only five G-invariant Einstein metrics, of which one is Kähler Einstein metric and four are non-Kähler Einstein metrics. PubDate: 2018-01-01 DOI: 10.1007/s11401-018-1054-8 Issue No:Vol. 39, No. 1 (2018)

Authors:Wael W. Mohammed Pages: 145 - 162 Abstract: The main goal of this paper is to approximate the Kuramoto-Shivashinsky (K-S for short) equation on an unbounded domain near a change of bifurcation, where a band of dominant pattern is changing stability. This leads to a slow modulation of the dominant pattern. Here we consider PDEs with quadratic nonlinearities and derive rigorously the modulation equation, which is called the Ginzburg-Landau (G-L for short) equation, for the amplitudes of the dominating modes. PubDate: 2018-01-01 DOI: 10.1007/s11401-018-1057-5 Issue No:Vol. 39, No. 1 (2018)