Description:    We consider an initial boundary value problem for a quantum version of the Zakharov system arising in plasma physics. We prove the global well-posedness of this problem in some Sobolev type classes and study properties of solutions. This means that the quantum Zakharov model is coherent with the observation of an absence of collapse of (quantum) Langmuir waves, hence might be a valid model for the description of electronic plasma waves. In the dissipative case the existence of a finite-dimensional global attractor is established, and regularity properties of this attractor are studied. For this we use the recently developed method of quasi-stability estimates. In the case when external forces are C ∞ functions, we show that every trajectory in the attractor is C ∞ in both time and spatial variables. This can be interpreted as the absence of sharp coherent structures in the limiting dynamics. Content Type Journal Article Pages 1-23 DOI 10.1007/s00033-012-0278-9 Authors Igor Chueshov, Department of Mechanics and Mathematics, Kharkov National University, 61077 Kharkov, Ukraine Journal Zeitschrift für Angewandte Mathematik und Physik (ZAMP) Online ISSN 1420-9039 Print ISSN 0044-2275

Description:    In shape optimization, recently the question arose, whether or not the cylindrical pipe has the optimal shape for the transport of an incompressible fluid. In this short note, a proof will be presented that a cylindrical pipe with Poiseuille’s flow inside indeed is optimal for the transportation of an incompressible fluid under the criterion “energy dissipated by the fluid.” The proof reduces the problem to the minimization of a two-dimensional Dirichlet’s integral. This simpler problem can be solved with a symmetrization argument. Content Type Journal Article Pages 1-9 DOI 10.1007/s00033-012-0277-x Authors Andreas Schulz, Karlsruhe Institute of Technology (KIT), Institute for Applied and Numerical Mathematics, Kaiserstraße 89-93, 76128 Karlsruhe, Germany Journal Zeitschrift für Angewandte Mathematik und Physik (ZAMP) Online ISSN 1420-9039 Print ISSN 0044-2275

Description:    By a sequence of rollings without slipping or twisting along segments of a straight line of the plane, a spherical ball of unit radius has to be transferred from an initial state to an arbitrary final state taking into account the orientation of the ball. We provide a new proof that with at most 3 moves, we can go from a given initial state to an arbitrary final state. The first proof of this result is due to Hammersley ( 1983 ). His proof is more algebraic than ours which is more geometric. We also showed that “generically” no one of the three moves, in any elimination of the spin discrepancy, may have length equal to an integral multiple of 2π. Content Type Journal Article Pages 1-13 DOI 10.1007/s00033-012-0279-8 Authors Laura M. O. Biscolla, Universidade Paulista, Rua Dr. Bacelar 1212, São Paulo, CEP 04026-002, Brazil Jaume Llibre, Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra Barcelona, Catalonia, Spain Waldyr M. Oliva, CAMGSD, ISR, Instituto Superior Técnico, UTL, Av. Rovisco Pais, 1049-001 Lisbon, Portugal Journal Zeitschrift für Angewandte Mathematik und Physik (ZAMP) Online ISSN 1420-9039 Print ISSN 0044-2275

Description:    In this paper, we propose a thermodynamically consistent model for superfluid-normal phase transition in liquid helium, accounting for variations of temperature and density. The phase transition is described by means of an order parameter, according to the Ginzburg–Landau theory, emphasizing the analogies between superfluidity and superconductivity. The normal component of the velocity is assumed to be compressible, and the usual phase diagram of liquid helium is recovered. Moreover, the continuity equation leads to a dependence between density and temperature in agreement with the experimental data. Content Type Journal Article Pages 1-11 DOI 10.1007/s00033-012-0280-2 Authors Alessia Berti, Facoltà di Ingegneria, Università e-Campus, 22060 Novedrate (CO), Italy Valeria Berti, Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy Journal Zeitschrift für Angewandte Mathematik und Physik (ZAMP) Online ISSN 1420-9039 Print ISSN 0044-2275

Description:    In the present work, we deal with the convergence of a class of numerical schemes for maximal monotone evolution systems in the particular case where the maximal monotone term is a subdifferential of a convex proper and lower semi-continuous function and the right-hand side depends on time and on solution. More precisely, we focus on an implicit Euler scheme and we show that the order of this scheme is one. Finally, some applications are given for a large class of rheological models. Content Type Journal Article Pages 1-12 DOI 10.1007/s00033-012-0276-y Authors Jérôme Bastien, Centre de Recherche et d’Innovation sur le Sport (CRIS), U.F.R.S.T.A.P.S., Université Claude Bernard-Lyon 1, 27-29, Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France Journal Zeitschrift für Angewandte Mathematik und Physik (ZAMP) Online ISSN 1420-9039 Print ISSN 0044-2275

Description:    This paper is concerned with the multidimensional stability of non-isothermal subsonic phase transitions in a steady supersonic flow of van der Waals type. For the sake of seeking physical admissible planar waves, the viscosity–capillarity criterion (Slemrod in Arch Ration Mech Anal 81(4):301–315, 1983 ) is chosen to be the admissible criterion. By showing the Lopatinski determinant being non-zero, we prove that subsonic phase transitions are uniformly stable in the sense of Majda (Mem Am Math Soc 41(275):1–95, 1983 ) under both one-dimensional and multidimensional perturbations. Content Type Journal Article Pages 1-16 DOI 10.1007/s00033-012-0275-z Authors Shu-Yi Zhang, Business Information Management School, Shanghai Institute of Foreign Trade, 1900 Wenxiang Rd., Shanghai, 201620 People’s Republic of China Journal Zeitschrift für Angewandte Mathematik und Physik (ZAMP) Online ISSN 1420-9039 Print ISSN 0044-2275

Description:    In this paper, we provide a complete stability analysis for an abstract system of coupled hyperbolic and parabolic equations ll      u tt = - Au + g A a q ,    q t = - g A a u t - kA b q , u (0) = u 0 ,     u t (0) = v 0 ,     q (0) = q 0 where A is a self-adjoint, positive definite operator on a Hilbert space H . For ( a , b ) Î [0,1] ×[0,1] , the region of exponential stability had been identified in Ammar-Khodja et al. (ESAIM Control Optim Calc Var 4:577–593, 1999 ). Our contribution is to show that the rest of the region can be classified as region of polynomial stability and region of instability. Moreover, we obtain the optimality of the order of polynomial stability. Content Type Journal Article Pages 1-15 DOI 10.1007/s00033-012-0274-0 Authors Jianghao Hao, School of Mathematical Sciences, Shanxi University, Taiyuan, 030006 Shanxi, People’s Republic of China Zhuangyi Liu, Department of Mathematics and Statistics, University of Minnesota, Duluth, MN 55812-2496, USA Journal Zeitschrift für Angewandte Mathematik und Physik (ZAMP) Online ISSN 1420-9039 Print ISSN 0044-2275

Description:    General solutions for the problems of an elastic half-space and an elastic half-plane, respectively, subjected to a symmetrically distributed normal force of arbitrary profile are analytically derived using a simplified strain gradient elasticity theory (SSGET) that contains one material length scale parameter. Mindlin’s potential function method and Fourier transforms are employed in the formulation, and the half-space and half-plane contact problems are solved in a unified manner. The specific solutions for the problems of a half-space/plane subjected to a concentrated normal force or a uniformly distributed normal force are obtained by directly applying the general solutions, which recover the existing classical elasticity-based solutions of the Flamant and Boussinesq problems as special cases. In addition, the indentation problems of an elastic half-space indented by a flat-ended cylindrical punch, a spherical punch, and a conical punch, respectively, are solved using the general solutions, leading to hardness formulas that are indentation size- and material microstructure-dependent. Numerical results reveal that the displacement and stress fields in a half-space/plane given by the current SSGET-based solutions are smoother than those predicted by the classical elasticity-based solutions and do not exhibit the discontinuity and/or singularity displayed by the latter. Also, the indentation hardness values based on the newly obtained half-space solution are found to increase with decreasing indentation radius and increasing material length scale parameter, thereby explaining the microstructure-dependent indentation size effect. Content Type Journal Article Pages 1-24 DOI 10.1007/s00033-012-0273-1 Authors Xin-Lin Gao, Department of Mechanical Engineering, University of Texas at Dallas, 800 West Campbell Road, Richardson, TX 75080-3021, USA Song-Sheng Zhou, Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, USA Journal Zeitschrift für Angewandte Mathematik und Physik (ZAMP) Online ISSN 1420-9039 Print ISSN 0044-2275

Description:    In this paper, we study the existence of minimizers for F ( u ) = \frac12 ó õ \mathbb R 3   | Ñ u | 2 d x + \frac14 ó õ \mathbb R 3   ó õ \mathbb R 3   \frac | u ( x ) | 2 | u ( y ) | 2 | x - y | d x d y - \frac1 p ó õ \mathbb R 3   | u | p d x on the constraint S ( c ) = { u Î H 1 (\mathbb R 3 ) : ó õ \mathbb R 3   | u | 2 d x = c } , where c 〉  0 is a given parameter. In the range p Î [3,\frac103] , we explicit a threshold value of c 〉  0 separating existence and nonexistence of minimizers. We also derive a nonexistence result of critical points of F ( u ) restricted to S ( c ) when c 〉  0 is sufficiently small. Finally, as a by-product of our approaches, we extend some results of Colin et al. (Nonlinearity 23(6):1353–1385, 2010 ) where a constrained minimization problem, associated with a quasi-linear equation, is considered. Content Type Journal Article Pages 1-18 DOI 10.1007/s00033-012-0272-2 Authors Louis Jeanjean, Laboratoire de Mathématiques (UMR 6623), Université de Franche-Comté, 16, Route de Gray, 25030 Besançon Cedex, France Tingjian Luo, Laboratoire de Mathématiques (UMR 6623), Université de Franche-Comté, 16, Route de Gray, 25030 Besançon Cedex, France Journal Zeitschrift für Angewandte Mathematik und Physik (ZAMP) Online ISSN 1420-9039 Print ISSN 0044-2275

Description:    The well-posedness of the hydrostatic equations is linked to long wave stability criteria for parallel shear flows. We revisit the Kelvin--Helmholtz instability with a free surface. In the wall-bounded case, the flow is unstable to all wave lengths. Short wave instabilities are localized and independent of boundary conditions. On the other hand, long waves are shown to be stable if the upper boundary is a free surface and gravity is sufficiently small. We also consider smooth velocity profiles of the base flow rather than a velocity jump. We show that stability of long waves for small gravity generally holds for monotone profiles U(y) . On the other hand, this need not be the case if U is not monotone. Content Type Journal Article Pages 1-11 DOI 10.1007/s00033-012-0270-4 Authors Didier Bresch, LAMA, UMR5127 CNRS, 73376 Le Bourget du Lac cedex, France Michael Renardy, Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, USA Journal Zeitschrift für Angewandte Mathematik und Physik (ZAMP) Online ISSN 1420-9039 Print ISSN 0044-2275