Authors:Van-Sang Ngo Pages: 1 - 42 Abstract: Abstract In this article, we study an anisotropic rotating system arising in magnetohydrodynamics (MHD) in the whole space \(\mathbb{R}^{3}\) , in the case where there are no diffusivity in the vertical direction and a vanishing diffusivity in the horizontal direction (when the rotation goes to infinity). We first prove the local existence and uniqueness of a strong solution and then, using Strichartz-type estimates, we prove that this solution exists globally in time for large initial data, when the rotation is fast enough. PubDate: 2017-08-01 DOI: 10.1007/s10440-016-0092-z Issue No:Vol. 150, No. 1 (2017)

Authors:Bernard Ducomet; Marek Kobera; Šárka Nečasová Pages: 43 - 65 Abstract: Abstract We consider a simplified model based on the Navier-Stokes-Fourier system coupled to a transport equation and the Maxwell system, proposed to describe radiative flows in stars. We establish global-in-time existence for the associated initial-boundary value problem in the framework of weak solutions. PubDate: 2017-08-01 DOI: 10.1007/s10440-016-0093-y Issue No:Vol. 150, No. 1 (2017)

Authors:Fuyi Xu; Xinguang Zhang; Yonghong Wu; Lishan Liu Pages: 67 - 80 Abstract: Abstract The present paper is dedicated to the study of the Cauchy problems for the three-dimensional compressible nematic liquid crystal flow. We obtain the global existence and the optimal decay rates of smooth solutions to the system under the condition that the initial data in lower regular spaces are close to the constant equilibrium state. Our main method is based on the spectral analysis and the smooth effect of dissipative operator. PubDate: 2017-08-01 DOI: 10.1007/s10440-017-0094-5 Issue No:Vol. 150, No. 1 (2017)

Authors:J.-P. Antoine; M. Speckbacher; C. Trapani Pages: 81 - 101 Abstract: Abstract We analyze the notion of reproducing pair of weakly measurable functions, which generalizes that of continuous frame. We show, in particular, that each reproducing pair generates two Hilbert spaces, conjugate dual to each other. Several examples, both discrete and continuous, are presented. PubDate: 2017-08-01 DOI: 10.1007/s10440-017-0095-4 Issue No:Vol. 150, No. 1 (2017)

Authors:Xinghong Pan Pages: 103 - 109 Abstract: Abstract In this paper, we study the regularity of 3d axisymmetric Navier-Stokes equations under a prior point assumption on \(v^{r}\) or \(v^{z}\) . That is, the weak solution of the 3d axisymmetric Navier-Stokes equations \(v\) is smooth if $$ rv^{r}\geq-1; \quad\mbox{or}\quad r\bigl v^{r}(t,x)\bigr \leq Cr^{\alpha}, \ \alpha\in(0,1];\quad\mbox{or} \quad r\bigl v^{z}(t,x)\bigr \leq Cr^{ \beta},\ \beta\in[0,1]; $$ where \(r\) is the distance from the point \(x\) to the symmetric axis. PubDate: 2017-08-01 DOI: 10.1007/s10440-017-0096-3 Issue No:Vol. 150, No. 1 (2017)

Authors:Ewa Zadrzyńska; Wojciech M. Zaja̧czkowski Abstract: Abstract In this paper we prove existence of global strong-weak two-dimensional solutions to the Navier-Stokes and heat equations coupled by the external force dependent on temperature and the heat dissipation, respectively. The existence is proved in a bounded domain with the Navier boundary conditions for velocity and the Dirichlet boundary condition for temperature. Next, we prove existence of 3d global strong solutions via stability. PubDate: 2017-08-03 DOI: 10.1007/s10440-017-0116-3

Authors:Roelof Bruggeman; Ferdinand Verhulst Abstract: Abstract A 4-particles chain with different masses represents a natural generalization of the classical Fermi-Pasta-Ulam chain. It is studied by identifying the mass ratios that produce prominent resonances. This is a technically complicated problem as we have to solve an inverse problem for the spectrum of the corresponding linearized equations of motion. In the case of such an inhomogeneous periodic chain with four particles each mass ratio determines a frequency ratio for the quadratic part of the Hamiltonian. Most prominent frequency ratios occur but not all. In general we find a one-dimensional variety of mass ratios for a given frequency ratio. A detailed study is presented of the resonance \(1:2:3\) . A small cubic term added to the Hamiltonian leads to a dynamical behaviour that shows a difference between the case that two opposite masses are equal and a striking difference with the classical case of four equal masses. For two equal masses and two different ones the normalized system is integrable and chaotic behaviour is small-scale. In the transition to four different masses we find a Hamiltonian-Hopf bifurcation of one of the normal modes leading to complex instability and Shilnikov-Devaney bifurcation. The other families of short-periodic solutions can be localized from the normal forms together with their stability characteristics. For illustration we use action simplices and examples of behaviour with time. PubDate: 2017-08-02 DOI: 10.1007/s10440-017-0115-4

Authors:Yi-rong Jiang; Nan-jing Huang; Donal O’Regan Abstract: Abstract The main purpose of this paper is to establish variational inequality theory in connection with demicontinuous \(\psi_{p}\) -dissipative maps in reflexive smooth Banach spaces by considering the convergence of approximants. As an application of this variational inequality theory, existence, uniqueness and convergence of approximants of positive weak solution for \(p\) -Laplacian elliptic inequalities are obtained under suitable conditions. PubDate: 2017-08-01 DOI: 10.1007/s10440-017-0118-1

Authors:Xiaojun Cui; Jian Cheng Abstract: Abstract On a smooth, non-compact, complete, boundaryless, connected Riemannian manifold there are two kinds of functions: Busemann functions with respect to rays and barrier functions with respect to lines (if there exists at least one). In this paper we collect some known properties on Busemann functions and introduce some new fundamental properties on barrier functions. Based on these properties of barrier functions, we could define some relations on the set of lines and thus classify them. With the equivalence relation we introduced, we present a generalization of a rigidity conjecture. PubDate: 2017-07-31 DOI: 10.1007/s10440-017-0114-5

Authors:Salih Djilali; Tarik Mohammed Touaoula; Sofiane El-Hadi Miri Abstract: Abstract We consider an age structured heroin epidemic model, in a population divided into three sub-populations: \(S\) the susceptible individuals, \(U_{1}\) the drug users and \(U_{2}\) the drug users under treatment, interacting as follows: $$ \left \{ \textstyle\begin{array}{l} S'=A-\mu S-F ( S,U_{1} ) , \\ U_{1}'=F ( S,U_{1} ) - ( \mu +\delta_{1}+p ) U_{1}+\int_{0}^{\infty }k ( a ) U_{2} ( t,a ) da, \\ \frac{\partial U_{2}}{\partial t}+\frac{\partial U_{2}}{\partial a}=- ( \mu +\delta_{2}+k ( a ) ) U_{2}. \end{array}\displaystyle \right . $$ Our main contribution consists in considering a nonlinear incidence function \(F(S,U_{1})\) in its very general form. Global dynamics of the obtained problem is analyzed. PubDate: 2017-07-31 DOI: 10.1007/s10440-017-0117-2

Authors:J. López-Salazar; G. Pérez-Villalón Abstract: Abstract Given a sequence of data \(\{ y_{n} \} _{n \in \mathbb{Z}}\) with polynomial growth and an odd number \(d\) , Schoenberg proved that there exists a unique cardinal spline \(f\) of degree \(d\) with polynomial growth such that \(f ( n ) =y_{n}\) for all \(n\in \mathbb{Z}\) . In this work, we show that this result also holds if we consider weighted average data \(f\ast h ( n ) =y_{n}\) , whenever the average function \(h\) satisfies some light conditions. In particular, the interpolation result is valid if we consider cell-average data \(\int_{n-a}^{n+a}f ( x ) dx=y_{n}\) with \(0< a\leq 1/2\) . The case of even degree \(d\) is also studied. PubDate: 2017-07-28 DOI: 10.1007/s10440-017-0112-7

Authors:Wenbin Yang Abstract: Abstract The paper is concerned with a predator-prey diffusive system subject to homogeneous Neumann boundary conditions, where the growth rate \((\frac{\alpha}{1+\beta v})\) of the predator population is nonlinear. We study the existence of equilibrium solutions and the long-term behavior of the solutions. The main tools used here include the super-sub solution method, the bifurcation theory and linearization method. PubDate: 2017-07-27 DOI: 10.1007/s10440-017-0111-8

Authors:Silvia Bertoluzza; Valérie Perrier Abstract: Abstract In this article we introduce a new mixed Lagrange–Hermite interpolating wavelet family on the interval, to deal with two types (Dirichlet and Neumann) of boundary conditions. As this construction is a slight modification of the interpolating wavelets on the interval of Donoho, it leads to fast decomposition, error estimates and norm equivalences. This new basis is then used in adaptive wavelet collocation schemes for the solution of one dimensional fourth order problems. Numerical tests conducted on the 1D Euler–Bernoulli beam problem, show the efficiency of the method. PubDate: 2017-07-13 DOI: 10.1007/s10440-017-0110-9

Authors:Kuo-Shou Chiu Abstract: Abstract In this paper we introduce an impulsive cellular neural network models with piecewise alternately advanced and retarded argument. The model with the advanced argument is system with strong anticipation. Some sufficient conditions are established for the existence and global exponential stability of a unique periodic solution. The approaches are based on employing Banach’s fixed point theorem and a new integral inequality of Gronwall type with impulses and deviating arguments. The criteria given are easily verifiable, possess many adjustable parameters, and depend on impulses and piecewise constant argument deviations, which provides flexibility for the design and analysis of cellular neural network models. Several numerical examples and simulations are also given to show the feasibility and effectiveness of our results. PubDate: 2017-06-16 DOI: 10.1007/s10440-017-0108-3

Authors:Lance Nielsen Abstract: We establish a comprehensive stability theory for Feynman’s operational calculus (informally, the forming of functions of several noncommuting operators) in the time-dependent setting. Indeed, the main theorem, Theorem 2, contains many of the current stability theorems for the operational calculus and allows the stability theory to be significantly extended. The assumptions needed for the main theorem, Theorem 2, are rather mild and fit in nicely with the current abstract theory of the operational calculus in the time-dependent setting. Moreover, Theorem 2 allows the use of arbitrary time-ordering measures, as long as the discrete parts of these measures are finitely supported. PubDate: 2017-06-16 DOI: 10.1007/s10440-017-0109-2

Authors:Claudianor O. Alves; Ailton R. da Silva Abstract: Abstract The aim of this work is to establish the existence of multi-peak solutions for the following class of quasilinear problems $$ - \mbox{div} \bigl(\epsilon^{2}\phi\bigl(\epsilon \nabla u \bigr)\nabla u \bigr) + V(x)\phi\bigl(\vert u\vert\bigr)u = f(u)\quad\mbox{in } \mathbb{R}^{N}, $$ where \(\epsilon\) is a positive parameter, \(N\geq2\) , \(V\) , \(f\) are continuous functions satisfying some technical conditions and \(\phi\) is a \(C^{1}\) -function. PubDate: 2017-06-14 DOI: 10.1007/s10440-017-0107-4

Authors:Tong Tang; Hongjun Gao Abstract: Abstract In this paper, we study the compressible Euler-Korteweg equations with free boundary condition in vacuum. Under physically assumptions of positive density and pressure, we introduce some physically quantities to show that the spreading diameter of regions grows linearly in time. This is an interesting result as one would expect that the capillary forces would prevent the boundary from spreading. Moreover, we construct a spherically symmetric global solution to support our theorem, followed by Sideris (J. Differ. Equ. 257:1–14, 2014). PubDate: 2017-06-05 DOI: 10.1007/s10440-017-0097-2

Authors:Huiling Li; Yang Zhang Abstract: Abstract This paper concerns global existence and finite time blow-up behavior of positive solutions for a nonlinear reaction-diffusion system with different diffusion coefficients. By use of algebraic matrix theory and modern analytical theory, we extend results of Wang (Z. Angew. Math. Phys. 51:160–167, 2000) to a more general system. Furthermore, we give a complete answer to the open problem which was brought forward in Wang (Z. Angew. Math. Phys. 51:160–167, 2000). PubDate: 2017-06-05 DOI: 10.1007/s10440-017-0105-6

Authors:Gergő Nemes Abstract: Abstract In this paper, we reconsider the large-argument asymptotic expansions of the Hankel, Bessel and modified Bessel functions and their derivatives. New integral representations for the remainder terms of these asymptotic expansions are found and used to obtain sharp and realistic error bounds. We also give re-expansions for these remainder terms and provide their error estimates. A detailed discussion on the sharpness of our error bounds and their relation to other results in the literature is given. The techniques used in this paper should also generalize to asymptotic expansions which arise from an application of the method of steepest descents. PubDate: 2017-05-17 DOI: 10.1007/s10440-017-0099-0

Authors:A. Ambrazevičius; V. Skakauskas Abstract: Abstract Coupled system of nonlinear parabolic equations for grain drying is proposed and the existence and uniqueness theorem of classical solutions is proved by using the upper and lower solutions technique. The long-time behaviour of the solution is also investigated. PubDate: 2017-05-15 DOI: 10.1007/s10440-017-0098-1