Authors:Rory Biggs; Claudiu C. Remsing Pages: 1 - 59 Abstract: Abstract Quadratic Hamilton–Poisson systems on three-dimensional Lie–Poisson spaces are considered. The homogeneous (positive) semidefinite systems are classified up to linear isomorphism; an exhaustive and nonredundant list of 23 normal forms is exhibited. For each normal form, the stability nature of the equilibria is determined. Each normal form is explicitly integrated, with the exception of three families of systems. Based on the analysis of the normal forms, some simple invariants are identified. PubDate: 2017-04-01 DOI: 10.1007/s10440-016-0074-1 Issue No:Vol. 148, No. 1 (2017)

Authors:Xia Ye Pages: 61 - 69 Abstract: Abstract In this paper, we consider the Cauchy problem of non-stationary motion of heat-conducting incompressible viscous fluids in \(\mathbb{R}^{2}\) , where the viscosity and heat-conductivity coefficient vary with the temperature. It is shown that the Cauchy problem has a unique global-in-time strong solution \((u, \theta)(x,t)\) on \(\mathbb{R}^{2}\times(0,\infty)\) , provided the initial norm \(\ \nabla u_{0}\ _{L^{2}}\) is suitably small, or the lower-bound of the coefficient of heat conductivity (i.e. \(\underline{\kappa}\) ) is large enough, or the derivative of viscosity (i.e. \( \mu'(\theta) \) ) is small enough. PubDate: 2017-04-01 DOI: 10.1007/s10440-016-0078-x Issue No:Vol. 148, No. 1 (2017)

Authors:Onur Gün; Atilla Yilmaz Pages: 71 - 102 Abstract: Abstract We propose a new model of permanent monogamous pair formation in zoological populations with multiple types of females and males. According to this model, animals randomly encounter members of the opposite sex at their so-called firing times to form temporary pairs which then become permanent if mating happens. Given the distributions of the firing times and the mating preferences upon encounter, we analyze the contingency table of permanent pair types in three cases: (i) definite mating upon encounter; (ii) Poisson firing times; and (iii) Bernoulli firing times. In the first case, the contingency table has a multiple hypergeometric distribution which implies panmixia. The other two cases generalize the encounter-mating models of Gimelfarb (Am. Nat. 131(6):865–884, 1988) who gives conditions that he conjectures to be sufficient for panmixia. We formulate adaptations of his conditions and prove that they not only characterize panmixia but also allow us to reduce the model to the first case by changing its underlying parameters. Finally, when there are only two types of females and males, we provide a full characterization of panmixia, homogamy and heterogamy. PubDate: 2017-04-01 DOI: 10.1007/s10440-016-0079-9 Issue No:Vol. 148, No. 1 (2017)

Authors:Hongling Jiang; Lijuan Wang Pages: 103 - 120 Abstract: Abstract This paper is concerned with a Variable-territory model with limited self-limitation of predator. By the bifurcation theorem, regular perturbation theorem and numerical simulation, the conditions of existence, stability and convergence of positive solutions are established. This work shows that prey and predator can be controlled by parameters in Variable-territory model, such as intrinsic growth rate, death rate, handling time and the self-limitation of predator. PubDate: 2017-04-01 DOI: 10.1007/s10440-016-0080-3 Issue No:Vol. 148, No. 1 (2017)

Authors:Ling Zhou; Shan Zhang; Zuhan Liu Pages: 121 - 142 Abstract: Abstract In this paper we consider the system of reaction-diffusion-advection equations with a free boundary, which arises in a competition ecological model in heterogeneous environment. In strong competition case, we study the influence of competition rates on the long time behavior of solutions and prove that two species spatially segregate as the competition rates become large. Besides, by using a blow up method, we obtain the uniform Hölder bounds for solutions of the system. PubDate: 2017-04-01 DOI: 10.1007/s10440-016-0081-2 Issue No:Vol. 148, No. 1 (2017)

Authors:Tuan Nguyen Huy; Mokhtar Kirane; Bessem Samet; Van Au Vo Pages: 143 - 155 Abstract: Abstract We study the backward problem for non-linear (semilinear) parabolic partial differential equations in Hilbert spaces. The problem is severely ill-posed in the sense of Hadamard. Under a weak a priori assumption on the exact solution, we propose a new Fourier truncated regularization method for stabilising the ill-posed problem. In comparison with previous studies on solving the nonlinear backward problem, our method shows a significant improvement. PubDate: 2017-04-01 DOI: 10.1007/s10440-016-0082-1 Issue No:Vol. 148, No. 1 (2017)

Authors:Pan Zheng; Chunlai Mu Pages: 157 - 177 Abstract: Abstract This paper deals with a two-competing-species chemotaxis system with two different chemicals $$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad}l} \displaystyle u_{t}=\Delta u-\chi_{1}\nabla \cdot (u\nabla v)+\mu_{1} u(1-u-a _{1}w), & (x,t)\in \varOmega \times (0,\infty ), \\ \displaystyle \tau v_{t}=\Delta v-v+w, & (x,t)\in \varOmega \times (0,\infty ), \\ \displaystyle w_{t}=\Delta w-\chi_{2}\nabla \cdot (w\nabla z)+\mu_{2}w(1-a_{2}u-w), & (x,t)\in \varOmega \times (0,\infty ), \\ \displaystyle \tau z_{t}=\Delta z-z+u, & (x,t)\in \varOmega \times (0,\infty ), \end{array}\displaystyle \right . \end{aligned}$$ under homogeneous Neumann boundary conditions in a smooth bounded domain \(\varOmega \subset \mathbb{R}^{n}\) \((n\geq 1)\) with the nonnegative initial data \((u_{0},\tau v_{0},w_{0},\tau z_{0})\in C^{0}(\overline{\varOmega }) \times W^{1,\infty }(\varOmega )\times C^{0}(\overline{\varOmega })\times W ^{1,\infty }(\varOmega )\) , where \(\tau \in \{0,1\}\) and the parameters \(\chi_{i},\mu_{i},a_{i}\) ( \(i=1,2\) ) are positive. When \(\tau =0\) , based on some a priori estimates and Moser-Alikakos iteration, it is shown that regardless of the size of initial data, the system possesses a unique globally bounded classical solution for any positive parameters if \(n=2\) . On the other hand, when \(\tau =1\) , relying on the maximal Sobolev regularity and semigroup technique, it is proved that the system admits a unique globally bounded classical solution provided that \(n\geq 1\) and there exists \(\theta_{0}>0\) such that \(\frac{\chi_{2}}{ \mu_{1}}<\theta_{0}\) and \(\frac{\chi_{1}}{\mu_{2}}<\theta_{0}\) . PubDate: 2017-04-01 DOI: 10.1007/s10440-016-0083-0 Issue No:Vol. 148, No. 1 (2017)

Authors:Pablo Braz e Silva; Wilberclay G. Melo; Paulo R. Zingano Pages: 1 - 17 Abstract: Abstract We obtain lower bounds on blow-up of solutions for the 3D magneto-micropolar equations. More precisely, we establish some estimates for the solution \((\mathbf{u},\mathbf{w},\mathbf{b}) (t)\) in its maximal interval \([0,T^{*})\) provided that \(T^{*}<\infty\) , which show for \(\delta\in(0,1)\) that \(\ (\mathbf{u},\mathbf{w},\mathbf{b})(t)\ _{\dot{H}^{s}}\) is at least of the order \((T^{*}-t)^{-(\delta s)/(1+2\delta)}\) for \(s\geq1/2+\delta\) . In particular, by choosing a suitable \(\delta\) , one concludes that \(\ (\mathbf{u},\mathbf{w},\mathbf{b})(t)\ _{\dot{H}^{s}}\) is at least of the order \((T^{*}-t)^{-s/4}\) , and \((T^{*}-t)^{1/4-s/2}\) for \(s\geq1\) , and \(1/2< s<3/2\) , respectively. We also show that \((T^{*}-t)^{-s/3}\) is a lower rate for \(\ (\mathbf{u},\mathbf{w},\mathbf{b})(t)\ _{\dot{H}^{s}}\) if \(s>3/2\) . PubDate: 2017-02-01 DOI: 10.1007/s10440-016-0065-2 Issue No:Vol. 147, No. 1 (2017)

Authors:Shengquan Liu; Shujuan Wang Pages: 39 - 62 Abstract: Abstract This paper is concerned with the Cauchy problem for compressible nematic liquid crystal flows in the two-dimensional space (2D). We establish a blow-up criterion in terms of the density only, provided the macroscopic average of the nematic liquid crystal orientation field satisfies a geometric condition. In particular, the initial vacuum is allowed and the compatibility condition is removed. PubDate: 2017-02-01 DOI: 10.1007/s10440-016-0067-0 Issue No:Vol. 147, No. 1 (2017)

Authors:Qiao Liu; Yemei Wei Pages: 63 - 80 Abstract: Abstract In this paper, we investigate blow up criteria for the local smooth solutions to the 3D incompressible nematic liquid crystal flows via the components of the gradient velocity field \(\nabla u\) and the gradient orientation field \(\nabla d\) . More precisely, we show that \(0< T_{ \ast}<+\infty\) is the maximal time interval if and only if $$\begin{aligned} & \int_{0}^{T_{\ast}} \bigl\Vert \Vert \partial_{i}u\Vert _{L_{x_{i}} ^{\gamma}} \bigr\Vert _{L_{x_{j}x_{k}}^{\alpha}}^{\beta}+ \ \nabla d\ _{L^{\infty}}^{\frac{8}{3}}\mathrm{d}t=\infty, \\ &\quad\text{ with } \frac{2}{\alpha}+\frac{2}{\beta}\leq\frac{3\alpha +2}{4\alpha}, \text{ and } 1\leq\gamma\leq\alpha,2< \alpha\leq+\infty, \end{aligned}$$ or $$\begin{aligned} \int_{0}^{T_{\ast}}\ \partial_{3}u_{3} \ ^{\beta}_{L^{\alpha}}+\ \nabla d\ ^{\frac{8}{3}}_{L^{\infty}} \mathrm{d}t=\infty,\quad\text{with } \frac{3}{\alpha}+\frac{2}{\beta}\leq \frac{3(\alpha+2)}{4 \alpha}, \text{ and } 2< \alpha\leq\infty, \end{aligned}$$ where \(i,j,k\in\{1,2,3\}\) , \(i\neq j\) , \(i\neq k\) , and \(j\neq k\) . PubDate: 2017-02-01 DOI: 10.1007/s10440-016-0068-z Issue No:Vol. 147, No. 1 (2017)

Authors:Renhui Wan; Yong Zhou Pages: 95 - 111 Abstract: Abstract In this paper, we obtain the local well-posedness for the 3D incompressible Hall-magnetohydrodynamics (Hall-MHD) system with \(\varLambda^{2\alpha }u\) and \(\varLambda^{2\beta }B\) , \(0<\alpha \le 1\) , \(\frac{1}{2}<\beta \le 1\) . Our results improve regularity conditions on the initial data of previous works. PubDate: 2017-02-01 DOI: 10.1007/s10440-016-0070-5 Issue No:Vol. 147, No. 1 (2017)

Authors:Xiuli Zhu; Zhonghai Xu; Huapeng Li Pages: 113 - 136 Abstract: Abstract In this paper, we consider an initial-value problem to the two-dimensional incompressible micropolar fluid equations. Our main purpose is to study the boundary layer effects as the angular and micro-rotational viscosities go to zero. It is also shown that the boundary layer thickness is of the order \(O(\gamma^{\beta })\) with \((0<\beta <\frac{2}{3})\) . In contrast with Chen et al. (Z. Angew. Math. Phys. 65:687–710, 2014), the BL-thickness we got is thinner than that in Chen et al. (Z. Angew. Math. Phys. 65:687–710, 2014). In addition, the convergence rates are also improved. PubDate: 2017-02-01 DOI: 10.1007/s10440-016-0071-4 Issue No:Vol. 147, No. 1 (2017)

Authors:Yan-Xia Ren; Renming Song; Rui Zhang Pages: 137 - 175 Abstract: Abstract In this paper, we establish some functional central limit theorems for a large class of general supercritical superprocesses with spatially dependent branching mechanisms satisfying a second moment condition. In the particular case when the state \(E\) is a finite set and the underlying motion is an irreducible Markov chain on \(E\) , our results are superprocess analogs of the functional central limit theorems of Janson (Stoch. Process. Appl. 110:177–245, 2004) for supercritical multitype branching processes. The results of this paper are refinements of the central limit theorems in Ren et al. (Stoch. Process. Appl. 125:428–457, 2015). PubDate: 2017-02-01 DOI: 10.1007/s10440-016-0072-3 Issue No:Vol. 147, No. 1 (2017)

Authors:Naoki Tsuge Pages: 177 - 186 Abstract: Abstract We are concerned with a scalar conservation law with a source term. This equation is proposed to describe the qualitative behavior of waves for a general system in resonance with the source term by T.P. Liu. In addition to this, the scalar conservation law is used in various areas such as fluid dynamics, traffic problems etc. In the present paper, we prove the global existence and stability of entropy solutions to the Cauchy problem. The difficult point is to obtain the bounded estimate of solutions. To solve it, we introduce some functions as the lower and upper bounds. Therefore, our bounded estimate depends on the space variable. This idea comes from the generalized invariant region theory for the compressible Euler equation. The method is also applicable to other nonlinear problems involving similar difficulties. Finally, we use the vanishing viscosity method to construct approximate solutions and derive the convergence by the compensated compactness. PubDate: 2017-02-01 DOI: 10.1007/s10440-016-0073-2 Issue No:Vol. 147, No. 1 (2017)

Authors:Shen Bian Pages: 187 - 195 Abstract: Abstract We consider a nonlocal Fisher-KPP reaction-diffusion model arising from population dynamics, consisting of a certain type reaction term \(u^{\alpha} ( 1-\int_{\varOmega}u^{\beta}dx ) \) , where \(\varOmega\) is a bounded domain in \(\mathbb{R}^{n}(n \ge1)\) . The energy method is applied to prove the global existence of the solutions and the results show that the long time behavior of solutions heavily depends on the choice of \(\alpha\) , \(\beta\) . More precisely, for \(1 \le\alpha <1+ ( 1-2/p ) \beta\) , where \(p\) is the exponent from the Sobolev inequality, the problem has a unique global solution. Particularly, in the case of \(n \ge3\) and \(\beta=1\) , \(\alpha<1+2/n\) is the known Fujita exponent (Fujita in J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 13:109–124, 1966). Comparing to Fujita equation (Fujita in J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 13:109–124, 1966), this paper will give an opposite result to our nonlocal problem. PubDate: 2017-02-01 DOI: 10.1007/s10440-016-0075-0 Issue No:Vol. 147, No. 1 (2017)

Authors:E. S. Baranovskii; M. A. Artemov Pages: 197 - 210 Abstract: Abstract We investigate the system of nonlinear partial differential equations governing the unsteady motion of an incompressible viscoelastic fluid of Oldroyd type in a bounded domain under Navier’s slip boundary condition. We prove the existence of global weak solutions for the corresponding initial-boundary value problem without assuming that the model constants, body force or the initial values of the velocity and the stress tensor are small. PubDate: 2017-02-01 DOI: 10.1007/s10440-016-0076-z Issue No:Vol. 147, No. 1 (2017)

Authors:Xinghong Pan 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: 2017-04-06 DOI: 10.1007/s10440-017-0096-3

Authors:Fuyi Xu; Xinguang Zhang; Yonghong Wu; Lishan Liu 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: 2017-03-10 DOI: 10.1007/s10440-017-0094-5

Authors:J.-P. Antoine; M. Speckbacher; C. Trapani 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: 2017-03-10 DOI: 10.1007/s10440-017-0095-4

Authors:Jiayin Liu; Duchao Liu; Peihao Zhao Abstract: Abstract This paper is concerned with a kind of quasilinear Schrödinger equation with combined nonlinearities, a convex term with any growth and a singular term, in a bounded smooth domain. Multiplicity results are obtained by critical point theory together with truncation arguments and the method of upper and lower solutions. PubDate: 2016-12-21 DOI: 10.1007/s10440-016-0084-z