doi: 10.1007/s10440-018-0201-2pmid: N/A
The main objective of this paper is to study some Ostrowski type inequalities for double integrals on Time Scales. Some other interesting inequalities are also given.
doi: 10.1007/s10440-018-0202-1pmid: N/A
This paper is concerned with the global well-posedness of strong and classical solutions for the 3D nonhomogeneous incompressible micropolar equations with vacuum. We prove that the problem (1.1)–(1.5) has a unique global strong/classical solution ( ρ , u , w ) $(\rho,u,w)$ , provided μ 1 $\mu_{1}$ is sufficiently large, or ∥ ρ 0 ∥ L ∞ $\|\rho_{0}\|_{L^{\infty}}$ or ∥ ρ 0 1 / 2 u 0 ∥ L 2 2 + ∥ ρ 0 1 / 2 w 0 ∥ L 2 2 $\|\rho_{0}^{1/2}u_{0}\| ^{2}_{L^{2}}+\|\rho_{0}^{1/2}w_{0}\|^{2}_{L^{2}}$ is small enough.
doi: 10.1007/s10440-018-0203-0pmid: N/A
In this paper we study the dynamics of a layer of incompressible viscous fluid bounded below by a rigid boundary and above by a free boundary, in the presence of a uniform gravitational field. We assume that a mass of surfactant is present both at the free surface and in the bulk of the fluid, and that conversion from one species to the other is possible. The surfactants couple to the fluid dynamics through the coefficient of surface tension, which depends on the surface density of surfactants. Gradients in this concentration give rise to Marangoni stress on the free surface. In turn, the fluid advects the surfactants and distorts their concentration through geometric distortions of the free surface. We model the surfactants in a way that allows absorption and desorption of surfactant between the surface and bulk. We prove that small perturbations of the equilibrium solutions give rise to global-in-time solutions that decay to equilibrium at an exponential rate. This establishes the asymptotic stability of the equilibrium solutions.
Burazin, Krešimir; Vrdoljak, Marko
doi: 10.1007/s10440-018-0204-zpmid: N/A
We consider two-phase multiple state optimal design problems for stationary diffusion equation. Both phases are taken to be isotropic, and the goal is to find the optimal distribution of materials within domain, with prescribed amounts, that minimizes a weighted sum of energies. In the case of one state equation, it is known that the proper relaxation of the problem via the homogenization theory is equivalent to a simpler relaxed problem, stated only in terms of the local proportion of given materials.
Zu, Li; Jiang, Daqing; O’Regan, Donal
doi: 10.1007/s10440-018-0205-ypmid: N/A
A biological population may be subjected to stochastic disturbance and exhibit periodicity. In this paper, a stochastic non-autonomous predator-prey system with Holling II functional response is proposed, and the existence of a unique positive solution is derived. We give sufficient conditions for extinction and strong persistence in the mean by analyzing a corresponding one-dimensional stochastic system. Also we establish the existence of positive periodic solutions for this stochastic non-autonomous predator-prey system. Finally, we use numerical simulations to illustrate our results and we present some conclusions and future directions. The results of this paper provide methods for other stochastic population models, which we hope to analyze in the future.
Meunier, Nicolas; Muller, Nicolas
doi: 10.1007/s10440-018-0206-xpmid: N/A
In this work we study the coupling of a nonlinear renewal equation to an ordinary differential equation. We start with existence and uniqueness issues for the coupled equations and, in particular cases, we study the long-time behaviour. The novelty here is the nonlinearity in the renewal equation. This model arises in the context of atherosclerosis. The renewal part accounts for the inflammatory process: leucocyte recruitment in the arterial wall, differentiation when internalizing low-density lipoprotein (LDL) and death. The ordinary differential equation describes the LDL dynamics in the arterial wall, leucocyte absorption and release in the blood.
Bramburger, Jason; Lutscher, Frithjof
doi: 10.1007/s10440-018-0207-9pmid: N/A
Integrodifference equations are a class of infinite-dimensional dynamical systems in discrete time that have recently received great attention as mathematical models of population dynamics in spatial ecology. The dispersal of individuals between generations is described by a ‘dispersal kernel’, a probability density function for the distance that an individual moves within a season. Previous authors recognized that the dynamics are reduced to a finite-dimensional problem when the dispersal kernel is separable. We prove some open questions from their work on the dynamics of a single population and then extend the idea to investigate the dynamics of two spatially distributed species in (i) a competitive relation, and (ii) a predator-prey relation. In all cases, we discuss how the dynamics of the population(s) depend on the amount of suitable space that is available to them. We find a number of bifurcations, such as period-doubling sequences and Naimark-Sacker bifurcations, which we illustrate through simulations.
Zhuang, Yuehong; Cui, Shangbin
doi: 10.1007/s10440-018-0208-8pmid: N/A
This paper is concerned with a free boundary problem modeling the growth of a spherically symmetric tumor with angiogenesis. The unknown nutrient concentration σ = σ ( r , t ) $\sigma =\sigma (r,t)$ occupies the unknown tumor region r < R ( t ) $r< R(t)$ and satisfies a nonlinear reaction diffusion equation, and the unknown tumor radius R = R ( t ) $R=R(t)$ satisfies a nonlinear integro-differential equation. Unlike existing literatures on this topic where Dirichlet boundary condition for σ $\sigma $ is imposed, in this paper the model uses the Robin boundary condition for σ $\sigma $ . We prove existence and uniqueness of a global in-time classical solution ( σ ( r , t ) , R ( t ) $\sigma (r,t),R(t)$ ) for arbitrary c > 0 $c>0$ and establish asymptotic stability of the unique stationary solution ( σ s ( r ) , R s $\sigma _{s}(r),R_{s}$ ) for sufficiently small c $c$ , where c $c$ is a positive constant reflecting the ratio between nutrient diffusion scale and the tumor cell-doubling scale.
Peitz, Sebastian; Ober-Blöbaum, Sina; Dellnitz, Michael
doi: 10.1007/s10440-018-0209-7pmid: N/A
In a wide range of applications it is desirable to optimally control a dynamical system with respect to concurrent, potentially competing goals. This gives rise to a multiobjective optimal control problem where, instead of computing a single optimal solution, the set of optimal compromises, the so-called Pareto set, has to be approximated. When the problem under consideration is described by a partial differential equation (PDE), as is the case for fluid flow, the computational cost rapidly increases and makes its direct treatment infeasible. Reduced order modeling is a very popular method to reduce the computational cost, in particular in a multi query context such as uncertainty quantification, parameter estimation or optimization. In this article, we show how to combine reduced order modeling and multiobjective optimal control techniques in order to efficiently solve multiobjective optimal control problems constrained by PDEs. We consider a global, derivative free optimization method as well as a local, gradient-based approach for which the optimality system is derived in two different ways. The methods are compared with regard to the solution quality as well as the computational effort and they are illustrated using the example of the flow around a cylinder and a backward-facing-step channel flow.
doi: 10.1007/s10440-018-0210-1pmid: N/A
Generalized conics are subsets in the space all of whose points have the same average distance from a given set of points (focal set). The function measuring the average distance is called the average distance function (or the generalized conic function). In general it is a convex function satisfying a kind of growth condition as the preliminary results of Sect. 2 show. Therefore any sublevel set is convex and compact. We can also conclude that such a function has a global minimizer.
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