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Discrete & Continuous Dynamical Systems - B

2012 , Volume 17 , Issue 3

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Two theorems on singularly perturbed semigroups with applications to models of applied mathematics
Adam Bobrowski and Radosław Bogucki
2012, 17(3): 735-757 doi: 10.3934/dcdsb.2012.17.735 +[Abstract](353) +[PDF](575.6KB)
Abstract:
We present two theorems on convergence of semigroups related to singularly perturbed abstract Cauchy problems, and apply them to some recent models of applied mathematics. The semigroups considered are related to piecewise deterministic Markov processes jumping between several copies of a rectangle in $\mathbb{R}^M$ and moving along deterministic integral curves of some ODEs between jumps. Our theorems describe limit behavior of the processes in the cases of frequent jumps and of fast motions in the direction of chosen variables. These results are motivated by Kepler--Elston's model of gene regulation and Lipniacki's model of gene expression. Application to other models, including those of mathematical economics, are also discussed.
Semiclassical wave packet dynamics in Schrödinger equations with periodic potentials
Rémi Carles and Christof Sparber
2012, 17(3): 759-774 doi: 10.3934/dcdsb.2012.17.759 +[Abstract](420) +[PDF](463.4KB)
Abstract:
We consider semiclassically scaled Schrödinger equations with an external potential and a highly oscillatory periodic potential. We construct asymptotic solutions in the form of semiclassical wave packets. These solutions are concentrated (both, in space and in frequency) around the effective semiclassical phase-space flow, and involve a slowly varying envelope whose dynamics is governed by a homogenized Schrödinger equation with time-dependent effective mass tensor. The corresponding adiabatic decoupling of the slow and fast degrees of freedom is shown to be valid up to Ehrenfest time scales.
Fragmentation and monomer lengthening of rod-like polymers, a relevant model for prion proliferation
Ionel Sorin Ciuperca, Erwan Hingant, Liviu Iulian Palade and Laurent Pujo-Menjouet
2012, 17(3): 775-799 doi: 10.3934/dcdsb.2012.17.775 +[Abstract](381) +[PDF](517.1KB)
Abstract:
The Greer, Pujo-Menjouet and Webb model [Greer et al., J. Theoret. Biol., 242 (2006), 598--606] for prion dynamics was found to be in good agreement with experimental observations under no-flow conditions. The objective of this work is to generalize the problem to the framework of general polymerization-fragmentation under flow motion, motivated by the fact that laboratory work often involves prion dynamics under flow conditions in order to observe faster processes. Moreover, understanding and modelling the microstructure influence of macroscopically monitored non-Newtonian behaviour is crucial for sensor design, with the goal to provide practical information about ongoing molecular evolution. This paper's results can then be considered as one step in the mathematical understanding of such models, namely the proof of positivity and existence of solutions in suitable functional spaces. To that purpose, we introduce a new model based on the rigid-rod polymer theory to account for the polymer dynamics under flow conditions. As expected, when applied to the prion problem, in the absence of motion it reduces to that in Greer et al. (2006). At the heart of any polymer kinetical theory there is a configurational probability diffusion partial differential equation (PDE) of Fokker-Planck-Smoluchowski type. The main mathematical result of this paper is the proof of existence of positive solutions to the aforementioned PDE for a class of flows of practical interest, taking into account the flow induced splitting/lengthening of polymers in general, and prions in particular.
Validity of the Reynolds equation for miscible fluids in microchannels
Mathieu Colin, Thierry Colin and Julien Dambrine
2012, 17(3): 801-834 doi: 10.3934/dcdsb.2012.17.801 +[Abstract](486) +[PDF](951.8KB)
Abstract:
In this paper, we consider asymptotic models for miscible flows in microchannels. The characteristics of the flows in microfluidics imply that usually the Hele-Shaw approximation is valid. We present asymptotic models in the Hele-Shaw regime for flows of miscible fluids in a channel in the case where the bottom and the top of the channels have been modified in two different ways. The first case concerns a flat bottom with slip boundary conditions obtained by chemical patterning. The second one is a non-flat bottom with a non-slipping surface. We derive in both cases 2.5D and 2D asymptotic models. We prove global well-posedness of the 2D model. We also prove that both approaches are asymptotically equivalent in the Hele-Shaw regime and we present direct 3D simulations showing that for passive mixing strategy, the Hele-Shaw approximation is not valid anymore.
Gravitational and electromagnetic properties of almost standing fields
Claude Elbaz
2012, 17(3): 835-848 doi: 10.3934/dcdsb.2012.17.835 +[Abstract](336) +[PDF](394.9KB)
Abstract:
For a scalar field propagating at light velocity $c$, kinematic properties of standing waves with constant frequency $\omega{}$ and velocity $v$, are formally identical with mechanic properties of isolated matter. They are both described by equations with the same mathematical structure, expressed by the Lorentz transformation with constant velocities $c$ and $v$. For almost standing waves, the variations of constant quantities lead to their dynamic properties. When they arise from adiabatic variations of the frequency $\Omega(x,t) = \omega \pm \delta\Omega(x,t)$, with $\omega{}$ constant, and $\delta\Omega(x,t) \ll \omega{}$, they lead to interactions which are formally identical with electromagnetic interactions. When they derive from variations of the field velocity $C(x,t) = c \pm \delta C(x,t)$, with $c$ constant, and $\delta C(x,t) \ll c$, they lead to interactions which are formally identical with gravitational interactions. The correspondence between almost standing waves of the field and matter, offers a common frame allowing an approach to investigate how gravitational and electromagnetic properties articulate together.
Quiescent phases with distributed exit times
Karl-Peter Hadeler and Frithjof Lutscher
2012, 17(3): 849-869 doi: 10.3934/dcdsb.2012.17.849 +[Abstract](363) +[PDF](549.2KB)
Abstract:
Diffusive coupling of a dynamical system to a quiescent (zero) phase, with the same rates for all variables, stabilizes against oscillations. When the coupling rates are increased then, at a stationary point, the eigenvalues of the Jacobian matrix with positive real parts and large imaginary parts may move towards the imaginary axis of the complex plane and eventually enter the left half-plane. Diffusive coupling means that holding times in the active and in the quiescent phase are exponentially distributed. Here, we ask whether similar phenomena occur if the exponential distributions are replaced by other distributions. A general stability result can be shown for arbitrary distributions, and several more specific results for Gamma distributions and delta peaks (leading to delay equations). Some of the results apply to traveling fronts in reaction diffusion equations with quiescent phase.
Fluctuations of solutions to Wigner equation with an Ornstein-Uhlenbeck potential
Tomasz Komorowski and Lenya Ryzhik
2012, 17(3): 871-914 doi: 10.3934/dcdsb.2012.17.871 +[Abstract](374) +[PDF](624.7KB)
Abstract:
We consider energy fluctuations for solutions of the Schrödinger equation with an Ornstein-Uhlenbeck random potential when the initial data is spatially localized. The limit of the fluctuations of the Wigner transform satisfies a kinetic equation with random initial data. This result generalizes that of [12] where the random potential was assumed to be white noise in time.
On the stability of homoclinic loops with higher dimension
Xingbo Liu and Deming Zhu
2012, 17(3): 915-932 doi: 10.3934/dcdsb.2012.17.915 +[Abstract](582) +[PDF](414.0KB)
Abstract:
In this paper the stability of homoclinic loops of saddle equilibrium states in high dimensional systems is analyzed. By constructing local moving frame along the unperturbed homoclinic orbit, the refined Poincaré map is well established, and simple criteria are given for the stability of the saddle homoclinic loop. Some known results are extended.
Spiral rotating waves of a geodesic curvature flow on the unit sphere
Bendong Lou
2012, 17(3): 933-942 doi: 10.3934/dcdsb.2012.17.933 +[Abstract](346) +[PDF](394.6KB)
Abstract:
This paper is concerned with a geodesic curvature flow on the unit sphere. In each zone between the equator and the circle with latitude $\theta_0 \in (0, \frac{\pi}{2} ]$, we give the existence and uniqueness of a spiral rotating wave of the geodesic curvature flow.
On the existence of doubly symmetric "Schubart-like" periodic orbits
Regina Martínez
2012, 17(3): 943-975 doi: 10.3934/dcdsb.2012.17.943 +[Abstract](333) +[PDF](536.7KB)
Abstract:
We give sufficient conditions to ensure the existence of symmetrical periodic orbits for a class of Hamiltonian systems having some singularities. The results are applied to different subproblems of the gravitational $n$-body problem where singularities appear due to collisions.
A constructive proof of the existence of a semi-conjugacy for a one dimensional map
Dyi-Shing Ou and Kenneth James Palmer
2012, 17(3): 977-992 doi: 10.3934/dcdsb.2012.17.977 +[Abstract](441) +[PDF](571.0KB)
Abstract:
A continuous map $f:[0,1]\rightarrow[0,1]$ is called an $n$-modal map if there is a partition $0=z_0 < z_1 < ... < z_n=1$ such that $f(z_{2i})=0$, $f(z_{2i+1})=1$ and, $f$ is (not necessarily strictly) monotone on each $[z_{i},z_{i+1}]$. It is well-known that such a map is topologically semi-conjugate to a piecewise linear map; however here we prove that the topological semi-conjugacy is unique for this class of maps; also our proof is constructive and yields a sequence of easily computable piecewise linear maps which converges uniformly to the semi-conjugacy. We also give equivalent conditions for the semi-conjugacy to be a conjugacy as in Parry's theorem. Related work was done by Fotiades and Boudourides and Banks, Dragan and Jones, who however only considered cases where a conjugacy exists. Banks, Dragan and Jones gave an algorithm to construct the conjugacy map but only for one-hump maps.
Global stability and convergence rate of traveling waves for a nonlocal model in periodic media
Zigen Ouyang and Chunhua Ou
2012, 17(3): 993-1007 doi: 10.3934/dcdsb.2012.17.993 +[Abstract](483) +[PDF](426.6KB)
Abstract:
In this paper, we study the stability and convergence rate of traveling wavefronts for a nonlocal population model in a periodic habitat \[ \left\{ \begin{array}{ll} \displaystyle\frac{\partial u(t,x)}{\partial t}=D(x)\frac{\partial ^2u(t,x)}{% \partial x^2}-d(x,u(t,x))+\int_R\Gamma (\tau ,x,y)b(y,u(t-\tau ,y))dy, & \\ u(\theta ,x)=\varphi (\theta ,x),\theta \in [-\tau ,0],& \end{array} \right. \] where $D(x), d(x,\cdot ), b(x,\cdot ), \Gamma (\tau ,x,y)$ are L-periodic functions with respect to space $x$ (and $y$) for some positive real constant $L $. Using the analysis of the principal eigenvalue of a non-local linear operator, we show that all noncritical wavefronts are globally exponentially stable, as long as the initial perturbation is uniformly bounded in a weighted space. This result can be generalized to n-dimensional case and three applications of our main results are also presented.
Bifurcation of a heterodimensional cycle with weak inclination flip
Zhiqin Qiao, Deming Zhu and Qiuying Lu
2012, 17(3): 1009-1025 doi: 10.3934/dcdsb.2012.17.1009 +[Abstract](454) +[PDF](594.1KB)
Abstract:
Local moving frame is constructed to analyze the bifurcation of a heterodimensional cycle with weak inclination flip in $\mathbb{R}^4$. Under some generic hypotheses, the existence conditions for the heteroclinic orbit, $1$-homoclinic orbit, $1$-periodic orbit and two-fold or three-fold $1$-periodic orbit are given, respectively.
Kinetic theories for biofilms
Qi Wang and Tianyu Zhang
2012, 17(3): 1027-1059 doi: 10.3934/dcdsb.2012.17.1027 +[Abstract](351) +[PDF](803.5KB)
Abstract:
We apply the kinetic theory formulation for binary complex fluids to develop a set of hydrodynamic models for the two-phase mixture of biofilms and solvent (water). It is aimed to model nonlinear growth and transport of the biomass in the mixture and the biomass-flow interaction. In the kinetic theory formulation of binary complex fluids, the biomass consisting of EPS (Extracellular Polymeric Substance) polymer networks and bacteria is coarse-grained into an effective fluid component, termed the effective polymer solution; while the other component, termed the effective solvent, is made up of the ensemble of nutrient substrates and the solvent. The mixture is modeled as an incompressible two-phase fluid in which the presence of the effective components are quantified by their respective volume fractions. The kinetic theory framework allows the incorporation of microscopic details of the biomass and its interaction with the coexisting effective solvent. The relative motion of the biomass and the solvent relative to an average velocity is described by binary mixing kinetics along with the intrinsic molecular elasticity of the EPS network strand modeled as an elastic dumbbell. This theory is valid in both the biofilm region which consists of the mixture of the biomass and solvent and the pure solvent region, making it convenient in numerical simulations of the biomass-flow interaction. Steady states and their stability are discussed under a growth condition. Nonlinear solutions of the three models developed in this study in simple shear are calculated and compared numerically in 1-D space.
Stability of the two dimensional magnetohydrodynamic flows in $\mathbb{R}^3$
Keyan Wang and Yi Du
2012, 17(3): 1061-1073 doi: 10.3934/dcdsb.2012.17.1061 +[Abstract](653) +[PDF](378.1KB)
Abstract:
We prove that two dimensional incompressible magnetohydrodynamic flows are stable in $\mathbb{R}^3$. As a corollary, we show the global existence of classical solutions to the three dimensional incompressible magnetohydrodynamic equations with small initial data. Furthermore, our smallness assumption of the perturbed initial data $(u_0, B_0)$ from that of the two dimensional case is only imposed on the scaling invariant quantity $\|u_0\|_{L^2}\|(\xi\cdot\nabla)u_0\|_{L^2} + \|B_0\|_{L^2}\|(\xi\cdot\nabla)B_0\|_{L^2}$ for one direction $\xi$, while $\|u_0\|_{L^2}\|\nabla u_0\|_{L^2} + \|B_0\|_{L^2}\|\nabla B_0\|_{L^2}$ may be arbitrarily large.
Nonlinear stability of combination of viscous contact wave with rarefaction waves for a 1D radiation hydrodynamics model
Feng Xie
2012, 17(3): 1075-1100 doi: 10.3934/dcdsb.2012.17.1075 +[Abstract](476) +[PDF](473.3KB)
Abstract:
In this paper we consider the large-time behavior of solutions for the Cauchy problem to a compressible radiating gas model, where the far field states are prescribed. This radiating gas model is represented by the one-dimensional system of gas dynamics coupled with an elliptic equation for radiation flux. When the corresponding Riemann problem for the compressible Euler system admits a solution consisting of a contact wave and two rarefaction waves, it is proved that for such a radiating gas model, the combination of viscous contact wave with rarefaction waves is asymptotically stable provided that the strength of combination wave is suitably small. This result is proved by a domain decomposition technique and elementary energy methods.

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