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Discrete & Continuous Dynamical Systems - A

2014 , Volume 34 , Issue 7

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A note on the Chern-Simons-Dirac equations in the Coulomb gauge
Nikolaos Bournaveas, Timothy Candy and Shuji Machihara
2014, 34(7): 2693-2701 doi: 10.3934/dcds.2014.34.2693 +[Abstract](415) +[PDF](402.0KB)
Abstract:
We prove that the Chern-Simons-Dirac equations in the Coulomb gauge are locally well-posed from initial data in $H^s$ with $s>\frac{1}{4}$. To study nonlinear Wave or Dirac equations at this regularity generally requires the presence of null structure. The novel point here is that we make no use of the null structure of the system. Instead we exploit the additional elliptic structure in the Coulomb gauge together with the bilinear Strichartz estimates of Klainerman-Tataru.
Bubbling solutions for the Chern-Simons gauged $O(3)$ sigma model in $\mathbb{R}^2$
Kwangseok Choe, Jongmin Han and Chang-Shou Lin
2014, 34(7): 2703-2728 doi: 10.3934/dcds.2014.34.2703 +[Abstract](575) +[PDF](510.1KB)
Abstract:
In this paper, we construct multivortex solutions of the elliptic governing equation for the self-dual Chern-Simons gauged $O(3)$ sigma model in $\mathbb{R}^2$ when the Chern-Simons coupling parameter is sufficiently small, and the location of singular points satisfy suitable conditions. Our solutions show concentration phenomena at some points of the singular points as the coupling parameter tends to zero, and they are locally asymptotically radial near each blow-up point.
Enveloping semigroups of systems of order d
Sebastián Donoso
2014, 34(7): 2729-2740 doi: 10.3934/dcds.2014.34.2729 +[Abstract](463) +[PDF](414.9KB)
Abstract:
In this paper we study the Ellis semigroup of a $d$-step nilsystem and the inverse limit of such systems. By using the machinery of cubes developed by Host, Kra and Maass, we prove that such a system has a $d$-step topologically nilpotent enveloping semigroup. In the case $d=2$, we prove that these notions are equivalent, extending a previous result by Glasner.
Rank as a function of measure
Tomasz Downarowicz, Yonatan Gutman and Dawid Huczek
2014, 34(7): 2741-2750 doi: 10.3934/dcds.2014.34.2741 +[Abstract](563) +[PDF](369.8KB)
Abstract:
We establish certain topological properties of rank understood as a function on the set of invariant measures on a topological dynamical system. To be exact, we show that rank is of Young class LU (i.e., it is the limit of an increasing sequence of upper semicontinuous functions).
Computability of the Julia set. Nonrecurrent critical orbits
Artem Dudko
2014, 34(7): 2751-2778 doi: 10.3934/dcds.2014.34.2751 +[Abstract](527) +[PDF](611.1KB)
Abstract:
We prove, that the Julia set of a rational function $f$ is computable in polynomial time, assuming that the postcritical set of $f$ does not contain any critical points or parabolic periodic orbits.
On the characterization of $p$-harmonic functions on the Heisenberg group by mean value properties
Fausto Ferrari, Qing Liu and Juan Manfredi
2014, 34(7): 2779-2793 doi: 10.3934/dcds.2014.34.2779 +[Abstract](423) +[PDF](371.1KB)
Abstract:
We characterize $p-$harmonic functions in the Heisenberg group in terms of an asymptotic mean value property, where $1 < p <\infty$, following the scheme described in [16] for the Euclidean case. The new tool that allows us to consider the subelliptic case is a geometric lemma, Lemma 3.2 below, that relates the directions of the points of maxima and minima of a function on a small subelliptic ball with the unit horizontal gradient of that function.
Schrödinger limit of weakly dissipative stochastic Klein--Gordon--Schrödinger equations and large deviations
Boling Guo, Yan Lv and Wei Wang
2014, 34(7): 2795-2818 doi: 10.3934/dcds.2014.34.2795 +[Abstract](554) +[PDF](448.2KB)
Abstract:
This paper derives a Schrödinger approximation for weakly dissipative stochastic Klein--Gordon--Schrödinger equations with a singular perturbation and scaled small noises on a bounded domain. Detail uniform estimates are given to pass the limit as perturbation and noise disappear. Approximation in two different spaces are considered. Furthermore a large deviation principe of solutions is derived by weak convergence approach.
Pointwise hyperbolicity implies uniform hyperbolicity
Boris Hasselblatt, Yakov Pesin and Jörg Schmeling
2014, 34(7): 2819-2827 doi: 10.3934/dcds.2014.34.2819 +[Abstract](446) +[PDF](384.9KB)
Abstract:
We provide a general mechanism for obtaining uniform information from pointwise data. For instance, a diffeomorphism of a compact Riemannian manifold with pointwise expanding and contracting continuous invariant cone families is an Anosov diffeomorphism, i.e., the entire manifold is uniformly hyperbolic.
Quantization coefficients for ergodic measures on infinite symbolic space
Mrinal Kanti Roychowdhury
2014, 34(7): 2829-2846 doi: 10.3934/dcds.2014.34.2829 +[Abstract](457) +[PDF](414.8KB)
Abstract:
In this paper we consider an ergodic measure with bounded distortion on a symbolic space generated by an infinite alphabet, and showed that for each $r\in (0, +\infty)$ there exists a unique $k_r \in (0, +\infty)$ such that both the $k_r$-dimensional lower and upper quantization coefficients for its image measure $m$ with the support lying on the limit set generated by an infinite conformal iterated function system satisfying the strong open set condition are finite and positive. In addition, it shows that $k_r$ can be expressed by a simple formula involving the temperature function of the system. The result extends and generalizes a similar result of Roychowdhury established for a finite conformal iterated function system [Bull. Polish Acad. Sci. Math. 57 (2009)].
Curves of equiharmonic solutions, and problems at resonance
Philip Korman
2014, 34(7): 2847-2860 doi: 10.3934/dcds.2014.34.2847 +[Abstract](385) +[PDF](395.7KB)
Abstract:
We consider the semilinear Dirichlet problem \[ \Delta u+kg(u)=\mu_1 \varphi_1+\cdots +\mu _n \varphi_n+e(x) \; \; for \; x \in \Omega, \; \; u=0 \; \; on \; \partial \Omega, \] where $\varphi_k$ is the $k$-th eigenfunction of the Laplacian on $\Omega$ and $e(x) \perp \varphi_k$, $k=1, \ldots,n$. Write the solution in the form $u(x)= \Sigma _{i=1}^n \xi _i \varphi_i+U(x)$, with $ U \perp \varphi_k$, $k=1, \ldots,n$. Starting with $k=0$, when the problem is linear, we continue the solution in $k$ by keeping $\xi =(\xi _1, \ldots,\xi _n)$ fixed, but allowing for $\mu =(\mu _1, \ldots,\mu _n)$ to vary. Studying the map $\xi \rightarrow \mu$ provides us with the existence and multiplicity results for the above problem. We apply our results to problems at resonance, at both the principal and higher eigenvalues. Our approach is suitable for numerical calculations, which we implement, illustrating our results.
Rotation-strain decomposition for the incompressible viscoelasticity in two dimensions
Zhen Lei
2014, 34(7): 2861-2871 doi: 10.3934/dcds.2014.34.2861 +[Abstract](393) +[PDF](364.4KB)
Abstract:
In this paper, we revisit the 2D rotation-strain model which was derived in [14] for the motion of incompressible viscoelastic materials and prove its global well-posedness theory without making use of the equation of the rotation angle. The proof relies on a new identity satisfied by the strain matrix. The smallness assumptions are only imposed on the $H^2$ norm of initial velocity field and the initial strain matrix, which implies that the deformation tensor is allowed being away from the equilibrium of 2 in the maximum norm.
Long-time behavior for a class of degenerate parabolic equations
Hongtao Li, Shan Ma and Chengkui Zhong
2014, 34(7): 2873-2892 doi: 10.3934/dcds.2014.34.2873 +[Abstract](655) +[PDF](455.5KB)
Abstract:
The long-time behavior of a class of degenerate parabolic equations in a bounded domain will be considered in the sense that the nonnegative diffusion coefficient $a(x)$ is allowed to vanish on a nonempty closed subset with zero measure. For this purpose, some appropriate weighted Sobolev spaces are introduced and the corresponding embedding theorem is established. Then, we show the global existence and uniqueness of weak solutions. Finally, we distinguish two cases (subcritical and supcritical) to prove the existence of compact attractors for the semigroup associated with this class of equations.
Generalized exact boundary synchronization for a coupled system of wave equations
Tatsien Li, Bopeng Rao and Yimin Wei
2014, 34(7): 2893-2905 doi: 10.3934/dcds.2014.34.2893 +[Abstract](415) +[PDF](359.5KB)
Abstract:
By means of Moore-Penrose generalized inverse, a general framework is presented to treat the generalized exact boundary synchronization for a coupled systems of wave equations.
Development of traveling waves in an interacting two-species chemotaxis model
Tai-Chia Lin and Zhi-An Wang
2014, 34(7): 2907-2927 doi: 10.3934/dcds.2014.34.2907 +[Abstract](541) +[PDF](510.1KB)
Abstract:
By constructing sub and super solutions, we establish the existence of traveling wave solutions to a two-species chemotaxis model, which describes two interacting species chemotactically reacting to a chemical signal that is degraded by the two species. We identify the full parameter regime in which the traveling wave solutions exist, derive the asymptotical decay rates of traveling wave solutions at far field and show that the traveling wave solutions are convergent as the chemical diffusion coefficient goes to zero.
Discrete admissibility and exponential trichotomy of dynamical systems
Adina Luminiţa Sasu and Bogdan Sasu
2014, 34(7): 2929-2962 doi: 10.3934/dcds.2014.34.2929 +[Abstract](565) +[PDF](497.0KB)
Abstract:
The aim of this paper is to present a new and complex study on the asymptotic behavior of dynamical systems, providing necessary and sufficient conditions for the existence of the exponential trichotomy. We associate to a nonautonomous discrete dynamical system an input-output system and we define a new admissibility concept called $(l^\infty(\mathbb{Z}, X), l^1(\mathbb{Z}, X))$-admissibility. First, we prove that the admissibility is a sufficient condition for the existence of the trichotomy projections, for their uniform boundedness, for their compatibility with the coefficients of the initial dynamical system and for certain reversibility properties. Assuming that the associated input-output operators satisfy a natural boundedness condition, we deduce that the admissibility is a necessary and sufficient condition for the existence of the uniform exponential trichotomy. Next, based on admissibility arguments, we obtain, for the first time in the literature, that all the trichotomic properties of a nonautonomous system can be completely recovered from the trichotomic behavior of the associated discrete dynamical system. Finally, we apply the main results in order to obtain a new characterization for uniform exponential trichotomy of evolution families in terms of discrete admissibility.
Hyperbolicity and types of shadowing for $C^1$ generic vector fields
Raquel Ribeiro
2014, 34(7): 2963-2982 doi: 10.3934/dcds.2014.34.2963 +[Abstract](425) +[PDF](466.8KB)
Abstract:
We study various types of shadowing properties and their implication for $C^1$ generic vector fields. We show that, generically, any of the following three hypotheses implies that an isolated set is topologically transitive and hyperbolic: (i) the set is chain transitive and satisfies the (classical) shadowing property, (ii) the set satisfies the limit shadowing property, or (iii) the set satisfies the (asymptotic) shadowing property with the additional hypothesis that stable and unstable manifolds of any pair of critical orbits intersect each other. In our proof we essentially rely on the property of chain transitivity and, in particular, show that it is implied by the limit shadowing property. We also apply our results to divergence-free vector fields.
The Aubry-Mather theorem for driven generalized elastic chains
Siniša Slijepčević
2014, 34(7): 2983-3011 doi: 10.3934/dcds.2014.34.2983 +[Abstract](442) +[PDF](801.6KB)
Abstract:
We consider uniformly (DC) or periodically (AC) driven generalized infinite elastic chains (a generalized Frenkel-Kontorova model) with gradient dynamics. We first show that the union of supports of all space-time invariant measures, denoted by $\mathcal{A}$, projects injectively to a dynamical system on a 2-dimensional cylinder. We also prove existence of space-time ergodic measures supported on a set of rotationaly ordered configurations with an arbitrary (rational or irrational) rotation number. This shows that the Aubry-Mather structure of ground states persists if an arbitrary AC or DC force is applied. The set $\mathcal{A}$ attracts almost surely (in probability) configurations with bounded spacing. In the DC case, $\mathcal{A}$ consists entirely of equilibria and uniformly sliding solutions. The key tool is a new weak Lyapunov function on the space of translationally invariant probability measures on the state space, which counts intersections.
On the higher-order b-family equation and Euler equations on the circle
Min Zhu
2014, 34(7): 3013-3024 doi: 10.3934/dcds.2014.34.3013 +[Abstract](504) +[PDF](388.2KB)
Abstract:
Considered herein is a geometric investigation on the higher-order b-family equation describing exponential curves of the manifold of smooth orientation-preserving diffeomorphisms of the unit circle in the plane. It is shown that the higher-order $b-$family equation can only be realized as an Euler equation on the Lie group Diff$(\mathbb{S}^1) $ of all smooth and orientation preserving diffeomorphisms on the circle if the parameter $b=2$ which corresponds to the higher-order Camassa-Holm equation with the metric $H^k, k\ge 1. $

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