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Discrete & Continuous Dynamical Systems - A

2017 , Volume 37 , Issue 9

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Notes on a theorem of Katznelson and Ornstein
Habibulla Akhadkulov , Akhtam Dzhalilov and  Konstantin Khanin
2017, 37(9): 4587-4609 doi: 10.3934/dcds.2017197 +[Abstract](63) +[HTML](1) +[PDF](521.8KB)
Abstract:

Let \begin{document} $\log f'$ \end{document} be an absolutely continuous and \begin{document} $f"/f'∈ L_{p}(S^{1}, d\ell)$ \end{document} for some \begin{document} $p>1, $ \end{document} where \begin{document} $\ell$ \end{document} is Lebesgue measure. We show that there exists a subset of irrational numbers of unbounded type, such that for any element \begin{document} $\widehat{ρ}$ \end{document} of this subset, the linear rotation \begin{document} $R_{\widehat{ρ}}$ \end{document} and the shift \begin{document} $f_{t}=f+t\mod 1, $ \end{document} \begin{document} $t∈ [0, 1)$ \end{document} with rotation number \begin{document} $\widehat{ρ}, $ \end{document} are absolutely continuously conjugate. We also introduce a certain Zygmund-type condition depending on a parameter \begin{document} $γ$ \end{document}, and prove that in the case \begin{document} $γ>\frac{1}{2}$ \end{document} there exists a subset of irrational numbers of unbounded type, such that every circle diffeomorphism satisfying the corresponding Zygmund condition is absolutely continuously conjugate to the linear rotation provided its rotation number belongs to the above set. Moreover, in the case of \begin{document} $γ> 1, $ \end{document} we show that the conjugacy is \begin{document} $C^{1}$ \end{document}-smooth.

Polynomial approximation of self-similar measures and the spectrum of the transfer operator
Christoph Bandt and  Helena Peña
2017, 37(9): 4611-4623 doi: 10.3934/dcds.2017198 +[Abstract](43) +[HTML](1) +[PDF](602.7KB)
Abstract:

We consider self-similar measures on \begin{document} $\mathbb{R}.$ \end{document} The Hutchinson operator \begin{document} $H$ \end{document} acts on measures and is the dual of the transfer operator \begin{document} $T$ \end{document} which acts on continuous functions. We determine polynomial eigenfunctions of \begin{document} $T.$ \end{document} As a consequence, we obtain eigenvalues of \begin{document} $H$ \end{document} and natural polynomial approximations of the self-similar measure. Bernoulli convolutions are studied as an example.

Analytic dependence on parameters for Evans' approximated Weak KAM solutions
Olga Bernardi and  Matteo Dalla Riva
2017, 37(9): 4625-4636 doi: 10.3934/dcds.2017199 +[Abstract](146) +[HTML](0) +[PDF](390.0KB)
Abstract:

We consider a variational principle for approximated Weak KAM solutions proposed by Evans. For Hamiltonians in quasi-integrable form \begin{document} $h(p)+\varepsilon f(\varphi,p)$ \end{document}, we prove that the map which takes the parameters \begin{document} $(\varepsilon,P,\varrho)$ \end{document} to Evans' approximated solution \begin{document} $u_{\varepsilon,P,\varrho}$ \end{document} is real analytic. In the mechanical case, we compute a recursive system of periodic partial differential equations identifying univocally the coefficients for the power series of the perturbative parameter \begin{document} $\varepsilon$ \end{document}.

Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach
Thomas Blanc , Mihai Bostan and  Franck Boyer
2017, 37(9): 4637-4676 doi: 10.3934/dcds.2017200 +[Abstract](37) +[HTML](0) +[PDF](619.3KB)
Abstract:

We perform the asymptotic analysis of parabolic equations with stiff transport terms. This kind of problem occurs, for example, in collisional gyrokinetic theory for tokamak plasmas, where the velocity diffusion of the collision mechanism is dominated by the velocity advection along the Laplace force corresponding to a strong magnetic field. This work appeal to the filtering techniques. Removing the fast oscillations associated to the singular transport operator, leads to a stable family of profiles. The limit profile comes by averaging with respect to the fast time variable, and still satisfies a parabolic model, whose diffusion matrix is completely characterized in terms of the original diffusion matrix and the stiff transport operator. Introducing first order correctors allows us to obtain strong convergence results, for general initial conditions (not necessarily well prepared).

Stability of stationary solutions to the compressible bipolar Euler--Poisson equations
Hong Cai and  Zhong Tan
2017, 37(9): 4677-4696 doi: 10.3934/dcds.2017201 +[Abstract](33) +[HTML](0) +[PDF](471.2KB)
Abstract:

In this paper, we study the compressible bipolar Euler-Poisson equations with a non-flat doping profile in three-dimensional space. The existence and uniqueness of the non-constant stationary solutions are established under the smallness assumption on the gradient of the doping profile. Then we show the global existence of smooth solutions to the Cauchy problem near the stationary state provided the $H^3$ norms of the initial density and velocity are small, but the higher derivatives can be arbitrarily large.

Spreading speeds and transition fronts of lattice KPP equations in time heterogeneous media
Feng Cao and  Wenxian Shen
2017, 37(9): 4697-4727 doi: 10.3934/dcds.2017202 +[Abstract](155) +[HTML](2) +[PDF](511.6KB)
Abstract:

The current paper is devoted to the study of spreading speeds and transition fronts of lattice KPP equations in time heterogeneous media. We first prove the existence, uniqueness, and stability of spatially homogeneous entire positive solutions. Next, we establish lower and upper bounds of the (generalized) spreading speed intervals. Then, by constructing appropriate sub-solutions and super-solutions, we show the existence and continuity of transition fronts with given front position functions. Also, we prove the existence of some kind of critical front.

On a semilinear Timoshenko-Coleman-Gurtin system: Quasi-stability and attractors
Baowei Feng
2017, 37(9): 4729-4751 doi: 10.3934/dcds.2017203 +[Abstract](109) +[HTML](0) +[PDF](482.7KB)
Abstract:

A semilinear Timoshenko-Coleman-Gurtin system is studied. The system describes a Timoshenko beam coupled with a temperature with Coleman-Gurtin law. Under some assumptions on nonlinear damping terms and nonlinear source terms, we establish the global well-posedness of the system. The main result is the long-time dynamics of the system. By using the methods developed by Chueshov and Lasiecka, we get the quasi-stability property of the system and obtain the existence of a global attractor which has finite fractal dimension. Result on exponential attractors of the system is also proved.

Entropy of diffeomorphisms of line
Baolin He
2017, 37(9): 4753-4766 doi: 10.3934/dcds.2017204 +[Abstract](33) +[HTML](1) +[PDF](437.7KB)
Abstract:

For diffeomorphisms of line, we set up the identities between their length growth rate and their entropy. Then, we prove that there is \begin{document}$C^0$\end{document}-open and \begin{document}$C^r$\end{document}-dense subset \begin{document}$\mathcal{U}$\end{document} of \begin{document}$\text{Diff}^r (\mathbb{R})$\end{document} with bounded first derivative, \begin{document}$r=1,2,\cdots$\end{document}, \begin{document}$+\infty$\end{document}, such that the entropy map with respect to strong \begin{document}$C^0$\end{document}-topology is continuous on \begin{document}$\mathcal{U}$\end{document}; moreover, for any \begin{document}$f \in \mathcal{U}$\end{document}, if it is uniformly expanding or \begin{document}$h(f)=0$\end{document}, then the entropy map is locally constant at \begin{document}$f$\end{document}.

Also, we construct two examples:

1. there exists open subset \begin{document}$\mathcal{U}$\end{document} of \begin{document}$\text{Diff}^{\infty} (\mathbb{R})$\end{document} such that for any \begin{document}$f \in \mathcal{U}$\end{document}, the entropy map with respect to strong \begin{document}$C^{\infty}$\end{document}-topology, is not locally constant at \begin{document}$f$\end{document}.

2. there exists \begin{document}$f \in \text{Diff}^{\infty}(\mathbb{R})$\end{document} such that the entropy map with respect to strong \begin{document}$C^{\infty}$\end{document}-topology, is neither lower semi-continuous nor upper semi-continuous at \begin{document}$f$\end{document}.

Infimum of the metric entropy of volume preserving Anosov systems
Huyi Hu , Miaohua Jiang and  Yunping Jiang
2017, 37(9): 4767-4783 doi: 10.3934/dcds.2017205 +[Abstract](173) +[HTML](2) +[PDF](402.8KB)
Abstract:

In this paper we continue our study [9] of the infimum of the metric entropy of the SRB measure in the space of hyperbolic dynamical systems on a smooth Riemannian manifold of higher dimension. We restrict our study to the space of volume preserving Anosov diffeomorphisms and the space of volume preserving expanding endomorphisms. In our previous paper, we use the perturbation method at a hyperbolic periodic point. It raises the question whether the volume can be preserved. In this paper, we answer this question affirmatively. We first construct a smooth path starting from any point in the space of volume preserving Anosov diffeomorphisms such that the metric entropy tends to zero as the path approaches the boundary of the space. Similarly, we construct a smooth path starting from any point in the space of volume preserving expanding endomorphisms with a fixed degree greater than one such that the metric entropy tends to zero as the path approaches the boundary of the space. Therefore, the infimum of the metric entropy as a functional is zero in both spaces.

Qualitative analysis on positive steady-states for an autocatalytic reaction model in thermodynamics
Yunfeng Jia , Yi Li and  Jianhua Wu
2017, 37(9): 4785-4813 doi: 10.3934/dcds.2017206 +[Abstract](145) +[HTML](0) +[PDF](844.8KB)
Abstract:

In this paper, a reaction-diffusion system known as an autocatalytic reaction model is considered. The model is characterized by a system of two differential equations which describe a type of complex biochemical reaction. Firstly, some basic characterizations of steady-state solutions of the model are presented. And then, the stability of positive constant steady-state solution and the non-existence, existence of non-constant positive steady-state solutions are discussed. Meanwhile, the bifurcation solution which emanates from positive constant steady-state is investigated, and the global analysis to the system is given in one dimensional case. Finally, a few numerical examples are provided to illustrate some corresponding analytic results.

Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure
Bum Ja Jin and  Kyungkeun Kang
2017, 37(9): 4815-4834 doi: 10.3934/dcds.2017207 +[Abstract](96) +[HTML](0) +[PDF](472.0KB)
Abstract:

We prove a Caccioppoli type inequality for the solution of a parabolic system related to the nonlinear Stokes problem. Using the method of Caccioppoli type inequality, we also establish the existence of weak solutions satisfying a local energy inequality without pressure for the non-Newtonian Navier-Stokes equations.

Connected components of meanders: I. bi-rainbow meanders
Anna Karnauhova and  Stefan Liebscher
2017, 37(9): 4835-4856 doi: 10.3934/dcds.2017208 +[Abstract](19) +[HTML](0) +[PDF](557.2KB)
Abstract:

Closed meanders are planar configurations of one or several disjoint closed Jordan curves intersecting a given line transversely. They arise as shooting curves of parabolic PDEs in one space dimension, as trajectories of Cartesian billiards, and as representations of elements of Temperley-Lieb algebras.

Given the configuration of intersections, for example as a permutation or an arc collection, the number of Jordan curves is unknown. We address this question in the special case of bi-rainbow meanders, which are given as non-branched families (rainbows) of nested arcs. Easily obtainable results for small bi-rainbow meanders containing at most four families in total (lower and upper rainbow families) suggest an expression of the number of curves by the greatest common divisor (gcd) of polynomials in the sizes of the rainbow families.We prove however, that this is not the case.

On the other hand, we provide a complexity analysis of nose-retraction algorithms. They determine the number of connected components of arbitrary bi-rainbow meanders in logarithmic time. In fact, the nose-retraction algorithms resemble the Euclidean algorithm.

Looking for a closed formula of the number of connected components, the nose-retraction algorithm is as good as a gcd-formula and therefore as good as we can possibly expect.

General decay of solutions of a Bresse system with viscoelastic boundary conditions
Ammar Khemmoudj and  Taklit Hamadouche
2017, 37(9): 4857-4876 doi: 10.3934/dcds.2017209 +[Abstract](101) +[HTML](0) +[PDF](397.5KB)
Abstract:

In this paper we are concerned with a multi-dimensional Bresse system, in a bounded domain, where the memory-type damping is acting on a portion of the boundary. We establish a general decay results, from which the usual exponential and polynomial decay rates are only special cases.

Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications
Youngwoo Koh and  Ihyeok Seo
2017, 37(9): 4877-4906 doi: 10.3934/dcds.2017210 +[Abstract](41) +[HTML](0) +[PDF](664.9KB)
Abstract:

We obtain weighted \begin{document}$L^2$\end{document} Strichartz estimates for Schrödinger equations \begin{document}$i\partial_tu+(-\Delta)^{a/2}u=F(x, t)$\end{document}, \begin{document}$u(x, 0)=f(x)$\end{document}, of general orders \begin{document}$a>1$\end{document} with radial data \begin{document}$f, F$\end{document} with respect to the spatial variable \begin{document}$x$\end{document}, whenever the weight is in a Morrey-Campanato type class. This is done by making use of a useful property of maximal functions of the weights together with frequency-localized estimates which follow from using bilinear interpolation and some estimates of Bessel functions. As consequences, we give an affirmative answer to a question posed in [1] concerning weighted homogeneous Strichartz estimates, and improve previously known Morawetz estimates. We also apply the weighted \begin{document}$L^2$\end{document} estimates to the well-posedness theory for the Schrödinger equations with time-dependent potentials in the class.

Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two
Xiaoli Li
2017, 37(9): 4907-4922 doi: 10.3934/dcds.2017211 +[Abstract](123) +[HTML](0) +[PDF](463.8KB)
Abstract:

This paper is devoted to the study of the initial-boundary value problem for density-dependent incompressible nematic liquid crystal flows with vacuum in a bounded smooth domain of \begin{document}$\mathbb{R}^2$\end{document}. The system consists of the Navier-Stokes equations, describing the evolution of an incompressible viscous fluid, coupled with various kinematic transport equations for the molecular orientations. Assuming the initial data are sufficiently regular and satisfy a natural compatibility condition, the existence and uniqueness are established for the global strong solution if the initial data are small. We make use of a critical Sobolev inequality of logarithmic type to improve the regularity of the solution. Our result relaxes the assumption in all previous work that the initial density needs to be bounded away from zero.

Hyperbolic sets that are not contained in a locally maximal one
Adriana da Luz
2017, 37(9): 4923-4941 doi: 10.3934/dcds.2017166 +[Abstract](31) +[HTML](0) +[PDF](291.2KB)
Abstract:

In this paper we study two properties related to the structure of hyperbolic sets. First we construct new examples answering in the negative the following question posed by Katok and Hasselblatt in [[12], p. 272]

Question. Let \begin{document}$\Lambda$\end{document} be a hyperbolic set, and let \begin{document}$V$\end{document} be an open neighborhood of \begin{document}$\Lambda$\end{document}. Does there exist a locally maximal hyperbolic set \begin{document}$\widetilde{\Lambda}$\end{document} such that \begin{document}$\Lambda \subset \widetilde{\Lambda} \subset V $\end{document}?

We show that such examples are present in linear Anosov diffeomorophisms of \begin{document}$\mathbb{T}^3$\end{document}, and are therefore robust.

Also we construct new examples of sets that are not contained in any locally maximal hyperbolic set. The examples known until now were constructed by Crovisier in [7] and by Fisher in [9], and these were either in dimension equal or bigger than 4 or they were not transitive. We give a transitive and robust example in \begin{document}$\mathbb{T}^3$\end{document}. And show that such examples cannot be build in dimension 2.

On averaged tracing of periodic average pseudo orbits
Piotr Oprocha and  Xinxing Wu
2017, 37(9): 4943-4957 doi: 10.3934/dcds.2017212 +[Abstract](55) +[HTML](0) +[PDF](395.1KB)
Abstract:

We propose a definition of average tracing of finite pseudo-orbits and show that in the case of this definition measure center has the same property as nonwandering set for the classical shadowing property. We also show that the average shadowing property trivializes in the case of mean equicontinuous systems, and that it implies distributional chaos when measure center is nondegenerate.

Fiber bunching and cohomology for Banach cocycles over hyperbolic systems
Victoria Sadovskaya
2017, 37(9): 4959-4972 doi: 10.3934/dcds.2017213 +[Abstract](30) +[HTML](0) +[PDF](392.8KB)
Abstract:

We consider Hölder continuous cocycles over hyperbolic dynamical systems with values in the group of invertible bounded linear operators on a Banach space. We show that two fiber bunched cocycles are Hölder continuously cohomologous if and only if they have Hölder conjugate periodic data. The fiber bunching condition means that non-conformality of the cocycle is dominated by the expansion and contraction in the base system. We show that this condition can be established based on the periodic data of a cocycle. We also establish Hölder continuity of a measurable conjugacy between a fiber bunched cocycle and one with values in a set which is compact in strong operator topology.

Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials
Xianhua Tang and  Sitong Chen
2017, 37(9): 4973-5002 doi: 10.3934/dcds.2017214 +[Abstract](222) +[HTML](0) +[PDF](548.0KB)
Abstract:

This paper is dedicated to studying the following Schrödinger-Poisson problem

where \begin{document}$V(x)$\end{document} is weakly differentiable and \begin{document}$f∈ \mathcal{C}(\mathbb{R}, \mathbb{R})$\end{document}. By introducing some new tricks, we prove the above problem admits a ground state solution of Nehari-Pohozaev type and a least energy solution under mild assumptions on \begin{document}$V$\end{document} and \begin{document}$f$\end{document}. Our results generalize and improve the ones in [D. Ruiz, J. Funct. Anal. 237 (2006) 655-674], [J.J. Sun, S.W. Ma, J. Differential Equations 260 (2016) 2119-2149] and some related literature.

Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation
Jingrui Wang and  Keyan Wang
2017, 37(9): 5003-5019 doi: 10.3934/dcds.2017215 +[Abstract](37) +[HTML](0) +[PDF](444.9KB)
Abstract:

In this paper we prove the almost sure existence of global weak solution to the 3D incompressible Navier-Stokes Equation for a set of large data in \begin{document}$\dot{H}^{-α}(\mathbb{R}^{3})$\end{document} or \begin{document}$\dot{H}^{-α}(\mathbb{T}^{3})$\end{document} with \begin{document}$0<α≤ 1/2$\end{document}. This is achieved by randomizing the initial data and showing that the energy of the solution modulus the linear part keeps finite for all \begin{document}$t≥0$\end{document}. Moreover, the energy of the solutions is also finite for all \begin{document}$t>0$\end{document}. This improves the recent result of Nahmod, Pavlović and Staffilani on (SIMA) in which \begin{document}$α$\end{document} is restricted to \begin{document}$0<α<\frac{1}{4}$\end{document}.

Boundedness in logistic Keller--Segel models with nonlinear diffusion and sensitivity functions
Qi Wang , Jingyue Yang and  Feng Yu
2017, 37(9): 5021-5036 doi: 10.3934/dcds.2017216 +[Abstract](139) +[HTML](0) +[PDF](758.0KB)
Abstract:

We consider the following fully parabolic Keller-Segel system

over a multi-dimensional bounded domain \begin{document}$Ω \subset \mathbb R^N$\end{document}, \begin{document}$N≥2$\end{document}. Here \begin{document}$D(u)$\end{document} and \begin{document}$S(u)$\end{document} are smooth functions satisfying: \begin{document}$D(0)>0$\end{document}, \begin{document}$D(u)≥ K_1u^{m_1}$\end{document} and \begin{document}$S(u)≤ K_2u^{m_2}$\end{document}, \begin{document}$\forall u≥0$\end{document}, for some constants \begin{document}$K_i∈\mathbb R^+$\end{document}, \begin{document}$m_i∈\mathbb R$\end{document}, \begin{document}$i=1, 2$\end{document}. It is proved that, when the parameter pair \begin{document}$(m_1, m_2)$\end{document} lies in some specific regions, the system admits global classical solutions and they are uniformly bounded in time. We cover and extend [22,28], in particular when \begin{document}$N≥3$\end{document} and \begin{document}$γ≥1$\end{document}, and [3,29] when \begin{document}$m_1>γ-\frac{2}{N}$\end{document} if \begin{document}$γ∈(0, 1)$\end{document} or \begin{document}$m_1>γ-\frac{4}{N+2}$\end{document} if \begin{document}$γ∈[1, ∞)$\end{document}. Moreover, according to our results, the index \begin{document}$\frac{2}{N}$\end{document} is, in contrast to the model without cellular growth, no longer critical to the global existence or collapse of this system.

Global solution for the $3D$ quadratic Schrödinger equation of $Q(u,\bar{u}$) type
Xuecheng Wang
2017, 37(9): 5037-5048 doi: 10.3934/dcds.2017217 +[Abstract](29) +[HTML](0) +[PDF](435.3KB)
Abstract:

We study a class of \begin{document}$3D$\end{document} quadratic Schrödinger equations as follows, \begin{document}$(\partial_t -i Δ) u = Q(u, \bar{u})$\end{document}. Different from nonlinearities of the \begin{document}$uu$\end{document} type and the \begin{document}$\bar{u}\bar{u}$\end{document} type, which have been studied by Germain-Masmoudi-Shatah in [2], the interaction of \begin{document}$u$\end{document} and \begin{document}$\bar{u}$\end{document} is very strong at the low frequency part, e.g., \begin{document}$1× 1 \to 0$\end{document} type interaction (the size of input frequency is "1" and the size of output frequency is "0"). It creates a growth mode for the Fourier transform of the profile of solution around a small neighborhood of zero. This growth mode will again cause the growth of profile in the medium frequency part due to the \begin{document}$1× 0\to 1$\end{document} type interaction. The issue of strong \begin{document}$1× 1\to 0$\end{document} type interaction makes the global existence problem very delicate.

In this paper, we show that, as long as there are "\begin{document}$ε$\end{document}" derivatives inside the quadratic term \begin{document}$Q (u, \bar{u})$\end{document}, there exists a global solution for small initial data. As a byproduct, we also give a simple proof for the almost global existence of the small data solution of \begin{document}$(\partial_t -i Δ)u = |u|^2 = u\bar{u}$\end{document}, which was first proved by Ginibre-Hayashi [3]. Instead of using vector fields, we consider this problem purely in Fourier space.

A characterization of Sierpiński carpet rational maps
Yan Gao , Jinsong Zeng and  Suo Zhao
2017, 37(9): 5049-5063 doi: 10.3934/dcds.2017218 +[Abstract](30) +[HTML](1) +[PDF](419.0KB)
Abstract:

In this paper we prove that a postcritically finite rational map with non-empty Fatou set is Thurston equivalent to an expanding Thurston map if and only if its Julia set is homeomorphic to the standard Sierpiński carpet.

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