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Discrete & Continuous Dynamical Systems - A

March 2017 , Volume 37 , Issue 3

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Necessity for isoperimetric inequality constraints
Jorge A. Becerril and Javier F. Rosenblueth
2017, 37(3): 1129-1158 doi: 10.3934/dcds.2017047 +[Abstract](527) +[HTML](3) +[PDF](286.0KB)
Abstract:

In this paper we deal with second order necessary conditions for the problem of Lagrange in the calculus of variations posed over piecewise smooth trajectories and involving inequality and equality isoperimetric constraints. We provide a review of different approaches to derive second order necessary conditions for this problem and prove that, surprisingly, though the solution set to the problem where the conditions hold may vary, all approaches impose the same strong assumption of normality relative to the set defined by equality constraints for active indices. Based on these approaches, we also give some applications to certain optimization problems with mixed constraints.

Asymptotic analysis of the scattering problem for the Helmholtz equations with high wave numbers
Daniel Bouche, Youngjoon Hong and Chang-Yeol Jung
2017, 37(3): 1159-1181 doi: 10.3934/dcds.2017048 +[Abstract](811) +[HTML](1) +[PDF](561.5KB)
Abstract:

We study the asymptotic behavior of the two dimensional Helmholtz scattering problem with high wave numbers in an exterior domain, the exterior of a circle. We impose the Dirichlet boundary condition on the obstacle, which corresponds to an incidental wave. For the outer boundary, we consider the Sommerfeld conditions. Using a polar coordinates expansion, the problem is reduced to a sequence of Bessel equations. Investigating the Bessel equations mode by mode, we find that the solution of the scattering problem converges to its limit solution at a specific rate depending on k.

Ergodic properties of folding maps on spheres
Almut Burchard, Gregory R. Chambers and Anne Dranovski
2017, 37(3): 1183-1200 doi: 10.3934/dcds.2017049 +[Abstract](412) +[HTML](7) +[PDF](429.8KB)
Abstract:

We consider the trajectories of points on \begin{document}$ \mathbb{S}^{d-1} $\end{document} under sequences of certain folding maps associated with reflections. The main result characterizes collections of folding maps that produce dense trajectories. The minimal number of maps in such a collection is d+1.

Strichartz estimates for charge transfer models
Gong Chen
2017, 37(3): 1201-1226 doi: 10.3934/dcds.2017050 +[Abstract](494) +[HTML](1) +[PDF](462.5KB)
Abstract:

In this note, we prove Strichartz estimates for scattering states of scalar charge transfer models in \begin{document} $\mathbb{R}^{3}$ \end{document}. Following the idea of Strichartz estimates based on [3,10], we also show that the energy of the whole evolution is bounded independently of time without using the phase space method, as for example, in [5]. One can easily generalize our arguments to \begin{document} $\mathbb{R}^{n}$ \end{document} for \begin{document} $n≥q3$ \end{document}. We also discuss the extension of these results to matrix charge transfer models in \begin{document} $\mathbb{R}^{3}$ \end{document}.

Functional envelopes relative to the point-open topology on a subset
Zhijing Chen and Yu Huang
2017, 37(3): 1227-1246 doi: 10.3934/dcds.2017051 +[Abstract](385) +[HTML](1) +[PDF](469.3KB)
Abstract:

If \begin{document}$(X, f)$\end{document} is a dynamical system given by a locally compact separable metric space \begin{document}$X$\end{document} without isolated points and a continuous map \begin{document}$f : X\to X $\end{document} , and \begin{document}$A$\end{document} is a countable dense subset of \begin{document}$X$\end{document} , then by the functional envelope of \begin{document}$(X, f)$\end{document} relative to $\mathcal{P}_A$ we mean the dynamical system $(S_A(X), F_f)$ whose phase space $S_A(X)$ is the space of all continuous selfmaps of \begin{document}$X$\end{document} endowed with the point-open topology on $A$ and the map $F_f : S_A(X)\to S_A(X)$ is defined by $F_f (\varphi)=fo\varphi$ for any \begin{document}$\varphi∈ S_A(X)$\end{document} .

In this paper, we mainly deal with the connection between the properties of a system and the properties of its functional envelope. We show that:(1) \begin{document}$(X, f)$\end{document} is weakly mixing if and only if there exists a countable dense subset \begin{document}$A$\end{document} of \begin{document}$X$\end{document} so that \begin{document}$\big(S_A(X), F_f\big)$\end{document} has a transitive point \begin{document}$φ∈ S(X)$\end{document} which is surjective; (2) \begin{document}$(X, f)$\end{document} is sensitive if and only if \begin{document}$\big(S_A(X), F_f\big)$\end{document} is sensitive for every countable dense subset \begin{document}$A$\end{document} of \begin{document}$X$\end{document} . Moreover, if \begin{document}$(X, f)$\end{document} is weakly mixing, then \begin{document}$\big(S_A(X), F_f\big)$\end{document} is Auslander-Yorke chaotic for many countable dense subsets \begin{document}$A$\end{document} of \begin{document}$X$\end{document} . As an application, we consider a class of one-dimensional wave equations with van der Pol boundary condition and show that if the boundary condition is weakly mixing, then there exists an initial condition such that the solutions of the equations exhibit complicated behaviours.

A note on the convergence of the solution of the high order Camassa-Holm equation to the entropy ones of a scalar conservation law
Giuseppe Maria Coclite and Lorenzo Di Ruvo
2017, 37(3): 1247-1282 doi: 10.3934/dcds.2017052 +[Abstract](620) +[HTML](7) +[PDF](475.3KB)
Abstract:

We consider the high order Camassa-Holm equation, which is a non linear dispersive equation of the fifth order. We prove that as the diffusion and dispersion parameters tends to zero, the solutions converge to the entropy ones of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the $L^p$ setting.

A remark on the partial regularity of a suitable weak solution to the Navier-Stokes Cauchy problem
Francesca Crispo and Paolo Maremonti
2017, 37(3): 1283-1294 doi: 10.3934/dcds.2017053 +[Abstract](423) +[HTML](7) +[PDF](416.7KB)
Abstract:

Starting from the partial regularity results for suitable weak solutions to the Navier-Stokes Cauchy problem by Caffarelli, Kohn and Nirenberg [1], as a corollary, under suitable assumptions of local character on the initial data, we investigate the behavior in time of the \begin{document}$L_{loc}^\infty$\end{document}-norm of the solution in a neighborhood of $t=0$. The behavior is the same as for the resolvent operator associated to the Stokes operator.

Modified energy functionals and the NLS approximation
Patrick Cummings and C. Eugene Wayne
2017, 37(3): 1295-1321 doi: 10.3934/dcds.2017054 +[Abstract](490) +[HTML](4) +[PDF](471.5KB)
Abstract:

We consider a model equation from [14] that captures important properties of the water wave equation. We give a new proof of the fact that wave packet solutions of this equation are approximated by the nonlinear Schrödinger equation. This proof both simplifies and strengthens the results of [14] so that the approximation holds for the full interval of existence of the approximate NLS solution rather than just a subinterval. Furthermore, the proof avoids the problems associated with inverting the normal form transform in [14] by working with a modified energy functional motivated by [1] and [8].

High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs
Deconinck Bernard and Olga Trichtchenko
2017, 37(3): 1323-1358 doi: 10.3934/dcds.2017055 +[Abstract](620) +[HTML](4) +[PDF](689.6KB)
Abstract:

Generalizing ideas of MacKay, and MacKay and Saffman, a necessary condition for the presence of high-frequency (i.e., not modulational) instabilities of small-amplitude periodic solutions of Hamiltonian partial differential equations is presented, entirely in terms of the Hamiltonian of the linearized problem. With the exception of a Krein signature calculation, the theory is completely phrased in terms of the dispersion relation of the linear problem. The general theory changes as the Poisson structure of the Hamiltonian partial differential equation is changed. Two important cases of such Poisson structures are worked out in full generality. An example not fitting these two important cases is presented as well, using a candidate Boussinesq-Whitham equation.

Discrete conley index theory for zero dimensional basic sets
Ketty A. De Rezende and Mariana G. Villapouca
2017, 37(3): 1359-1387 doi: 10.3934/dcds.2017056 +[Abstract](511) +[HTML](1) +[PDF](610.9KB)
Abstract:

In this article the discrete Conley index theory is used to study diffeomorphisms on closed differentiable n-manifolds with zero dimensional hyperbolic chain recurrent set. A theorem is established for the computation of the discrete Conley index of these basic sets in terms of the dynamical information contained in their associated structure matrices. Also, a classification of the reduced homology Conley index of these basic sets is presented using its Jordan real form. This, in turn, is essential to obtain a characterization of a pair of connection matrices for a Morse decomposition of zero-dimensional basic sets of a diffeomorphism.

A leading term for the velocity of stationary viscous incompressible flow around a rigid body performing a rotation and a translation
Paul Deuring, Stanislav Kračmar and Šárka Nečasová
2017, 37(3): 1389-1409 doi: 10.3934/dcds.2017057 +[Abstract](538) +[HTML](1) +[PDF](545.9KB)
Abstract:

We consider the Navier-Stokes system with Oseen and rotational terms describing the stationary flow of a viscous incompressible fluid around a rigid body moving at a constant velocity and rotating at a constant angular velocity. In a previous paper, we proved a representation formula for Leray solutions of this system. Here the representation formula is used as starting point for splitting the velocity into a leading term and a remainder, and for establishing pointwise decay estimates of the remainder and its gradient.

Minimal subshifts of arbitrary mean topological dimension
Dou Dou
2017, 37(3): 1411-1424 doi: 10.3934/dcds.2017058 +[Abstract](478) +[HTML](4) +[PDF](384.2KB)
Abstract:

Let $G$ be a countable infinite amenable group and $P$ be a polyhedron. We give a construction of minimal subshifts of $P^G$ with arbitrary mean topological dimension less than $\dim P$.

Multiple periodic solutions of Hamiltonian systems confined in a box
Alessandro Fonda and Andrea Sfecci
2017, 37(3): 1425-1436 doi: 10.3934/dcds.2017059 +[Abstract](545) +[HTML](1) +[PDF](371.1KB)
Abstract:

We consider a nonautonomous Hamiltonian system, $T$-periodic in time, possibly defined on a bounded space region, the boundary of which consists of singularity points which can never be attained. Assuming that the system has an interior equilibrium point, we prove the existence of infinitely many $T$-periodic solutions, by the use of a generalized version of the Poincaré-Birkhoff theorem.

Homogenization of second order discrete model with local perturbation and application to traffic flow
Nicolas Forcadel, Wilfredo Salazar and Mamdouh Zaydan
2017, 37(3): 1437-1487 doi: 10.3934/dcds.2017060 +[Abstract](397) +[HTML](1) +[PDF](648.2KB)
Abstract:

The goal of this paper is to derive a traffic flow macroscopic model from a second order microscopic model with a local perturbation. At the microscopic scale, we consider a Bando model of the type following the leader, i.e the acceleration of each vehicle depends on the distance of the vehicle in front of it. We consider also a local perturbation like an accident at the roadside that slows down the vehicles. After rescaling, we prove that the "cumulative distribution functions" of the vehicles converges towards the solution of a macroscopic homogenized Hamilton-Jacobi equation with a flux limiting condition at junction which can be seen as a LWR (Lighthill-Whitham-Richards) model.

Qualitative description of the particle trajectories for the N-solitons solution of the Korteweg-de Vries equation
Ludovick Gagnon
2017, 37(3): 1489-1507 doi: 10.3934/dcds.2017061 +[Abstract](530) +[HTML](2) +[PDF](632.4KB)
Abstract:

The qualitative properties of the particle trajectories of the $N$-solitons solution of the KdV equation are recovered from the first order velocity field by the introduction of the stream function. Numerical simulations show an accurate depth dependance of the particles trajectories for solitary waves. Failure of the free surface kinematic boundary condition for the first order type velocity field is highlighted.

On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators
Huijun He and Zhaoyang Yin
2017, 37(3): 1509-1537 doi: 10.3934/dcds.2017062 +[Abstract](535) +[HTML](3) +[PDF](636.1KB)
Abstract:

In this paper, we mainly consider the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators: $m=(1-\partial_x^2)^su, s>1$. By Littlewood-Paley theory and transport equation theory, we first establish the local well-posedness of the generalized b-equation with fractional higher-order inertia operators which is the subsystem of the generalized two-component water wave system. Then we prove the local well-posedness of the generalized two-component water wave system with fractional higher-order inertia operators. Next, we present the blow-up criteria for these systems. Moreover, we obtain some global existence results for these systems.

Effect of cross-diffusion in the diffusion prey-predator model with a protection zone
Shanbing Li and Jianhua Wu
2017, 37(3): 1539-1558 doi: 10.3934/dcds.2017063 +[Abstract](603) +[HTML](3) +[PDF](521.9KB)
Abstract:

In this work, we continue the mathematical study started in [K. Oeda, J. Differential Equations 250 (2011) 3988-4009] on the analytic aspects of the diffusion prey-predator system with a protection zone and cross-diffusion. For small birth rates of two species and large cross-diffusion for the prey, the detailed structure of positive solutions is established by the bifurcation theory and the Lyapunov-Schmidt reduction, which is determined by a finite dimensional limiting system. Moreover, we prove that the stability of positive solutions changes only at every turning point by a spectral analysis for the linearized eigenvalue problem of the limiting system and its perturbation.

Subharmonic solutions and minimal periodic solutions of first-order Hamiltonian systems with anisotropic growth
Chungen Liu and Xiaofei Zhang
2017, 37(3): 1559-1574 doi: 10.3934/dcds.2017064 +[Abstract](570) +[HTML](2) +[PDF](465.3KB)
Abstract:

Using a homologically link theorem in variational theory and iteration inequalities of Maslov-type index, we show the existence of a sequence of subharmonic solutions of non-autonomous Hamiltonian systems with the Hamiltonian functions satisfying some anisotropic growth conditions, i.e., the Hamiltonian functions may have simultaneously, in different components, superquadratic, subquadratic and quadratic behaviors. Moreover, we also consider the minimal period problem of some autonomous Hamiltonian systems with anisotropic growth.

On an $N$-Component Camassa-Holm equation with peakons
Yongsheng Mi, Boling Guo and Chunlai Mu
2017, 37(3): 1575-1601 doi: 10.3934/dcds.2017065 +[Abstract](595) +[HTML](1) +[PDF](560.1KB)
Abstract:

In this paper, we are concerned with \begin{document}$N$\end{document}-Component Camassa-Holm equation with peakons. Firstly, we establish the local well-posedness in a range of the Besov spaces by using Littlewood-Paley decomposition and transport equation theory. Secondly, we present a precise blowup scenario and several blowup results for strong solutions to that system, we then obtain the blowup rate of strong solutions when a blowup occurs. Next, we investigate the persistence property for the strong solutions. Finally, we consider the initial boundary value problem, our approach is based on sharp extension results for functions on the half-line and several symmetry preserving properties of the equations under discussion.

Critical mass of degenerate Keller-Segel system with no-flux and Neumann boundary conditions
Yoshifumi Mimura
2017, 37(3): 1603-1630 doi: 10.3934/dcds.2017066 +[Abstract](442) +[HTML](5) +[PDF](517.7KB)
Abstract:

We prove the existence of solutions of degenerate parabolic-parabolic Keller-Segel system with no-flux and Neumann boundary conditions for each variable respectively, under the assumption that the total mass of the first variable is below a certain constant. The proof relies on the interpretation of the system as a gradient flow in the product space of the Wasserstein space and the standard \begin{document}$L^2$\end{document}-space. More precisely, we apply the ''minimizing movement'' scheme and show a certain critical mass appears in the application of this scheme to our problem.

Nonlocal Schrödinger-Kirchhoff equations with external magnetic field
Mingqi Xiang, Patrizia Pucci, Marco Squassina and Binlin Zhang
2017, 37(3): 1631-1649 doi: 10.3934/dcds.2017067 +[Abstract](599) +[HTML](1) +[PDF](525.8KB)
Abstract:

The paper deals with the existence and multiplicity of solutions of the fractional Schrödinger-Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case

where \begin{document}$s∈ (0,1)$\end{document}, \begin{document}$N>2s$\end{document}, \begin{document}$a∈ \mathbb{R}^+_0$\end{document}, \begin{document}$b∈ \mathbb{R}^+_0$\end{document}, \begin{document}$θ∈[1,N/(N-2s))$\end{document}, \begin{document}$A:\mathbb{R}^N\to\mathbb{R}^N$\end{document} is a magnetic potential, \begin{document}$V:\mathbb{R}^N\to \mathbb{R}^+$\end{document} is an electric potential, \begin{document}$(-Δ )_A^s$\end{document} is the fractional magnetic operator. In the super-and sub-linear cases, the existence of least energy solutions for the above problem is obtained by the mountain pass theorem, combined with the Nehari method, and by the direct methods respectively. In the superlinear-sublinear case, the existence of infinitely many solutions is investigated by the symmetric mountain pass theorem.

Renormalizable Expanding Baker Maps: Coexistence of strange attractors
Antonio Pumariño, José Ángel Rodríguez and Enrique Vigil
2017, 37(3): 1651-1678 doi: 10.3934/dcds.2017068 +[Abstract](422) +[HTML](1) +[PDF](796.7KB)
Abstract:

We introduce the concept of Expanding Baker Maps and renormalizable Expanding Baker Maps in a two-dimensional scenario. For a one-parameter family of Expanding Baker Maps we prove the existence of an interval of parameters for which the respective transformation is renormalizable. Moreover, we show the existence of intervals of parameters for which coexistence of strange attractors takes place.

Dynamical canonical systems and their explicit solutions
Alexander Sakhnovich
2017, 37(3): 1679-1689 doi: 10.3934/dcds.2017069 +[Abstract](373) +[HTML](1) +[PDF](377.5KB)
Abstract:

Dynamical canonical systems and their connections with the classical (spectral) canonical systems are considered. We construct Bäcklund-Darboux transformation and explicit solutions of the dynamical canonical systems. We study also those properties of the solutions, which are of interest in evolution and control theories.

A perturbation result for a critical elliptic equation with zero Dirichlet boundary condition
Khadijah Sharaf
2017, 37(3): 1691-1706 doi: 10.3934/dcds.2017070 +[Abstract](462) +[HTML](1) +[PDF](477.0KB)
Abstract:

In this paper we consider the following nonlinear critical problem: \begin{document} $-Δ u= (1+\varepsilon_0 K_0(x)) u^\frac{n+2}{n-2}$ \end{document}, \begin{document} $u>0$ \end{document} in \begin{document} $Ω$ \end{document}, \begin{document} $u=0$ \end{document}, on \begin{document} $\partial Ω$ \end{document}, where \begin{document} $Ω$ \end{document} is a bounded domain of \begin{document} $\mathbb{R}^n$ \end{document}, \begin{document} $K_0$ \end{document} is a given function and \begin{document} $\varepsilon_0$ \end{document} is a small parameter. Under the assumption that \begin{document} $K_0$ \end{document} is flat near its critical points, we prove an existence result in terms of the Euler-Hopf index. We believe that it is the very first result in this direction that we do not need any restrictions on the flatness coefficient.

Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential
Liping Wang and Chunyi Zhao
2017, 37(3): 1707-1731 doi: 10.3934/dcds.2017071 +[Abstract](556) +[HTML](2) +[PDF](548.0KB)
Abstract:

In the paper we prove the multiplicity existence of both nonlinear Schrödinger equation and Schrödinger system with slow decaying rate of electric potential at infinity. Namely, for any \begin{document} $\mathit{\boldsymbol{m}},\mathit{\boldsymbol{n > }}{\bf{0}}$ \end{document}, the potentials \begin{document} $P, Q$ \end{document} have the asymptotic behavior

then Schrödinger equation and Schrödinger system have infinitely many solutions with arbitrarily large energy, which extends the results of [37] for single Schrödinger equation and [30] for Schrödinger system.

Conserved quantities, global existence and blow-up for a generalized CH equation
Long Wei, Zhijun Qiao, Yang Wang and Shouming Zhou
2017, 37(3): 1733-1748 doi: 10.3934/dcds.2017072 +[Abstract](533) +[HTML](8) +[PDF](448.2KB)
Abstract:

In this paper, we study conserved quantities, blow-up criterions, and global existence of solutions for a generalized CH equation. We investigate the classification of self-adjointness, conserved quantities for this equation from the viewpoint of Lie symmetry analysis. Then, based on these conserved quantities, blow-up criterions and global existence of solutions are presented.

Asymptotic properties of standing waves for mass subcritical nonlinear Schrödinger equations
Xiaoyu Zeng
2017, 37(3): 1749-1762 doi: 10.3934/dcds.2017073 +[Abstract](578) +[HTML](3) +[PDF](450.5KB)
Abstract:

We study the following minimization problem:

where the functional \begin{document} $E_{q,a_q}(·)$ \end{document} is given by

Here \begin{document} $a_q>0, \ q∈(0,2)$ \end{document} and \begin{document} $V(x)$ \end{document} is some type of trapping potential. Let \begin{document} $a^*:= \|Q\|_2^2$ \end{document}, where \begin{document} $Q$ \end{document} is the unique (up to translations) positive radial solution of \begin{document} $Δ u-u+u^3=0$ \end{document} in \begin{document} $\mathbb{R}^2$ \end{document}. We prove that if \begin{document} $\lim_{q\nearrow2}a_q=a<a^*$ \end{document}, then minimizers of \begin{document} $d_{a_q}(q)$ \end{document} is compact in a suitable space as \begin{document} $q\nearrow2$ \end{document}. On the contraty, if \begin{document} $\lim_{q\nearrow2}a_q=a≥q a^*$ \end{document}, by directly using asymptotic analysis, we prove that all minimizers must blow up and give the detailed asymptotic behavior of minimizers. These conclusions extend the results of Guo-Zeng-Zhou [Concentration behavior of standing waves for almost mass critical nonlinear Schrödinger equations, J. Differential Equations. 256, (2014), 2079-2100].

Analytic results for the linear stability of the equilibrium point in Robe's restricted elliptic three-body problem
Qinglong Zhou and Yongchao Zhang
2017, 37(3): 1763-1787 doi: 10.3934/dcds.2017074 +[Abstract](633) +[HTML](3) +[PDF](495.8KB)
Abstract:

We study the Robe's restricted three-body problem. Such a motion was firstly studied by A. G. Robe in [13], which is used to model small oscillations of the earth's inner core taking into account the moon's attraction. Earlier results for the linear stability of the elliptic equilibrium point in Robe's restricted problem depend on a lot of numerical computations, while we give an analytic approach to it. The linearized Hamiltonian system near the elliptic equilibrium point in our problem coincides with the linearized system near the Euler elliptic relative equilibria in the classical three-body problem except for the range of the mass parameter. We first establish some relations of the linear stability problem to the properties of some symplectic paths and some corresponding linear operators. Then using the Maslov-type \begin{document} $ω$ \end{document}-index theory of symplectic paths and the theory of linear operators, we compute \begin{document} $ω$ \end{document}-indices and obtain certain properties of the linear stability of the elliptic equilibrium point of Robe's restricted three-body problem.

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