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Discrete & Continuous Dynamical Systems - A

2018 , Volume 38 , Issue 3

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Non-formally integrable centers admitting an algebraic inverse integrating factor
Antonio Algaba , Natalia Fuentes , Cristóbal García and  Manuel Reyes
2018, 38(3): 967-988 doi: 10.3934/dcds.2018041 +[Abstract](121) +[HTML](17) +[PDF](519.69KB)

We study the existence of a class of inverse integrating factor for a family of non-formally integrable systems whose lowest-degree quasi-homogeneous term is a Hamiltonian vector field. Once the existence of an inverse integrating factor is established, we study the systems having a center. Among others, we characterize the centers of the perturbations of the system \begin{document}$ -y^3\partial_x+x^3\partial_y$\end{document} having an algebraic inverse integrating factor.

Exponential multiple mixing for some partially hyperbolic flows on products of $ {\rm{PSL}}(2, \mathbb{R})$
James Tanis
2018, 38(3): 989-1006 doi: 10.3934/dcds.2018042 +[Abstract](63) +[HTML](13) +[PDF](461.54KB)

We prove a sharp estimate up to a logarithmic factor on the rate of equidistribution of coordinate horocycle flows on \begin{document}$ Γ \backslash{\rm{PSL}}(2, \mathbb{R})^d$\end{document}, where \begin{document}$ d ∈ \mathbb{N}_{≥2}$\end{document} and \begin{document}$ Γ \subset {\rm{PSL}}(2, \mathbb{R})^d$\end{document} is a cocompact and irreducible lattice. As a consequence, we prove exponential multiple mixing for partially hyperbolic coordinate geodesic flows on these manifolds.

What is topological about topological dynamics?
Chris Good and  Sergio Macías
2018, 38(3): 1007-1031 doi: 10.3934/dcds.2018043 +[Abstract](104) +[HTML](6) +[PDF](483.41KB)

We consider various notions from the theory of dynamical systems from a topological point of view. Many of these notions can be sensibly defined either in terms of (finite) open covers or uniformities. These Hausdorff or uniform versions coincide in compact Hausdorff spaces and are equivalent to the standard definition stated in terms of a metric in compact metric spaces.

We show for example that in a Tychonoff space, transitivity and dense periodic points imply (uniform) sensitivity to initial conditions. We generalise Bryant's result that a compact Hausdorff space admitting a $c$-expansive homeomorphism in the obvious uniform sense is metrizable. We study versions of shadowing, generalising a number of well-known results to the topological setting, and internal chain transitivity, showing for example that $ω$-limit sets are (uniform) internally chain transitive and weak incompressibility is equivalent to (uniform) internal chain transitivity in compact spaces.

Dynamics in dimension zero A survey
Tomasz Downarowicz and  Olena Karpel
2018, 38(3): 1033-1062 doi: 10.3934/dcds.2018044 +[Abstract](97) +[HTML](15) +[PDF](577.44KB)

The goal of this paper is to put together several techniques in handling dynamical systems on zero-dimensional spaces, such as array representation, inverse limit representation, or Bratteli-Vershik representation. We describe how one can switch from one representation to another. We also briefly review some more recent related notions: symbolic extensions, symbolic extensions with an embedding, and uniform generators. We devote a great deal of attention to marker techniques and we use them to prove two types of results: one concerning entropy and vertical data compression, and another, about the existence of isomorphic minimal models for aperiodic systems. We also introduce so-called decisiveness of Bratteli-Vershik systems and give for it a sufficient condition.

The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations
Huicheng Yin and  Lin Zhang
2018, 38(3): 1063-1102 doi: 10.3934/dcds.2018045 +[Abstract](87) +[HTML](4) +[PDF](555.11KB)

We concern with the global existence and large time behavior of compressible fluids (including the inviscid gases, viscid gases, and Boltzmann gases) in an infinitely expanding ball. Such a problem is one of the interesting models in studying the theory of global smooth solutions to multidimensional compressible gases with time dependent boundaries and vacuum states at infinite time. Due to the conservation of mass, the fluid in the expanding ball becomes rarefied and eventually tends to a vacuum state meanwhile there are no appearances of vacuum domains in any part of the expansive ball, which is easily observed in finite time. In this paper, as the second part of our three papers, we will confirm this physical phenomenon for the compressible viscid fluids by obtaining the exact lower and upper bound on the density function.

Remarks on the convergence of an algorithm for curvature-dependent motions of hypersurfaces
Katsuyuki Ishii and  Takahiro Izumi
2018, 38(3): 1103-1125 doi: 10.3934/dcds.2018046 +[Abstract](63) +[HTML](7) +[PDF](482.49KB)

We consider a threshold-type algorithm for curvature-dependent motions (CDM for short) of hypersurfaces. This algorithm was numerically studied by Kimura - Notsu [13], Esedoḡlu - Ruuth - Tsai [7] and Mohammad - Švadlenka [16], where they used the signed distance function as the level set function for CDM. The convergence of this algorithm and its optimal rate have been considered in Ishii - Kimura [12]. In this paper we give different approaches to the optimal rate of convergence to the smooth and compact CDM from [12]. As for the optimality, we give a more precise estimate than that in [12].

On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces
Van Duong Dinh
2018, 38(3): 1127-1143 doi: 10.3934/dcds.2018047 +[Abstract](69) +[HTML](4) +[PDF](448.14KB)

We proved the local well-posedness for the power-type nonlinear semi-relativistic or half-wave equation (NLHW) in Sobolev spaces. Our proofs are mainly based on the contraction mapping argument using Strichartz estimates. We also apply the technique of Christ-Colliander-Tao in [6] to prove the ill-posedness for the (NLHW) in a certain range of the super-critical case.

Uniform hyperbolicity in nonflat billiards
Mickaël Kourganoff
2018, 38(3): 1145-1160 doi: 10.3934/dcds.2018048 +[Abstract](68) +[HTML](9) +[PDF](422.07KB)

Uniform hyperbolicity is a strong chaotic property which holds, in particular, for Sinai billiards. In this paper, we consider the case of a nonflat billiard, that is, a Riemannian surface with boundary. Each trajectory follows the geodesic flow in the interior of the billiard, and bounces when it meets the boundary. We give a sufficient condition for a nonflat billiard to be uniformly hyperbolic. As a particular case, we obtain a new criterion to show that a closed surface has an Anosov geodesic flow.

Soliton solutions for the elastic metric on spaces of curves
Martin Bauer , Martins Bruveris , Philipp Harms and  Peter W. Michor
2018, 38(3): 1161-1185 doi: 10.3934/dcds.2018049 +[Abstract](84) +[HTML](25) +[PDF](420.54KB)

In this article we investigate a first order reparametrization-invariant Sobolev metric on the space of immersed curves. Motivated by applications in shape analysis where discretizations of this infinite-dimensional space are needed, we extend this metric to the space of Lipschitz curves, establish the wellposedness of the geodesic equation thereon, and show that the space of piecewise linear curves is a totally geodesic submanifold. Thus, piecewise linear curves are natural finite elements for the discretization of the geodesic equation. Interestingly, geodesics in this space can be seen as soliton solutions of the geodesic equation, which were not known to exist for reparametrization-invariant Sobolev metrics on spaces of curves.

Nonlocal stabilization by starting control of the normal equation generated by Helmholtz system
Andrei Fursikov and  Lyubov Shatina
2018, 38(3): 1187-1242 doi: 10.3934/dcds.2018050 +[Abstract](61) +[HTML](3) +[PDF](870.29KB)

Let \begin{document}$ y(t,x;y_0) $\end{document} be a solution to the semilinear parabolic equation of normal type generated by the 3D Helmholtz system with periodic boundary conditions and arbitrary initial datum \begin{document}$ y_0(x) $\end{document}. The problem of stabilization to zero of the solution \begin{document}$ y(t,x;y_0) $\end{document} by starting control is studied. This problem is reduced to establishing three inequalities connected with starting control, one of which has been proved in [10], [15]. The proof for the other two is given here.

A convergent Crank-Nicolson Galerkin scheme for the Benjamin-Ono equation
Sondre Tesdal Galtung
2018, 38(3): 1243-1268 doi: 10.3934/dcds.2018051 +[Abstract](84) +[HTML](10) +[PDF](672.87KB)

In this paper we prove the convergence of a Crank-Nicolson type Galerkin finite element scheme for the initial value problem associated to the Benjamin-Ono equation. The proof is based on a recent result for a similar discrete scheme for the Korteweg-de Vries equation and utilizes a local smoothing effect to bound the \begin{document}$ H^{1/2} $\end{document}-norm of the approximations locally. This enables us to show that the scheme converges strongly in \begin{document}$ L^{2}(0,T;L^{2}_{\text{loc}}(\mathbb{R})) $\end{document} to a weak solution of the equation for initial data in $L^{2}(\mathbb{R})$ and some \begin{document}$ T > 0 $\end{document}. Finally we illustrate the method with some numerical examples.

Weak regularization by stochastic drift : Result and counter example
Paul-Eric Chaudru De Raynal
2018, 38(3): 1269-1291 doi: 10.3934/dcds.2018052 +[Abstract](76) +[HTML](3) +[PDF](519.75KB)

In this paper, weak uniqueness of hypoelliptic stochastic differential equation with Hölder drift is proved when the Hölder exponent is strictly greater than 1/3. This result then "extends" to a weak framework the previous works [4,23,10], where strong uniqueness was proved when the regularity index of the drift is strictly greater than 2/3. Part of the result is also shown to be almost sharp thanks to a counter example when the Hölder exponent of the degenerate component is just below 1/3.

The approach is based on martingale problem formulation of Stroock and Varadhan and so on smoothing properties of the associated PDE which is, in the current setting, degenerate.

Topological solutions in the Maxwell-Chern-Simons model with anomalous magnetic moment
Youngae Lee
2018, 38(3): 1293-1314 doi: 10.3934/dcds.2018053 +[Abstract](86) +[HTML](10) +[PDF](544.58KB)

In this paper, we consider a Maxwell-Chern-Simons model with anomalous magnetic moment. Our main goal is to show the existence and uniqueness of topological type solutions to this problem on a flat two torus for any configuration of vortex points. Moreover, we also discuss about the stability of topological solutions.

Long-time behaviour of a radially symmetric fluid-shell interaction system
Tamara Fastovska
2018, 38(3): 1315-1348 doi: 10.3934/dcds.2018054 +[Abstract](71) +[HTML](10) +[PDF](547.34KB)

We study the long-time dynamics of a coupled system consisting of the 2D Navier-Stokes equations and full von Karman elasticity equations. We show that this problem generates an evolution semigroup $S_t$ possessing a compact finite-dimensional global attractor.

Pointwise wave behavior of the Navier-Stokes equations in half space
Linglong Du and  Haitao Wang
2018, 38(3): 1349-1363 doi: 10.3934/dcds.2018055 +[Abstract](72) +[HTML](7) +[PDF](398.07KB)

In this paper, we investigate the pointwise behavior of the solution for the compressible Navier-Stokes equations with mixed boundary condition in half space. Our results show that the leading order of Green's function for the linear system in half space are heat kernels propagating with sound speed in two opposite directions and reflected heat kernel (due to the boundary effect) propagating with positive sound speed. With the strong wave interactions, the nonlinear analysis exhibits the rich wave structure: the diffusion waves interact with each other and consequently, the solution decays with algebraic rate.

Hölder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian
Carlos Lizama and  Luz Roncal
2018, 38(3): 1365-1403 doi: 10.3934/dcds.2018056 +[Abstract](99) +[HTML](2) +[PDF](642.28KB)

We study the equations

\begin{document}$\begin{align}\partial_t u(t, n) = L u(t, n) + f(u(t, n), n); \partial_t u(t, n) = iL u(t, n) + f(u(t, n), n)\end{align}$ \end{document}


\begin{document}$\begin{align}\partial_{tt} u(t, n) =Lu(t, n) + f(u(t, n), n), \end{align}$ \end{document}

where \begin{document} $n∈ \mathbb{Z}$ \end{document}, \begin{document} $t∈ (0, ∞)$ \end{document}, and \begin{document} $L$ \end{document} is taken to be either the discrete Laplacian operator \begin{document} $Δ_\mathrm{d} f(n)=f(n+1)-2f(n)+f(n-1)$ \end{document}, or its fractional powers \begin{document} $-(-Δ_{\mathrm{d}})^{σ}$ \end{document}, \begin{document} $0<σ<1$ \end{document}. We combine operator theory techniques with the properties of the Bessel functions to develop a theory of analytic semigroups and cosine operators generated by \begin{document} $Δ_\mathrm{d}$ \end{document} and \begin{document} $-(-Δ_\mathrm{d})^{σ}$ \end{document}. Such theory is then applied to prove existence and uniqueness of almost periodic solutions to the above-mentioned equations. Moreover, we show a new characterization of well-posedness on periodic Hölder spaces for linear heat equations involving discrete and fractional discrete Laplacians. The results obtained are applied to Nagumo and Fisher-KPP models with a discrete Laplacian. Further extensions to the multidimensional setting \begin{document} $\mathbb{Z}^N$ \end{document} are also accomplished.

Improved energy methods for nonlocal diffusion problems
José A. Cañizo and  Alexis Molino
2018, 38(3): 1405-1425 doi: 10.3934/dcds.2018057 +[Abstract](56) +[HTML](21) +[PDF](502.71KB)

We prove an energy inequality for nonlocal diffusion operators of the following type, and some of its generalisations:

where \begin{document} $L$ \end{document} acts on a real function \begin{document} $u$ \end{document} defined on \begin{document} $\mathbb{R}^N$ \end{document}, and we assume that \begin{document} $K(x, y)$ \end{document} is uniformly strictly positive in a neighbourhood of \begin{document} $x=y$ \end{document}. The inequality is a nonlocal analogue of the Nash inequality, and plays a similar role in the study of the asymptotic decay of solutions to the nonlocal diffusion equation \begin{document} $\partial_t u = L u$ \end{document} as the Nash inequality does for the heat equation. The inequality allows us to give a precise decay rate of the \begin{document} $L^p$ \end{document} norms of \begin{document} $u$ \end{document} and its derivatives. As compared to existing decay results in the literature, our proof is perhaps simpler and gives new results in some cases.

On interior $C^2$-estimates for the Monge-Ampère equation
Diego Maldonado
2018, 38(3): 1427-1440 doi: 10.3934/dcds.2018058 +[Abstract](92) +[HTML](19) +[PDF](401.89KB)

An approach towards apriori interior \begin{document} $C^2$ \end{document}-estimates for the Monge-Ampère equation based on a mean-value inequality for nonnegative subsolutions to the linearized Monge-Ampère equation is implemented.

On the positive solutions for a perturbed negative exponent problem on $\mathbb{R}^3$
Sanjiban Santra
2018, 38(3): 1441-1460 doi: 10.3934/dcds.2018059 +[Abstract](277) +[HTML](16) +[PDF](443.08KB)

In this paper, we study the following fourth order elliptic problem with a negative nonlinearity :

\begin{document}$\begin{align}\left\{\begin{aligned} Δ^2 u&=-\frac{15}{16}(1+ \varepsilon Q)u^{-7} &&\text{ in } \mathbb R^3\\ u &>0 &&\text{ in } \mathbb R^3,\\ u(x) &\sim |x| \text{ as }{|x|\to ∞}. & \end{aligned} \right.\end{align}$ \end{document}

Here \begin{document} $Q$ \end{document} is a \begin{document} $C^{1}$ \end{document} bounded function on \begin{document} $\mathbb{R}^3$ \end{document} and \begin{document} $\varepsilon >0$ \end{document} is a small parameter. We prove the existence, uniqueness of positive solutions for the above perturbed fourth order problem.

Pullback attractor and invariant measures for the three-dimensional regularized MHD equations
Zeqi Zhu and  Caidi Zhao
2018, 38(3): 1461-1477 doi: 10.3934/dcds.2018060 +[Abstract](88) +[HTML](5) +[PDF](502.13KB)

This article studies the three-dimensional regularized Magnetohydrodynamics (MHD) equations. Using the approach of energy equations, the authors prove that the associated process possesses a pullback attractor. Then they establish the unique existence of the family of invariant Borel probability measures which is supported by the pullback attractor.

Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in 2D
Shinya Kinoshita
2018, 38(3): 1479-1504 doi: 10.3934/dcds.2018061 +[Abstract](49) +[HTML](31) +[PDF](498.15KB)

This paper is concerned with the Cauchy problem of the Klein-Gordon-Zakharov system with very low regularity initial data. We prove the bilinear estimates which are crucial to get the local in time well-posedness. The estimates are established by the Fourier restriction norm method. We utilize the nonlinear version of the classical Loomis-Whitney inequality.

Existence of nonnegative solutions to singular elliptic problems, a variational approach
Tomas Godoy and  Alfredo Guerin
2018, 38(3): 1505-1525 doi: 10.3934/dcds.2018062 +[Abstract](129) +[HTML](9) +[PDF](533.58KB)

We consider the problem \begin{document}$-Δ u = χ_{\{ u>0\} }g( .,u) +f( .,u) $\end{document} in \begin{document}$Ω,$\end{document} \begin{document}$u = 0$\end{document} on \begin{document}$\partialΩ,$\end{document} \begin{document}$ u≥0$\end{document} in \begin{document}$Ω,$\end{document} where \begin{document}$Ω$\end{document} is a bounded domain in \begin{document}$\mathbb{R}^{n}$\end{document}, \begin{document}$f:Ω×[ 0,∞) →\mathbb{R}$\end{document} and \begin{document}$ g:Ω×( 0,∞) →[ 0,∞) $\end{document} are Carathéodory functions, with \begin{document}$g( x,.) $\end{document} nonnegative, nonincreasing, and singular at the origin. We establish sufficient conditions for the existence of a nonnegative weak solution \begin{document}$0\not \equiv u∈ H_{0}^{1}( Ω) $\end{document} to the stated problem. We also provide conditions that guarantee that the found solution is positive \begin{document}$a.e.$\end{document} in \begin{document}$ Ω$\end{document}. The problem with a parameter \begin{document}$Δ u = χ_{\{ u>0\} }g( .,u) +λ f( .,u) $\end{document} in \begin{document}$Ω,$\end{document} \begin{document}$u = 0$\end{document} on \begin{document}$ \partialΩ,$\end{document} \begin{document}$u≥0$\end{document} in \begin{document}$Ω$\end{document} is also studied. For both problems, the special case when \begin{document}$g( x,s) : = a( x) s^{-α( x) },$\end{document} i.e., a singularity with variable exponent, is also considered.

Phase transition layers for Fife-Greenlee problem on smooth bounded domain
Feifei Tang , Suting Wei and  Jun Yang
2018, 38(3): 1527-1552 doi: 10.3934/dcds.2018063 +[Abstract](55) +[HTML](3) +[PDF](556.84KB)

We consider the Fife-Greenlee problem

\begin{document}$ε^2\triangle u + \bigl(u-\mathbf{a}(y)\bigr)(1-u^2) =0 ~~~ \mbox{in}\ Ω,~~~~~~~\frac{\partial u}{\partialν} = 0 ~~~ \mbox{on}\ \partialΩ,$ \end{document}

where \begin{document}$Ω$\end{document} is a bounded domain in \begin{document}${\mathbb R}^2$\end{document} with smooth boundary, \begin{document}$\epsilon>0$\end{document} is a small parameter, \begin{document}$ν$\end{document} denotes the unit outward normal of \begin{document}$\partialΩ$\end{document}. Let \begin{document}$Γ = \{y∈ Ω: \mathbf{a}(y) = 0 \}$\end{document} be a simple smooth curve intersecting orthogonally with \begin{document}$\partialΩ$\end{document} at exactly two points and dividing \begin{document}$Ω$\end{document} into two disjoint nonempty components. We assume that \begin{document}$-1\,<\,\mathbf{a}(y)\,<1$\end{document} on \begin{document}$Ω$\end{document} and \begin{document}$\triangledown\mathbf{a}≠ 0$\end{document} on \begin{document}$Γ$\end{document}, and also some admissibility conditions between the curves \begin{document}$Γ$\end{document}, \begin{document}$\partialΩ$\end{document} and the inhomogeneity \begin{document}${\mathbf a}$\end{document} hold at the connecting points. We can prove that there exists a solution \begin{document}$u_{\epsilon}$\end{document} such that: as \begin{document}$\epsilon → 0$\end{document}, \begin{document}$u_{\epsilon}$\end{document} approaches \begin{document}$+1$\end{document} in one part, while tends to \begin{document}$-1$\end{document} in the other part, except a small neighborhood of \begin{document}$Γ$\end{document}.

On the universality of the incompressible Euler equation on compact manifolds
Terence Tao
2018, 38(3): 1553-1565 doi: 10.3934/dcds.2018064 +[Abstract](68) +[HTML](7) +[PDF](422.44KB)

The incompressible Euler equations on a compact Riemannian manifold \begin{document}$(M,g)$\end{document} take the form

\begin{document}$\partial_t u + \nabla_u u =- \mathrm{grad}_g p \\\mathrm{div}_g u =0.$ \end{document}

We show that any quadratic ODE \begin{document}$\partial_t y =B(y,y)$\end{document}, where \begin{document}$B \colon \mathbb{R}^n × \mathbb{R}^n \to \mathbb{R}^n$\end{document} is a symmetric bilinear map, can be linearly embedded into the incompressible Euler equations for some manifold \begin{document}$M$\end{document} if and only if \begin{document}$B$\end{document} obeys the cancellation condition \begin{document}$\langle B(y,y), y \rangle =0$\end{document} for some positive definite inner product \begin{document}$\langle,\rangle$\end{document} on \begin{document}$\mathbb{R}^n$\end{document}. This allows one to construct explicit solutions to the Euler equations with various dynamical features, such as quasiperiodic solutions, or solutions that transition from one steady state to another, and provides evidence for the "Turing universality" of such Euler flows.

Traveling wave solutions of a highly nonlinear shallow water equation
Anna Geyer and  Ronald Quirchmayr
2018, 38(3): 1567-1604 doi: 10.3934/dcds.2018065 +[Abstract](169) +[HTML](38) +[PDF](12373.99KB)

Motivated by the question whether higher-order nonlinear model equations, which go beyond the Camassa-Holm regime of moderate amplitude waves, could point us to new types of waves profiles, we study the traveling wave solutions of a quasilinear evolution equation which models the propagation of shallow water waves of large amplitude. The aim of this paper is a complete classification of its traveling wave solutions. Apart from symmetric smooth, peaked and cusped solitary and periodic traveling waves, whose existence is well-known for moderate amplitude equations like Camassa-Holm, we obtain entirely new types of singular traveling waves: periodic waves which exhibit singularities on both crests and troughs simultaneously, waves with asymmetric peaks, as well as multi-crested smooth and multi-peaked waves with decay. Our approach uses qualitative tools for dynamical systems and methods for integrable planar systems.

Wave breaking of periodic solutions to the Fornberg-Whitham equation
Günther Hörmann
2018, 38(3): 1605-1613 doi: 10.3934/dcds.2018066 +[Abstract](50) +[HTML](10) +[PDF](324.89KB)

Based on recent well-posedness results in Sobolev (or Besov spaces) for periodic solutions to the Fornberg-Whitham equations we investigate here the questions of wave breaking and blow-up for these solutions. We show first that finite maximal life time of a solution necessarily leads to wave breaking. Second, we prove that for a certain class of initial wave profiles the corresponding solutions do indeed blow-up in finite time.

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