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Discrete & Continuous Dynamical Systems - A

September 2020 , Volume 40 , Issue 9

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Renormalizing an infinite rational IET
W. Patrick Hooper, Kasra Rafi and Anja Randecker
2020, 40(9): 5105-5116 doi: 10.3934/dcds.2020220 +[Abstract](291) +[HTML](76) +[PDF](452.5KB)

We study an interval exchange transformation of \begin{document}$ [0, 1] $\end{document} formed by cutting the interval at the points \begin{document}$ \frac{1}{n} $\end{document} and reversing the order of the intervals. We find that the transformation is periodic away from a Cantor set of Hausdorff dimension zero. On the Cantor set, the dynamics are nearly conjugate to the \begin{document}$ 2 $\end{document}–adic odometer.

Notes on the values of volume entropy of graphs
Wooyeon Kim and Seonhee Lim
2020, 40(9): 5117-5129 doi: 10.3934/dcds.2020221 +[Abstract](269) +[HTML](81) +[PDF](363.85KB)

Volume entropy is an important invariant of metric graphs as well as Riemannian manifolds. In this note, we calculate the change of volume entropy when an edge is added to a metric graph or when a vertex and edges around it are added. In the second part, we estimate the value of the volume entropy which can be used to suggest an algorithm for calculating the persistent volume entropy of graphs.

Exponential convergence in the Wasserstein metric $ W_1 $ for one dimensional diffusions
Lingyan Cheng, Ruinan Li and Liming Wu
2020, 40(9): 5131-5148 doi: 10.3934/dcds.2020222 +[Abstract](257) +[HTML](81) +[PDF](390.07KB)

In this paper, we find some general and efficient sufficient conditions for the exponential convergence \begin{document}$ W_{1,d}(P_t(x,\cdot), P_t(y,\cdot) )\le Ke^{-\delta t}d(x,y) $\end{document} for the semigroup \begin{document}$ (P_t) $\end{document} of one-dimensional diffusion. Moreover, some sharp estimates of the involved constants \begin{document}$ K\ge 1, \delta>0 $\end{document} are provided. Those general results are illustrated by a series of examples.

Contributions to the study of Anosov geodesic flows in non-compact manifolds
Ítalo Melo and Sergio Romaña
2020, 40(9): 5149-5171 doi: 10.3934/dcds.2020223 +[Abstract](259) +[HTML](60) +[PDF](1021.13KB)

In this paper, we study the relations between curvature and Anosov geodesic flow. More specifically, we prove that when the geodesic flow of a complete manifold without conjugate points is of the Anosov type, then the average of the sectional curvature in tangent planes along geodesics is negative and uniformly away from zero. Moreover, if a surface has no focal points, then the latter condition is sufficient to obtain that the geodesic flow is of Anosov type.

Limit theorems for additive functionals of path-dependent SDEs
Jianhai Bao, Feng-Yu Wang and Chenggui Yuan
2020, 40(9): 5173-5188 doi: 10.3934/dcds.2020224 +[Abstract](201) +[HTML](62) +[PDF](359.87KB)

By using limit theorems of uniform mixing Markov processes and martingale difference sequences, the strong law of large numbers, central limit theorem, and the law of iterated logarithm are established for additive functionals of path-dependent stochastic differential equations.

On the Cauchy problem for higher dimensional Benjamin-Ono and Zakharov-Kuznetsov equations
Robert Schippa
2020, 40(9): 5189-5215 doi: 10.3934/dcds.2020225 +[Abstract](142) +[HTML](60) +[PDF](435.98KB)

A family of dispersive equations is considered, which links a higher-dimensional Benjamin-Ono equation and the Zakharov-Kuznetsov equation. For these fractional Zakharov-Kuznetsov equations new well-posedness results are proved using transversality and time localization to small frequency dependent time intervals.

A topological study of planar vector field singularities
Robert Roussarie
2020, 40(9): 5217-5245 doi: 10.3934/dcds.2020226 +[Abstract](149) +[HTML](69) +[PDF](931.42KB)

In this paper one extends results of Bendixson [1] and Dumortier [2] about the germs of vector fields at the origin of \begin{document}$ {{I\kern-0.3emR}}^2, $\end{document} which is assumed to be an singularity isolated from other singularities and periodic orbits as well. As a new tool, one uses minimal centred curves, which are curves surrounding the origin, with a minimal number of contact points with the vector field. A similar notion was introduced by Le Roux in [4]. It is noticeable that the arguments are essentially topological, with no use of a desingularization theory, as in [2] for instance.

Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices
Nimish Shah and Lei Yang
2020, 40(9): 5247-5287 doi: 10.3934/dcds.2020227 +[Abstract](133) +[HTML](62) +[PDF](580.95KB)

In this paper, we study an analytic curve \begin{document}$ \varphi: I = [a, b]\rightarrow \mathrm{M}(m\times n, \mathbb{R}) $\end{document} in the space of \begin{document}$ m $\end{document} by \begin{document}$ n $\end{document} real matrices, and show that if \begin{document}$ \varphi $\end{document} satisfies certain geometric condition, then for almost every point on the curve, the Diophantine approximation given by Dirichlet's Theorem can not be improved. To do this, we embed the curve into a homogeneous space \begin{document}$ G/\Gamma $\end{document}, and prove that under the action of some expanding diagonal subgroup \begin{document}$ A = \{a(t): t \in \mathbb{R}\} $\end{document}, the translates of the curve tend to be equidistributed in \begin{document}$ G/\Gamma $\end{document}, as \begin{document}$ t \rightarrow +\infty $\end{document}. The proof relies on the linearization technique and representation theory.

Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces
Hongjie Dong and Kunrui Wang
2020, 40(9): 5289-5323 doi: 10.3934/dcds.2020228 +[Abstract](152) +[HTML](66) +[PDF](491.55KB)

We study regularity criteria for the \begin{document}$ d $\end{document}-dimensional incompressible Navier-Stokes equations. We prove if \begin{document}$ u\in L_{\infty}^tL_d^x((0,T)\times{\mathbb{R}}^d_+) $\end{document} is a Leray-Hopf weak solution vanishing on the boundary, then \begin{document}$ u $\end{document} is regular up to the boundary in \begin{document}$ (0,T)\times {\mathbb{R}}^d_+ $\end{document}. Furthermore, with a stronger uniform local condition on the pressure \begin{document}$ p $\end{document}, we prove \begin{document}$ u $\end{document} is unique and tends to zero as \begin{document}$ t\rightarrow \infty $\end{document} if \begin{document}$ T = \infty $\end{document}. This generalizes a result by Escauriaza, Seregin, and Šverák [14] to higher dimensions and domains with boundary. We also study the local problem in half unit cylinder \begin{document}$ Q^+ $\end{document} and prove that if \begin{document}$ u\in L^t_{\infty}L^x_d(Q^+) $\end{document} and \begin{document}$ p\in L_{2-1/d}(Q^+) $\end{document}, then \begin{document}$ u $\end{document} is Hölder continuous in the closure of the set \begin{document}$ Q^+(1/4) $\end{document}.

Reducibility of quasi-periodically forced circle flows
Saša Kocić and João Lopes Dias
2020, 40(9): 5325-5345 doi: 10.3934/dcds.2020229 +[Abstract](139) +[HTML](53) +[PDF](393.05KB)

We develop a renormalization group approach to the problem of reducibility of quasi-periodically forced circle flows. We apply the method to prove a reducibility theorem for such flows.

Polynomial 3-mixing for smooth time-changes of horocycle flows
Adam Kanigowski and Davide Ravotti
2020, 40(9): 5347-5371 doi: 10.3934/dcds.2020230 +[Abstract](114) +[HTML](56) +[PDF](504.61KB)

Let \begin{document}$ (h_t)_{t\in {\mathbb{R}}} $\end{document} be the horocycle flow acting on \begin{document}$ (M,\mu) = (\Gamma \backslash \operatorname{SL}(2,{\mathbb{R}}), \mu) $\end{document}, where \begin{document}$ \Gamma $\end{document} is a co-compact lattice in \begin{document}$ \operatorname{SL}(2,{\mathbb{R}}) $\end{document} and \begin{document}$ \mu $\end{document} is the homogeneous probability measure locally given by the Haar measure on \begin{document}$ \operatorname{SL}(2,{\mathbb{R}}) $\end{document}. Let \begin{document}$ \tau\in W^6(M) $\end{document} be a strictly positive function and let \begin{document}$ \mu^{\tau} $\end{document} be the measure equivalent to \begin{document}$ \mu $\end{document} with density \begin{document}$ \tau $\end{document}. We consider the time changed flow \begin{document}$ (h_t^\tau)_{t\in {\mathbb{R}}} $\end{document} and we show that there exists \begin{document}$ \gamma = \gamma(M,\tau)>0 $\end{document} and a constant \begin{document}$ C>0 $\end{document} such that for any \begin{document}$ f_0, f_1, f_2\in W^6(M) $\end{document} and for all \begin{document}$ 0 = t_0<t_1<t_2 $\end{document}, we have

With the same techniques, we establish polynomial mixing of all orders under the additional assumption of \begin{document}$ \tau $\end{document} being fully supported on the discrete series.

Liouville theorems on the upper half space
Lei Wang and Meijun Zhu
2020, 40(9): 5373-5381 doi: 10.3934/dcds.2020231 +[Abstract](172) +[HTML](58) +[PDF](303.71KB)

In this paper we shall establish some Liouville theorems for solutions bounded from below to certain linear elliptic equations on the upper half space. In particular, we show that for \begin{document}$ a \in (0, 1) $\end{document} constants are the only \begin{document}$ C^1 $\end{document} up to the boundary positive solutions to \begin{document}$ div(x_n^a \nabla u) = 0 $\end{document} on the upper half space.

Super fast vanishing solutions of the fast diffusion equation
Shu-Yu Hsu
2020, 40(9): 5383-5414 doi: 10.3934/dcds.2020232 +[Abstract](115) +[HTML](56) +[PDF](439.27KB)

We will extend a recent result of B. Choi, P. Daskalopoulos and J. King [5]. For any \begin{document}$ n\ge 3 $\end{document}, \begin{document}$ 0<m<\frac{n-2}{n+2} $\end{document} and \begin{document}$ \gamma>0 $\end{document}, we will construct subsolutions and supersolutions of the fast diffusion equation \begin{document}$ u_t = \frac{n-1}{m}\Delta u^m $\end{document} in \begin{document}$ \mathbb{R}^n\times (t_0, T) $\end{document}, \begin{document}$ t_0<T $\end{document}, which decay at the rate \begin{document}$ (T-t)^{\frac{1+\gamma}{1-m}} $\end{document} as \begin{document}$ t\nearrow T $\end{document}. As a consequence we obtain the existence of unique solution of the Cauchy problem \begin{document}$ u_t = \frac{n-1}{m}\Delta u^m $\end{document} in \begin{document}$ \mathbb{R}^n\times (t_0, T) $\end{document}, \begin{document}$ u(x, t_0) = u_0(x) $\end{document} in \begin{document}$ \mathbb{R}^n $\end{document}, which decay at the rate \begin{document}$ (T-t)^{\frac{1+\gamma}{1-m}} $\end{document} as \begin{document}$ t\nearrow T $\end{document} when \begin{document}$ u_0 $\end{document} satisfies appropriate decay condition.

Time periodic solution to a coupled chemotaxis-fluid model with porous medium diffusion
Jiapeng Huang and Chunhua Jin
2020, 40(9): 5415-5439 doi: 10.3934/dcds.2020233 +[Abstract](168) +[HTML](59) +[PDF](403.06KB)

This paper is concerned with the time periodic problem to a coupled chemotaxis-fluid model with porous medium diffusion \begin{document}$ \Delta n^m $\end{document}. The global existence of solutios for the initial and boundary value problem of this model have been studied by many authors, and in particular, the global solvability is established for \begin{document}$ m>\frac65 $\end{document} in dimension 3. Here, taking advantage of a double-level approximation scheme, we establish the existence of uniformly bounded time periodic solution for any \begin{document}$ m\ge \frac 65 $\end{document} and any large periodic source \begin{document}$ g(x, t) $\end{document}. In particular, the energy estimates techniques we used also applicable to the proof of global existence of the initial-boundary value problem, and one can supply the existence of global solutions for \begin{document}$ m = \frac65 $\end{document} by this method.

Multiplicity of solutions for critical quasilinear Schrödinger equations using a linking structure
Edcarlos D. Silva and Jefferson S. Silva
2020, 40(9): 5441-5470 doi: 10.3934/dcds.2020234 +[Abstract](142) +[HTML](62) +[PDF](414.4KB)

It is established multiplicity of solutions for critical quasilinear Schrödinger equations defined in the whole space using a linking structure. The main difficulty comes from the lack of compactness of Sobolev embedding into Lebesgue spaces. Moreover, the potential is bounded from below and above by positive constants. In order to overcome these difficulties we employ Lions Concentration Compactness Principle together with some fine estimates for the energy functional restoring some kind of compactness.

Global existence and convergence of nondimensionalized incompressible Navier-Stokes equations in low Froude number regime
Stefano Scrobogna
2020, 40(9): 5471-5511 doi: 10.3934/dcds.2020235 +[Abstract](126) +[HTML](57) +[PDF](574.92KB)

We prove that the incompressible, density dependent, Navier-Stokes equations are globally well posed in a low Froude number regime. The density profile is supposed to be increasing in depth and linearized around a stable state. Moreover if the Froude number tends to zero we prove that such system converges (strongly) to a two-dimensional, stratified Navier-Stokes equations with full diffusivity. No smallness assumption is considered on the initial data.

Nonexistence of global solutions for the semilinear Moore – Gibson – Thompson equation in the conservative case
Wenhui Chen and Alessandro Palmieri
2020, 40(9): 5513-5540 doi: 10.3934/dcds.2020236 +[Abstract](154) +[HTML](53) +[PDF](450.48KB)

In this work, the Cauchy problem for the semilinear Moore – Gibson – Thompson (MGT) equation with power nonlinearity \begin{document}$ |u|^p $\end{document} on the right – hand side is studied. Applying \begin{document}$ L^2 - L^2 $\end{document} estimates and a fixed point theorem, we obtain local (in time) existence of solutions to the semilinear MGT equation. Then, the blow - up of local in time solutions is proved by using an iteration method, under certain sign assumption for initial data, and providing that the exponent of the power of the nonlinearity fulfills \begin{document}$ 1<p\leqslant p_{\mathrm{Str}}(n) $\end{document} for \begin{document}$ n\geqslant2 $\end{document} and \begin{document}$ p>1 $\end{document} for \begin{document}$ n = 1 $\end{document}. Here the Strauss exponent \begin{document}$ p_{\mathrm{Str}}(n) $\end{document} is the critical exponent for the semilinear wave equation with power nonlinearity. In particular, in the limit case \begin{document}$ p = p_{\mathrm{Str}}(n) $\end{document} a different approach with a weighted space average of a local in time solution is considered.

Global existence and scattering of equivariant defocusing Chern-Simons-Schrödinger system
Jianjun Yuan
2020, 40(9): 5541-5570 doi: 10.3934/dcds.2020237 +[Abstract](137) +[HTML](53) +[PDF](431.25KB)

In this paper, we consider the following equivariant defocusing Chern-Simons-Schrödinger system,

where \begin{document}$ \phi(t, x_1, x_2): \mathbb{R}^{1+2}\rightarrow \mathbb{R} $\end{document} is a complex scalar field, \begin{document}$ A_\mu(t, x_1, x_2): \mathbb{R}^{1+2}\rightarrow \mathbb{R} $\end{document} is the gauge field for \begin{document}$ \mu = 0, 1, 2 $\end{document}, \begin{document}$ A_r = \frac{x_1}{|x|}A_1+\frac{x_2}{|x|}A_2 $\end{document}, \begin{document}$ A_{\theta} = -x_2A_1+x_1A_2 $\end{document}, \begin{document}$ \lambda<0 $\end{document} and \begin{document}$ p>4 $\end{document}.

When \begin{document}$ p>4 $\end{document}, the system is in the mass supercritical and energy subcrtical range. By using the conservation law of the system and the concentration compactness method introduced in [17], we show that the solution of the system exists globally and scatters.

Linearization of a nonautonomous unbounded system with nonuniform contraction: A spectral approach
Ignacio Huerta
2020, 40(9): 5571-5590 doi: 10.3934/dcds.2020238 +[Abstract](141) +[HTML](50) +[PDF](431.37KB)

For a nonautonomous linear system with nonuniform contraction, we construct a topological conjugacy between this system and an unbounded nonlinear perturbation. This topological conjugacy is constructed as a composition of homeomorphisms. The first one is set up by considering the fact that linear system is almost reducible to diagonal system with a small enough perturbation where the diagonal entries belong to spectrum of the nonuniform exponential dichotomy; and the second one is constructed in terms of the crossing times with respect to unit sphere of an adequate Lyapunov function associated to the linear system.

Infinitely many solutions for quasilinear equations with critical exponent and Hardy potential in $ \mathbb{R}^N $
Fengshuang Gao and Yuxia Guo
2020, 40(9): 5591-5616 doi: 10.3934/dcds.2020239 +[Abstract](152) +[HTML](53) +[PDF](518.08KB)

In this paper, we consider the following critical quasilinear equation with Hardy potential:

where \begin{document}$ a_{ij}(u)\!\in \!C^1\!(\mathbb{R},\mathbb{R}) $\end{document}, \begin{document}$ \nu\!>\!0 $\end{document}, \begin{document}$ 0\!\leq\!\mu\!<\!\alpha\bar{\mu} $\end{document}, and \begin{document}$ \max\!\left\{\!\frac{\alpha\bar{\mu}\gamma}{\alpha\bar{\mu}-\mu}\!+\!2,2^*\!\!-\!\frac{2}{N-2}\sqrt{\!\bar{\mu}\!-\!\frac{\mu}{\alpha}}\right\} \!\!<q<2^* $\end{document}, \begin{document}$ \alpha, \gamma>0 $\end{document}, \begin{document}$ \bar{\mu} = \frac{(N-2)^2}{4} $\end{document}, \begin{document}$ 2^\ast = \frac{2N}{N-2} $\end{document} is the Sobolev critical exponent. And \begin{document}$ a(x) $\end{document} is a finite, positive potential function satisfying suitable decay assumptions. By using truncation method combining with the regularization approximation approach and compactness arguments, we prove the existence of infinitely many solutions for this equation.

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