# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

November 2019 , Volume 39 , Issue 11

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2019, 39(11): 6175-6206 doi: 10.3934/dcds.2019269 +[Abstract](420) +[HTML](93) +[PDF](508.74KB)
Abstract:

For the system of polyconvex adiabatic thermoelasticity, we define a notion of dissipative measure-valued solution, which can be considered as the limit of a viscosity approximation. We embed the system into a symmetrizable hyperbolic one in order to derive the relative entropy. Exploiting the weak-stability properties of the transport and stretching identities, we base our analysis in the original variables, instead of the symmetric ones (in which the entropy is convex) and we prove measure-valued weak versus strong uniqueness using the averaged relative entropy inequality.

2019, 39(11): 6207-6230 doi: 10.3934/dcds.2019270 +[Abstract](453) +[HTML](85) +[PDF](414.71KB)
Abstract:

We introduce a new formulation for differential equation describing dynamics of measures on an Euclidean space, that we call Measure Differential Equations with sources. They mix two different phenomena: on one side, a transport-type term, in which a vector field is replaced by a Probability Vector Field, that is a probability distribution on the tangent bundle; on the other side, a source term. Such new formulation allows to write in a unified way both classical transport and diffusion with finite speed, together with creation of mass.

The main result of this article shows that, by introducing a suitable Wasserstein-like functional, one can ensure existence of solutions to Measure Differential Equations with sources under Lipschitz conditions. We also prove a uniqueness result under the following additional hypothesis: the measure dynamics needs to be compatible with dynamics of measures that are sums of Dirac masses.

2019, 39(11): 6231-6239 doi: 10.3934/dcds.2019271 +[Abstract](501) +[HTML](99) +[PDF](290.34KB)
Abstract:

In this paper, we devote to the study of the Bowen's entropy for fixed-point free flows and show that the Bowen entropy of the whole compact space is equal to the topological entropy. To obtain this result, we establish the Brin-Katok's local entropy formula for fixed-point free flows in ergodic case.

2019, 39(11): 6241-6260 doi: 10.3934/dcds.2019272 +[Abstract](546) +[HTML](85) +[PDF](455.71KB)
Abstract:

We study the structure of the escape orbits for a certain class of interval maps. This structure is encoded in the escape transition matrix \begin{document}$\widehat{A}_f$\end{document} of an interval map \begin{document}$f$\end{document}, extending the traditional matrix \begin{document}$A_f$\end{document} which considers the transition among the Markov subintervals. We show that the escape transition matrix is a topological conjugacy invariant. We then characterize the \begin{document}$0$\end{document}\begin{document}$1$\end{document} matrices that can be fabricated as escape transition matrices of Markov interval maps \begin{document}$f$\end{document} with escape sets. This shows the richness of this class of interval maps.

2019, 39(11): 6261-6276 doi: 10.3934/dcds.2019273 +[Abstract](283) +[HTML](105) +[PDF](370.02KB)
Abstract:

The material conservation of vorticity in fluid flows confined to a thin layer on the surface of a large rotating sphere, is a central result of geophysical fluid dynamics. In this paper we revisit the conservation of vorticity in the context of global scale flows on a rotating sphere. Starting from the vorticity equation instead of the Euler equation, we examine the kinematical and dynamical assumptions that are necessary to arrive at this result. We argue that, in contrast to the planar case, a two-dimensional velocity field does not lead to a single component vorticity equation on the sphere. The shallow fluid approximation is then used to argue that only one component of the vorticity equation is significant for global scale flows. Spherical coordinates are employed throughout, and no planar approximation is used.

2019, 39(11): 6277-6298 doi: 10.3934/dcds.2019274 +[Abstract](368) +[HTML](80) +[PDF](517.49KB)
Abstract:

We develop a Thermodynamic Formalism for bounded continuous potentials defined on the sequence space \begin{document}$X\equiv E^{\mathbb{N}}$\end{document}, where \begin{document}$E$\end{document} is a general standard Borel space. In particular, we introduce meaningful concepts of entropy and pressure for shifts acting on \begin{document}$X$\end{document} and obtain the existence of equilibrium states as finitely additive probability measures for any bounded continuous potential. Furthermore, we establish convexity and other structural properties of the set of equilibrium states, prove a version of the Perron-Frobenius-Ruelle theorem under additional assumptions on the regularity of the potential and show that the Yosida-Hewitt decomposition of these equilibrium states does not have a purely finite additive part.

We then apply our results to the construction of invariant measures of time-homogeneous Markov chains taking values on a general Borel standard space and obtain exponential asymptotic stability for a class of Markov operators. We also construct conformal measures for an infinite collection of interacting random paths which are associated to a potential depending on infinitely many coordinates. Under an additional differentiability hypothesis, we show how this process is related after a proper scaling limit to a certain infinite-dimensional diffusion.

2019, 39(11): 6299-6353 doi: 10.3934/dcds.2019275 +[Abstract](286) +[HTML](70) +[PDF](645.97KB)
Abstract:

We consider a mass-critical system of nonlinear Schrödinger equations

where \begin{document}$(u,v)$\end{document} is a \begin{document}$\mathbb{C}^2$\end{document}-valued unknown function and \begin{document}$\kappa >0$\end{document} is a constant. If \begin{document}$\kappa = 1/2$\end{document}, we say the equation satisfies mass-resonance condition. We are interested in the scattering problem of this equation under the condition \begin{document}$M(u,v)<M(\phi ,\psi)$\end{document}, where \begin{document}$M(u,v)$\end{document} denotes the mass and \begin{document}$(\phi ,\psi)$\end{document} is a ground state. In the mass-resonance case, we prove scattering by the argument of Dodson [5]. Scattering is also obtained without mass-resonance condition under the restriction that \begin{document}$(u,v)$\end{document} is radially symmetric.

2019, 39(11): 6355-6389 doi: 10.3934/dcds.2019276 +[Abstract](262) +[HTML](71) +[PDF](1660.16KB)
Abstract:

Global feedback stabilizability results are derived for nonautonomous coupled systems arising from the linearization around a given time-dependent trajectory of FitzHugh-CNagumo type systems. The feedback is explicit and is based on suitable oblique (nonorthogonal) projections in Hilbert spaces. The actuators are, typically, a finite number of indicator functions and act only in the parabolic equation. Subsequently, local feedback stabilizability to time-dependent trajectories results are derived for nonlinear coupled parabolic-ODE systems of the FitzHugh-CNagumo type. Simulations are presented showing the stabilizing performance of the feedback control.

2019, 39(11): 6391-6417 doi: 10.3934/dcds.2019277 +[Abstract](264) +[HTML](75) +[PDF](459.6KB)
Abstract:

We obtain spectral estimates for the iterations of Ruelle operators \begin{document}$L_{f + (a + {\bf i} b)\tau + (c + {\bf i} d) g}$\end{document} with two complex parameters and Hölder continuous functions \begin{document}$f,\: g$\end{document} generalizing the case \begin{document}${\rm{Pr}}(f) = 0$\end{document} studied in [9]. As an application we prove a sharp large deviation theorem concerning exponentially shrinking intervals which improves the result in [8].

2019, 39(11): 6419-6440 doi: 10.3934/dcds.2019278 +[Abstract](264) +[HTML](75) +[PDF](481.26KB)
Abstract:

For ${\mathbb{Z}}_2-$symmetric analytic deformation of the circle (with certain Fourier decaying rate), the necessary condition for the corresponding billiard map to keep the coexistence of period 2, 3 caustics is that the deformation has to be an isometric transformation.

2019, 39(11): 6441-6465 doi: 10.3934/dcds.2019279 +[Abstract](332) +[HTML](81) +[PDF](432.66KB)
Abstract:

In this paper, we provide a technical result on the existence of Gibbs $cu$-states for diffeomorphisms with dominated splittings. More precisely, for given $C^2$ diffeomorphim $f$ with dominated splitting $T_{\Lambda}M = E\oplus F$ on an attractor $\Lambda$, by considering some suitable random perturbation of $f$, we show that for any zero noise limit of ergodic stationary measures, if it has positive integrable Lyapunov exponents along invariant sub-bundle $E$, then its ergodic components contain Gibbs $cu$-states associated to $E$. With this technique, we show the existence of SRB measures and physical measures for some systems exhibiting dominated splittings and weak hyperbolicity.

2019, 39(11): 6467-6483 doi: 10.3934/dcds.2019280 +[Abstract](354) +[HTML](110) +[PDF](442.75KB)
Abstract:

In this paper, we focus on the exponential stability of stochastic differential equations driven by fractional Brownian motion (fBm) with Hurst parameter $H\in(1/2, 1)$. Based on the generalized Itô formula and representation of the fBm, some sufficient conditions for exponential stability of a class of SDEs with additive fractional noise are given. Besides, we present a criterion on the exponential stability for the fractional Ornstein-Uhlenbeck process with Markov switching. A numerical example is provided to illustrate our results.

2019, 39(11): 6485-6506 doi: 10.3934/dcds.2019281 +[Abstract](288) +[HTML](74) +[PDF](374.5KB)
Abstract:

In this paper, we prove the propagation of the Gevrey regularity of solutions to the three-dimensional incompressible ideal magnetohydrodynamics (MHD) equations. We also obtain an uniform estimate of Gevrey radius for the solution of MHD equation.

2019, 39(11): 6507-6522 doi: 10.3934/dcds.2019282 +[Abstract](256) +[HTML](69) +[PDF](395.68KB)
Abstract:

For a positive integer $M$ and a real base $q\in(1, M+1]$, let ${\mathcal{U}}_q$ denote the set of numbers having a unique expansion in base $q$ over the alphabet $\{0, 1, \dots, M\}$, and let $\mathbf{U}_q$ denote the corresponding set of sequences in $\{0, 1, \dots, M\}^ {\mathbb{N}}$. Komornik et al. [ Adv. Math. 305 (2017), 165–196] showed recently that the Hausdorff dimension of ${\mathcal{U}}_q$ is given by $h(\mathbf{U}_q)/\log q$, where $h(\mathbf{U}_q)$ denotes the topological entropy of $\mathbf{U}_q$. They furthermore showed that the function $H: q\mapsto h(\mathbf{U}_q)$ is continuous, nondecreasing and locally constant almost everywhere. The plateaus of $H$ were characterized by Alcaraz Barrera et al. [ Trans. Amer. Math. Soc., 371 (2019), 3209–3258]. In this article we reinterpret the results of Alcaraz Barrera et al. by introducing a notion of composition of fundamental words, and use this to obtain new information about the structure of the function $H$. This method furthermore leads to a more streamlined proof of their main theorem.

2019, 39(11): 6523-6539 doi: 10.3934/dcds.2019283 +[Abstract](347) +[HTML](67) +[PDF](390.17KB)
Abstract:

In this paper, we consider the following fractional Laplacian system with one critical exponent and one subcritical exponent

where \begin{document}$(-\Delta)^{s}$\end{document} is the fractional Laplacian, \begin{document}$0<s<1,\ N>2s, \ \lambda <\sqrt{\mu\nu },\ 1<p<2^{\ast}-1\; \text{and}\; \ 2^{\ast} = \frac{2N}{N-2s}$\end{document} is the Sobolev critical exponent. By using the Nehari\ manifold, we show that there exists a \begin{document}$\mu_{0}\in(0,1)$\end{document}, such that when \begin{document}$0<\mu\leq\mu_{0}$\end{document}, the system has a positive ground state solution. When \begin{document}$\mu>\mu_{0}$\end{document}, there exists a \begin{document}$\lambda_{\mu,\nu}\in[\sqrt{(\mu-\mu_{0})\nu},\sqrt{\mu\nu})$\end{document} such that if \begin{document}$\lambda>\lambda_{\mu,\nu}$\end{document}, the system has a positive ground state solution, if \begin{document}$\lambda<\lambda_{\mu,\nu}$\end{document}, the system has no ground state solution.

2019, 39(11): 6541-6564 doi: 10.3934/dcds.2019284 +[Abstract](316) +[HTML](66) +[PDF](440.39KB)
Abstract:

We consider \begin{document}$C^2$\end{document} vector fields in the three dimensional sphere with an attracting heteroclinic cycle between two periodic hyperbolic solutions with real Floquet multipliers. The proper basin of this attracting set exhibits historic behavior and from the asymptotic properties of its orbits we obtain a complete set of invariants under topological conjugacy in a neighborhood of the cycle. As expected, this set contains the periods of the orbits involved in the cycle, a combination of their angular speeds, the rates of expansion and contraction in linearizing neighborhoods of them, besides information regarding the transition maps and the transition times between these neighborhoods. We conclude with an application of this result to a class of cycles obtained by the lifting of an example of R. Bowen.

2019, 39(11): 6565-6583 doi: 10.3934/dcds.2019285 +[Abstract](313) +[HTML](70) +[PDF](343.44KB)
Abstract:

In this paper the geodesic flow on the 2-torus in a non-zero magnetic field is considered. Suppose that this flow admits an additional first integral \begin{document}$F$\end{document} on \begin{document}$N+2$\end{document} different energy levels which is polynomial in momenta of an arbitrary degree \begin{document}$N$\end{document} with analytic periodic coefficients. It is proved that in this case the magnetic field and the metric are functions of one variable and there exists a linear in momenta first integral on all energy levels.

2019, 39(11): 6585-6597 doi: 10.3934/dcds.2019286 +[Abstract](296) +[HTML](62) +[PDF](370.98KB)
Abstract:

We establish almost sure invariance principles (ASIP), a strong form of approximation by Brownian motion, for non-stationary time series arising as observations on sequential maps possessing an indifferent fixed point. These transformations are obtained by perturbing the slope in the Pomeau-Manneville map. Quenched ASIP for random compositions of these maps is also obtained.

2019, 39(11): 6599-6630 doi: 10.3934/dcds.2019287 +[Abstract](333) +[HTML](74) +[PDF](521.56KB)
Abstract:

In this paper, we study the following nonlinear Schrödinger equation

where \begin{document}$V*$\end{document} is the Fourier multiplier defined by \begin{document}$\widehat{(V* u})_n = V_{n}\widehat{u}_n, V_n\in[-1, 1]$\end{document} and \begin{document}$f(x)$\end{document} is Gevrey smooth. It is shown that for \begin{document}$0\leq|\epsilon|\ll1$\end{document}, there is some \begin{document}$(V_n)_{n\in\mathbb{Z}}$\end{document} such that, the equation (1) admits a time almost periodic solution (i.e., full dimensional KAM torus) in the Gevrey space. This extends results of Bourgain [7] and Cong-Liu-Shi-Yuan [8] to the case that the nonlinear perturbation depends explicitly on the space variable \begin{document}$x$\end{document}. The main difficulty here is the absence of zero momentum of the equation.

2019, 39(11): 6631-6642 doi: 10.3934/dcds.2019288 +[Abstract](329) +[HTML](73) +[PDF](331.76KB)
Abstract:

We introduce the notion of relative topological entropy dimension to classify the different intermediate levels of relative complexity for factor maps. By considering the dimension or ''density" of special class of sequences along which the entropy is encountered, we provide equivalent definitions of relative entropy dimension. As applications, we investigate the corresponding localization theory and obtain a disjointness theorem involving relative entropy dimension.

2019, 39(11): 6643-6668 doi: 10.3934/dcds.2019289 +[Abstract](299) +[HTML](58) +[PDF](429.98KB)
Abstract:

We consider the focusing nonlinear Schrödinger equation \begin{document}$i u_t + \Delta u + |u|^{p-1}u = 0$\end{document}, \begin{document}$p>1,$\end{document} and the generalized Hartree equation \begin{document}$iv_t + \Delta v + (|x|^{-(N-\gamma)}\ast |v|^p)|v|^{p-2}u = 0$\end{document}, \begin{document}$p\geq2$\end{document}, \begin{document}$\gamma<N$\end{document}, in the mass-supercritical and energy-subcritical setting. With the initial data \begin{document}$u_0\in H^1( \mathbb R^N)$\end{document} the characterization of solutions behavior under the mass-energy threshold is known for the NLS case from the works of Holmer and Roudenko in the radial [15] and Duyckaerts, Holmer and Roudenko in the nonradial setting [10] and further generalizations (see [1,11,13]); for the generalized Hartree case it is developed in [2]. In particular, scattering is proved following the road map developed by Kenig and Merle [16], using the concentration compactness and rigidity approach, which is now standard in the dispersive problems.

In this work we give an alternative proof of scattering for both NLS and gHartree equations in the radial setting in the inter-critical regime, following the approach of Dodson and Murphy [8] for the focusing 3d cubic NLS equation, which relies on the scattering criterion of Tao [26], combined with the radial Sobolev and Morawetz-type estimates. We first generalize it in the NLS case, and then extend it to the nonlocal Hartree-type potential. This method provides a simplified way to prove scattering, which may be useful in other contexts.

2019, 39(11): 6669-6682 doi: 10.3934/dcds.2019290 +[Abstract](273) +[HTML](66) +[PDF](311.67KB)
Abstract:

In this paper, the global smooth solution of Cauchy's problem of incompressible, resistive, viscous Hall-magnetohydrodynamics (Hall-MHD) is studied. By exploring the nonlinear structure of Hall-MHD equations, a class of large initial data is constructed, which can be arbitrarily large in \begin{document}$H^3(\mathbb{R}^3)$\end{document}. Our result may also be considered as the extension of work of Lei-Lin-Zhou [15] from the second-order semilinear equations to the second-order quasilinear equations, because the Hall term elevates the Hall-MHD system to the quasilinear level.

2019, 39(11): 6683-6712 doi: 10.3934/dcds.2019291 +[Abstract](260) +[HTML](66) +[PDF](426.09KB)
Abstract:

We investigate the system consisting of the the nonlinear Schrödinger equation coupled with Bopp-Podolsky electrodynamics in the Proca setting in the context of closed \begin{document}$3$\end{document}-dimensional manifolds. We prove existence of solutions up to the gauge, and compactness of the system both in the subcritical and in the critical case.

2019, 39(11): 6713-6745 doi: 10.3934/dcds.2019292 +[Abstract](349) +[HTML](88) +[PDF](476.43KB)
Abstract:

This paper concerns the Cauchy problem of the two-dimensional density-dependent Boussinesq equations on the whole space \begin{document}$\mathbb{R}^{2}$\end{document} with zero density at infinity. We prove that there exists a unique global strong solution provided the initial density and the initial temperature decay not too slow at infinity. In particular, the initial data can be arbitrarily large and the initial density may contain vacuum states and even have compact support. Moreover, there is no need to require any Cho-Choe-Kim type compatibility conditions. Our proof relies on the delicate weighted estimates and a lemma due to Coifman-Lions-Meyer-Semmes [J. Math. Pures Appl., 72 (1993), pp. 247-286].

2019, 39(11): 6747-6760 doi: 10.3934/dcds.2019293 +[Abstract](322) +[HTML](117) +[PDF](372.11KB)
Abstract:

We consider a jumping problem for nonlocal singular problems. We apply a recent variational approach for nonlocal singular problem, together with a minimax method in the framework of nonsmooth critical point theory.

2019, 39(11): 6761-6784 doi: 10.3934/dcds.2019294 +[Abstract](298) +[HTML](81) +[PDF](452.59KB)
Abstract:

We consider radial solutions of a general elliptic equation involving a weighted Laplace operator. We establish the uniqueness of the radial bound state solutions to

\begin{document}$n>2$\end{document}, where \begin{document}$\mathsf A$\end{document} and \begin{document}$\mathsf B$\end{document} are two positive, radial, smooth functions defined on \begin{document}$\mathbb R^n\setminus\{0\}$\end{document}. We assume that the nonlinearity \begin{document}$f\in C(-c, c)$\end{document}, \begin{document}$0<c\le\infty$\end{document} is an odd function satisfying some convexity and growth conditions, and has a zero at \begin{document}$b>0$\end{document}, is non positive and not identically 0 in \begin{document}$(0, b)$\end{document}, positive in \begin{document}$(b, c)$\end{document}, and is differentiable in \begin{document}$(0, c)$\end{document}.

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