# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

October 2018 , Volume 38 , Issue 10

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2018, 38(10): 4819-4835 doi: 10.3934/dcds.2018211 +[Abstract](1569) +[HTML](81) +[PDF](721.11KB)
Abstract:

TWe prove the existence of \begin{document}$T$\end{document}-periodic solutions for the second order non-linear equation

where the non-linear term \begin{document}$g$\end{document} has two singularities and the weight function \begin{document}$h$\end{document} changes sign. We find a relation between the degeneracy of the zeroes of the weight function and the order of one of the singularities of the non-linear term. The proof is based on the classical Leray-Schauder continuation theorem. Some applications to important mathematical models are presented.

2018, 38(10): 4837-4873 doi: 10.3934/dcds.2018212 +[Abstract](9972) +[HTML](86) +[PDF](583.68KB)
Abstract:

In this paper we consider a class of continuity equations that are conditioned to stay in general space-time domains, which is formulated as a continuum limit of interacting particle systems. Firstly, we study the well-posedness of the solutions and provide examples illustrating that the stability of solutions is strongly related to the decay of initial data at infinity. In the second part, we consider the vanishing viscosity approximation of the system, given with the co-normal boundary data. If the domain is spatially convex, the limit coincides with the solution of our original system, giving another interpretation to the equation.

2018, 38(10): 4875-4913 doi: 10.3934/dcds.2018213 +[Abstract](1475) +[HTML](118) +[PDF](582.92KB)
Abstract:

We study the emergent collective behaviors for an ensemble of identical Kuramoto oscillators under the effect of inertia. In the absence of inertial effects, it is well known that the generic initial Kuramoto ensemble relaxes to the phase-locked states asymptotically (emergence of complete synchronization) in a large coupling regime. Similarly, even for the presence of inertial effects, similar collective behaviors are observed numerically for generic initial configurations in a large coupling strength regime. However, this phenomenon has not been verified analytically in full generality yet, although there are several partial results in some restricted set of initial configurations. In this paper, we present several improved complete synchronization estimates for the Kuramoto ensemble with inertia in two frameworks for a finite system. Our improved frameworks describe the emergence of phase-locked states and its structure. Additionally, we show that as the number of oscillators tends to infinity, the Kuramoto ensemble with infinite size can be approximated by the corresponding kinetic mean-field model uniformly in time. Moreover, we also establish the global existence of measure-valued solutions for the Kuramoto equation and its large-time asymptotics.

2018, 38(10): 4915-4927 doi: 10.3934/dcds.2018214 +[Abstract](1276) +[HTML](89) +[PDF](395.1KB)
Abstract:

We prove that on a large family of metric measure spaces, if the \begin{document}$L^p$\end{document}-gradient estimate for heat flows holds for some \begin{document}$p>2$\end{document}, then the \begin{document}$L^1$\end{document}-gradient estimate also holds. This result extends Savaré's result on metric measure spaces, and provides a new proof to von Renesse-Sturm theorem on smooth metric measure spaces. As a consequence, we propose a new analysis object based on Gigli's measure-valued Ricci tensor, to characterize the Ricci curvature of RCD space in a local way. In the proof we adopt an iteration technique based on non-smooth Bakry-Émery theory, which is a new method to study the curvature dimension condition of metric measure spaces.

2018, 38(10): 4929-4950 doi: 10.3934/dcds.2018215 +[Abstract](1514) +[HTML](65) +[PDF](491.32KB)
Abstract:

We study the nonlinear Fokker-Planck equation on graphs, which is the gradient flow in the space of probability measures supported on the nodes with respect to the discrete Wasserstein metric. The energy functional driving the gradient flow consists of a Boltzmann entropy, a linear potential and a quadratic interaction energy. We show that the solution converges to the Gibbs measures exponentially fast. The continuous analog of this asymptotic rate is related to the Yano's formula.

2018, 38(10): 4951-4977 doi: 10.3934/dcds.2018216 +[Abstract](1264) +[HTML](106) +[PDF](570.45KB)
Abstract:

We consider in this work a system of two stochastic differential equations named the perturbed compositional gradient flow. By introducing a separation of fast and slow scales of the two equations, we show that the limit of the slow motion is given by an averaged ordinary differential equation. We then demonstrate that the deviation of the slow motion from the averaged equation, after proper rescaling, converges to a stochastic process with Gaussian inputs. This indicates that the slow motion can be approximated in the weak sense by a standard perturbed gradient flow or the continuous-time stochastic gradient descent algorithm that solves the optimization problem for a composition of two functions. As an application, the perturbed compositional gradient flow corresponds to the diffusion limit of the Stochastic Composite Gradient Descent (SCGD) algorithm for minimizing a composition of two expected-value functions in the optimization literatures. For the strongly convex case, such an analysis implies that the SCGD algorithm has the same convergence time asymptotic as the classical stochastic gradient descent algorithm. Thus it validates, at the level of continuous approximation, the effectiveness of using the SCGD algorithm in the strongly convex case.

2018, 38(10): 4979-4996 doi: 10.3934/dcds.2018217 +[Abstract](1126) +[HTML](52) +[PDF](432.33KB)
Abstract:

We prove the existence of solutions for a coupled system modeling the flow of a suspension of fluid and negatively buoyant non-colloidal particles in the thin film limit. The equations take the form of a fourth-order non-linear degenerate parabolic equation for the film height \begin{document} $h$ \end{document} coupled to a second-order degenerate parabolic equation for the particle density \begin{document} $ψ$ \end{document}. We prove the existence of physically relevant solutions, which satisfy the uniform bounds \begin{document} $0 ≤ ψ/h ≤ 1$ \end{document} and \begin{document} $h ≥ 0$ \end{document}.

2018, 38(10): 4997-5010 doi: 10.3934/dcds.2018218 +[Abstract](1217) +[HTML](61) +[PDF](413.93KB)
Abstract:

Consider \begin{document}$T(x) = d \, x$\end{document} (mod 1) acting on \begin{document}$S^1$\end{document}, a Lipschitz potential \begin{document}$A:S^1 \to \mathbb{R}$\end{document}, \begin{document}zhongwenzy<\lambda<1$\end{document} and the unique function \begin{document}$b_\lambda:S^1 \to \mathbb{R}$\end{document} satisfying \begin{document}$ b_\lambda(x) = \max_{T(y) = x} \{ \lambda \, b_\lambda(y) + A(y)\}. $\end{document} We will show that, when \begin{document}$\lambda \to 1$\end{document}, the function \begin{document}$b_\lambda- \frac{m(A)}{1-\lambda}$\end{document} converges uniformly to the calibrated subaction \begin{document}$V(x) = \max_{\mu \in \mathcal{ M}} \int S(y, x) \, d \mu(y)$\end{document}, where \begin{document}$S$\end{document} is the Mañe potential, \begin{document}$\mathcal{ M}$\end{document} is the set of invariant probabilities with support on the Aubry set and \begin{document}$m(A) = \sup_{\mu \in \mathcal{M}} \int A\, d\mu$\end{document}. For \begin{document}$\beta>0$\end{document} and \begin{document}$\lambda \in (0, 1)$\end{document}, there exists a unique fixed point \begin{document}$u_{\lambda, \beta} :S^1\to \mathbb{R}$\end{document} for the equation \begin{document}$e^{u_{\lambda, \beta}(x)} = \sum_{T(y) = x}e^{\beta A(y) +\lambda u_{\lambda, \beta}(y)}$\end{document}. It is known that as \begin{document}$\lambda \to 1$\end{document} the family \begin{document}$e^{[u_{\lambda, \beta}- \sup u_{\lambda, \beta}]}$\end{document} converges uniformly to the main eigenfuntion \begin{document}$\phi_\beta $\end{document} for the Ruelle operator associated to \begin{document}$\beta A$\end{document}. We consider \begin{document}$\lambda = \lambda(\beta)$\end{document}, \begin{document}$\beta(1-\lambda(\beta))\to+\infty$\end{document} and \begin{document}$\lambda(\beta) \to 1$\end{document}, as \begin{document}$\beta \to\infty$\end{document}. Under these hypotheses we will show that \begin{document}$\frac{1}{\beta}(u_{\lambda, \beta}-\frac{P(\beta A)}{1-\lambda})$\end{document} converges uniformly to the above \begin{document}$V$\end{document}, as \begin{document}$\beta\to \infty$\end{document}. The parameter \begin{document}$\beta$\end{document} represents the inverse of temperature in Statistical Mechanics and \begin{document}$\beta \to \infty$\end{document} means that we are considering that the temperature goes to zero. Under these conditions we get selection of subaction when \begin{document}$\beta \to \infty$\end{document}. 2018, 38(10): 5011-5019 doi: 10.3934/dcds.2018219 +[Abstract](1219) +[HTML](73) +[PDF](436.15KB) Abstract: Let \begin{document}$f$\end{document} be a \begin{document}$C^1$\end{document} diffeomorphism on a compact Riemannian manifold without boundary and \begin{document}$\mu$\end{document} an ergodic \begin{document}$f$\end{document}-invariant measure whose Oseledets splitting admits domination. We give a hybrid estimate from above for the metric entropy of \begin{document}$\mu$\end{document} in terms of Lyapunov exponents and volume growth. Furthermore, for any \begin{document}$C^1$\end{document} diffeomorphism away from tangencies, its topological entropy is bounded by the volume growth. 2018, 38(10): 5021-5037 doi: 10.3934/dcds.2018220 +[Abstract](1239) +[HTML](66) +[PDF](837.19KB) Abstract: We consider a space \begin{document}$\mathcal{U}$\end{document} of 3-dimensional diffeomorphisms \begin{document}$f$\end{document} with hyperbolic fixed points \begin{document}$p$\end{document} the stable and unstable manifolds of which have quadratic tangencies and satisfying some open conditions and such that \begin{document}$Df(p)$\end{document} has non-real expanding eigenvalues and a real contracting eigenvalue. The aim of this paper is to study moduli of diffeomorphisms in \begin{document}$\mathcal{U}$\end{document}. We show that, for a generic element \begin{document}$f$\end{document} of \begin{document}$\mathcal{U}$\end{document}, all the eigenvalues of \begin{document}$Df(p)$\end{document} are moduli and the restriction of a conjugacy homeomorphism to a local unstable manifold is a uniquely determined linear conformal map. 2018, 38(10): 5039-5066 doi: 10.3934/dcds.2018221 +[Abstract](1246) +[HTML](77) +[PDF](553.47KB) Abstract: We study the nonlinear Schrödinger equation (NLS) on a star graph \begin{document}$\mathcal{G}$\end{document}. At the vertex an interaction occurs described by a boundary condition of delta type with strength \begin{document}$\alpha\in \mathbb{R}$\end{document}. We investigate the orbital instability of the standing waves \begin{document}$e^{i\omega t}{\bf \Phi}(x)$\end{document} of the NLS-\begin{document}$\delta$\end{document} equation with attractive power nonlinearity on \begin{document}$\mathcal{G}$\end{document} when the profile \begin{document}${\bf \Phi}(x)$\end{document} has mixed structure (i.e. has bumps and tails). In our approach we essentially use the extension theory of symmetric operators by Krein - von Neumann, and the analytic perturbations theory, avoiding the variational techniques standard in the stability study. We also prove the orbital stability of the unique standing wave solution to the NLS-\begin{document}$\delta$\end{document} equation with repulsive nonlinearity. 2018, 38(10): 5067-5083 doi: 10.3934/dcds.2018222 +[Abstract](1204) +[HTML](104) +[PDF](448.08KB) Abstract: This paper deals with a parabolic-elliptic chemotaxis system with nonlocal type of source in the whole space. It's proved that the initial value problem possesses a unique global solution which is uniformly bounded. Here we identify the exponents regimes of nonlinear reaction and aggregation in such a way that their scaling and the diffusion term coincide (see Introduction). Comparing to the classical KS model (without the source term), it's shown that how energy estimates give natural conditions on the nonlinearities implying the absence of blow-up for the solution without any restriction on the initial data. 2018, 38(10): 5085-5103 doi: 10.3934/dcds.2018223 +[Abstract](1367) +[HTML](72) +[PDF](374.54KB) Abstract: We consider the correlation functions of binary pattern sequences of degree 3 as well as those with general degrees and special patterns and obtain necessary and sufficient conditions to be noncorrelated. We also obtain the correlation dimensions for those with degree 2. 2018, 38(10): 5105-5118 doi: 10.3934/dcds.2018224 +[Abstract](1205) +[HTML](56) +[PDF](394.98KB) Abstract: We consider linear cocycles over non-uniformly hyperbolic dynamical systems. The base system is a diffeomorphism \begin{document}$f$\end{document} of a compact manifold \begin{document}$X$\end{document} preserving a hyperbolic ergodic probability measure \begin{document}$μ$\end{document}. The cocycle \begin{document}$\mathcal{A}$\end{document} over \begin{document}$f$\end{document} is Hölder continuous and takes values in \begin{document}$GL(d, \mathbb{R})$\end{document} or, more generally, in the group of invertible bounded linear operators on a Banach space. For a \begin{document}$GL(d, \mathbb{R})$\end{document}-valued cocycle \begin{document}$\mathcal{A}$\end{document} we prove that the Lyapunov exponents of \begin{document}$\mathcal{A}$\end{document} with respect to \begin{document}$μ$\end{document} can be approximated by the Lyapunov exponents of \begin{document}$\mathcal{A}$\end{document} with respect to measures on hyperbolic periodic orbits of \begin{document}$f$\end{document}. In the infinite-dimensional setting one can define the upper and lower Lyapunov exponents of \begin{document}$\mathcal{A}$\end{document} with respect to \begin{document}$μ$\end{document}, but they cannot always be approximated by the exponents of \begin{document}$\mathcal{A}$\end{document} on periodic orbits. We prove that they can be approximated in terms of the norms of the return values of \begin{document}$\mathcal{A}$\end{document} on hyperbolic periodic orbits of \begin{document}$f$\end{document}. 2018, 38(10): 5119-5128 doi: 10.3934/dcds.2018225 +[Abstract](1343) +[HTML](87) +[PDF](449.03KB) Abstract: We study chaotic properties of uniformly convergent nonautonomous dynamical systems. We show that, contrary to the autonomous systems on the compact interval, positivity of topological sequence entropy and occurrence of Li-Yorke chaos are not equivalent, more precisely, neither of the two possible implications is true. 2018, 38(10): 5129-5144 doi: 10.3934/dcds.2018226 +[Abstract](1537) +[HTML](63) +[PDF](403.93KB) Abstract: In this paper, we construct the Brin-Katok formula of conditional entropy for invariant measures of continuous maps on a compact metric space by replacing the Bowen metrics with the corresponding mean metrics. Additionally, this paper is also devoted to establishing the Katok's entropy formula of conditional entropy for ergodic measures in the case of mean metrics. 2018, 38(10): 5145-5161 doi: 10.3934/dcds.2018227 +[Abstract](1183) +[HTML](71) +[PDF](535.66KB) Abstract: Let \begin{document}$α ∈ \mathbb{N}$\end{document}, \begin{document}$α ≥ 1$\end{document} and \begin{document}$(-Δ)^{α} = -Δ((-Δ)^{α-1})$\end{document} be the polyharmonic operator. We prove existence and non-existence results for the following Hamiltonian systems of polyharmonic equations under Dirichlet boundary conditions where \begin{document}$Ω$\end{document} is a sufficiently smooth bounded domain, \begin{document}$N >2α$\end{document}, \begin{document}$ν$\end{document} is the outward pointing normal to \begin{document}$\partial Ω$\end{document} and the Hamiltonian \begin{document}$H ∈ C^1 (\mathbb{R}^2; \mathbb{R})$\end{document} satisfies suitable growth assumptions. 2018, 38(10): 5163-5188 doi: 10.3934/dcds.2018228 +[Abstract](1267) +[HTML](55) +[PDF](719.92KB) Abstract: We consider \begin{document}$ \mathit{Sp}\left( 2\mathit{d},\mathbb{R} \right)$\end{document} cocycles over two classes of partially hyperbolic diffeomorphisms: having compact center leaves and time one maps of Anosov flows. We prove that the Lyapunov exponents are non-zero in an open and dense set in the Hölder topology. 2018, 38(10): 5189-5204 doi: 10.3934/dcds.2018229 +[Abstract](1043) +[HTML](56) +[PDF](376.37KB) Abstract: Let \begin{document}$ R_α$\end{document} be an irrational rotation of the circle, and code the orbit of any point \begin{document}$ x$\end{document} by whether \begin{document}$ R_α^i(x) $\end{document} belongs to \begin{document}$ [0,α)$\end{document} or \begin{document}$ [α, 1)$\end{document} - this produces a Sturmian sequence. A point is undetermined at step \begin{document}$ j$\end{document} if its coding up to time \begin{document}$ j$\end{document} does not determine its coding at time \begin{document}$ j+1$\end{document}. We prove a pair of results on the asymptotic frequency of a point being undetermined, for full measure sets of \begin{document}$ α$\end{document} and \begin{document}$ x$\end{document}. 2018, 38(10): 5205-5220 doi: 10.3934/dcds.2018230 +[Abstract](1536) +[HTML](68) +[PDF](428.57KB) Abstract: It was showed that the generalized Camassa-Holm equation possible development of singularities in finite time, and beyond the occurrence of wave breaking which exists either global conservative or dissipative solutions. In present paper, we will further investigate the uniqueness of global conservative solutions to it based on the characteristics. From a given conservative solution \begin{document}$u = u(t,x)$\end{document}, an equation is introduced to single out a unique characteristic curve through each initial point. By analyzing the evolution of the quantities \begin{document}$u$\end{document} and \begin{document}$v = 2 \arctan u_x$\end{document} along each characteristic, it is obtained that the Cauchy problem with general initial data \begin{document}$u_0∈ H^1(\mathbb{R})$\end{document} has a unique global conservative solution. 2018, 38(10): 5221-5243 doi: 10.3934/dcds.2018231 +[Abstract](1348) +[HTML](61) +[PDF](525.05KB) Abstract: We study whether the solutions of a parabolic equation with diffusion given by the fractional Laplacian and a dominating gradient term satisfy Dirichlet boundary data in the classical sense or in the generalized sense of viscosity solutions. The Dirichlet problem is well posed globally in time when boundary data is assumed to be satisfied in the latter sense. Thus, our main results are a) the existence of solutions which satisfy the boundary data in the classical sense for a small time, for all Hölder-continuous initial data, with Hölder exponent above a critical a value, and b) the nonexistence of solutions satisfying the boundary data in the classical sense for all time. In this case, the phenomenon of loss of boundary conditions occurs in finite time, depending on a largeness condition on the initial data. 2018, 38(10): 5245-5260 doi: 10.3934/dcds.2018232 +[Abstract](1138) +[HTML](64) +[PDF](461.57KB) Abstract: Collision avoidance is an interesting feature of the Cucker-Smale (CS) model of flocking that has been studied in many works, e.g. [2,1,4,6,7,20,21,22]. In particular, in the case of singular interactions between agents, as is the case of the CS model with communication weights of the type \begin{document}$ψ(s) = s^{-α}$\end{document} for \begin{document}$α ≥ 1$\end{document}, it is important for showing global well-posedness of the underlying particle dynamics. In [4], a proof of the non-collision property for singular interactions is given in the case of the linear CS model, i.e. when the velocity coupling between agents \begin{document}$i,j$\end{document} is \begin{document}$v_{j}-v_{i}$\end{document}. This paper can be seen as an extension of the analysis in [4]. We show that particles avoid collisions even when the linear coupling in the CS system has been substituted with the nonlinear term \begin{document}$Γ(·)$\end{document} introduced in [12] (typical examples being \begin{document}$Γ(v) = v|v|^{2(γ -1)}$\end{document} for \begin{document}$γ ∈ (\frac{1}{2},\frac{3}{2})$\end{document}), and prove that no collisions can happen in finite time when \begin{document}$α ≥ 1$\end{document}. We also show uniform estimates for the minimum inter-particle distance, for a communication weight with expanded singularity \begin{document}$ψ_{δ}(s) = (s-δ)^{-α}$\end{document}, when \begin{document}$α ≥ 2γ$\end{document}, \begin{document}$δ ≥ 0$\end{document}. 2018, 38(10): 5261-5295 doi: 10.3934/dcds.2018233 +[Abstract](1484) +[HTML](104) +[PDF](603.35KB) Abstract: We consider the Cauchy problem of the \begin{document}$N$\end{document}-dimensional incompressible viscoelastic fluids with \begin{document}$N≥2$\end{document}. It is shown that, in the low frequency part, this system possesses some dispersive properties derived from the one parameter group \begin{document}$e^{± it\Lambda}$\end{document}. Based on this dispersive effect, we construct global solutions with large initial velocity concentrating on the low frequency part. Such kind of solution has never been seen before in the literature even for the classical incompressible Navier-Stokes equations. 2018, 38(10): 5297-5337 doi: 10.3934/dcds.2018234 +[Abstract](1230) +[HTML](66) +[PDF](634.12KB) Abstract: This paper deals with the initial value problem for a class of degenerate nonlinear parabolic equations on a bounded domain in \begin{document}$ \mathbb{R}^N$\end{document} for \begin{document}$ N≥2$\end{document} with the Dirichlet boundary condition. The assumptions ensure that \begin{document}$ u\equiv0\$\end{document} is a stationary solution and its stability is analysed. Amongst other things the results show that, in the case of critical degeneracy, the principle of linearized stability fails for some simple smooth nonlinearities. It is also shown that for levels of degeneracy less than the critical one linearized stability is justified for a broad class of nonlinearities including those for which it fails in the critical case.

2018, 38(10): 5339-5349 doi: 10.3934/dcds.2018235 +[Abstract](1590) +[HTML](85) +[PDF](354.15KB)
Abstract:

Chen, Li and Li [Adv. Math., 308(2017), pp. 404-437] developed a direct method of moving planes for the fractional Laplacian. In this paper, we extend their method to the logarithmic Laplacian. We consider both the logarithmic equation and the system. To carry out the method, we establish two kinds of narrow region principle for the equation and the system separately. Then using these narrow region principles, we give the radial symmetry results for the solutions to semi-linear logarithmic Laplacian equations and systems on the ball.

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