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Discrete & Continuous Dynamical Systems - A

October 2020 , Volume 40 , Issue 10

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Existence and uniqueness of very weak solution of the MHD type system
Yong Zeng
2020, 40(10): 5617-5638 doi: 10.3934/dcds.2020240 +[Abstract](172) +[HTML](58) +[PDF](407.58KB)

This paper studies the very weak solution to the steady MHD type system in a bounded domain. We prove the existence of very weak solutions to the MHD type system for arbitrary large external forces \begin{document}$ ({\bf f},{\bf{g}}) $\end{document} in \begin{document}$ L^r({\Omega})\times [X_{\theta',q'}({\Omega})]' $\end{document} and suitable boundary data \begin{document}$ ({\mathcal B}^0,{\mathcal U}^0) $\end{document} in \begin{document}$ W^{-1/p,p}({\partial}{\Omega})\times W^{-1/q,q}({\partial}{\Omega}) $\end{document}, under certain assumptions on \begin{document}$ p,q,r,\theta $\end{document}. The uniqueness of very weak solution for small data \begin{document}$ ({\bf f},{\bf{g}},{\mathcal B}^0,{\mathcal U}^0) $\end{document} is also studied.

Review of local and global existence results for stochastic pdes with Lévy noise
Justin Cyr, Phuong Nguyen, Sisi Tang and Roger Temam
2020, 40(10): 5639-5710 doi: 10.3934/dcds.2020241 +[Abstract](61) +[HTML](22) +[PDF](922.2KB)

This article is a review of Lévy processes, stochastic integration and existence results for stochastic differential equations and stochastic partial differential equations driven by Lévy noise. An abstract PDE of the typical type encountered in fluid mechanics is considered in a stochastic setting driven by a general Lévy noise. Existence and uniqueness of a local pathwise solution is established as a demonstration of general techniques in the area.

On the Yudovich's type solutions for the 2D Boussinesq system with thermal diffusivity
Marius Paicu and Ning Zhu
2020, 40(10): 5711-5728 doi: 10.3934/dcds.2020242 +[Abstract](69) +[HTML](16) +[PDF](412.68KB)

The goal of this paper is to study the two-dimensional inviscid Boussinesq equations with temperature-dependent thermal diffusivity. Firstly we establish the global existence theory and regularity estimates for this system with Yudovich's type initial data. Then we investigate the vortex patch problem, and proving that the patch remains in Hölder class \begin{document}$ C^{1+s}\; (0<s<1) $\end{document} for all the time.

A gradient flow approach of propagation of chaos
Samir Salem
2020, 40(10): 5729-5754 doi: 10.3934/dcds.2020243 +[Abstract](72) +[HTML](20) +[PDF](517.03KB)

We provide an estimation of the dissipation of the Wasserstein 2 distance between the law of some interacting \begin{document}$ N $\end{document}-particle system, and the \begin{document}$ N $\end{document} times tensorized product of the solution to the corresponding limit nonlinear conservation law. It then enables to recover classical propagation of chaos results [20] in the case of Lipschitz coefficients, uniform in time propagation of chaos in [17] in the case of strictly convex coefficients. And some recent results [7] as the case of particle in a double well potential.

A simple proof of the Adams type inequalities in $ {\mathbb R}^{2m} $
Van Hoang Nguyen
2020, 40(10): 5755-5764 doi: 10.3934/dcds.2020244 +[Abstract](52) +[HTML](20) +[PDF](319.65KB)

We provide the simple proof of the Adams type inequalities in whole space \begin{document}$ {\mathbb R}^{2m} $\end{document}. The main tools are the Fourier rearrangement technique introduced by Lenzmann and Sok [16], a Hardy–Rellich type inequality for radial functions, and the sharp Moser–Trudinger type inequalities in \begin{document}$ {\mathbb R}^2 $\end{document}.

On the effects of firing memory in the dynamics of conjunctive networks
Eric Goles, Pedro Montealegre and Martín Ríos-Wilson
2020, 40(10): 5765-5793 doi: 10.3934/dcds.2020245 +[Abstract](82) +[HTML](26) +[PDF](697.12KB)

A boolean network is a map \begin{document}$ F:\{0,1\}^n \to \{0,1\}^n $\end{document} that defines a discrete dynamical system by the subsequent iterations of \begin{document}$ F $\end{document}. Nevertheless, it is thought that this definition is not always reliable in the context of applications, especially in biology. Concerning this issue, models based in the concept of adding asynchronicity to the dynamics were propose. Particularly, we are interested in a approach based in the concept of delay. We focus in a specific type of delay called firing memory and it effects in the dynamics of symmetric (non-directed) conjunctive networks. We find, in the caseis in which the implementation of the delay is not uniform, that all the complexity of the dynamics is somehow encapsulated in the component in which the delay has effect. Thus, we show, in the homogeneous case, that it is possible to exhibit attractors of non-polynomial period. In addition, we study the prediction problem consisting in, given an initial condition, determinate if a fixed coordinate will eventually change its state. We find again that in the non-homogeneous case all the complexity is determined by the component that is affected by the delay and we conclude in the homogeneous case that this problem is PSPACE-complete.

Entropies of commuting transformations on Hilbert spaces
Zhiming Li and Yujun Zhu
2020, 40(10): 5795-5814 doi: 10.3934/dcds.2020246 +[Abstract](67) +[HTML](19) +[PDF](427.19KB)

By establishing Multiplicative Ergodic Theorem for commutative transformations on a separable infinite dimensional Hilbert space, in this paper, we investigate Pesin's entropy formula and SRB measures of a finitely generated random transformations on such space via its commuting generators. Moreover, as an application, we give a formula of Friedland's entropy for certain \begin{document}$ C^{2} $\end{document} \begin{document}$ \mathbb{N}^2 $\end{document}-actions.

Evolution of dispersal in advective homogeneous environments
Li Ma and De Tang
2020, 40(10): 5815-5830 doi: 10.3934/dcds.2020247 +[Abstract](66) +[HTML](18) +[PDF](316.0KB)

The effects of weak and strong advection on the dynamics of reaction-diffusion models have long been investigated. In contrast, the role of intermediate advection still remains poorly understood. This paper is devoted to studying a two-species competition model in a one-dimensional advective homogeneous environment, where the two species are identical except their diffusion rates and advection rates. Zhou (P. Zhou, On a Lotka-Volterra competition system: diffusion vs advection, Calc. Var. Partial Differential Equations, 55 (2016), Art. 137, 29 pp) considered the system under the no-flux boundary conditions. It is pointed that, in this paper, we focus on the case where the upstream end has the Neumann boundary condition and the downstream end has the hostile condition. By employing a new approach, we firstly determine necessary and sufficient conditions for the persistence of the corresponding single species model, in forms of the critical diffusion rate and critical advection rate. Furthermore, for the two-species model, we find that (i) the strategy of slower diffusion together with faster advection is always favorable; (ii) two species will also coexist when the faster advection with appropriate faster diffusion.

Existence of positive solutions of Schrödinger equations with vanishing potentials
Eduard Toon and Pedro Ubilla
2020, 40(10): 5831-5843 doi: 10.3934/dcds.2020248 +[Abstract](81) +[HTML](12) +[PDF](330.67KB)

We prove the existence of at least one positive solution for a Schrödinger equation in \begin{document}$ \mathbb{R}^N $\end{document} of type

with a vanishing potential at infinity and subcritical nonlinearity \begin{document}$ f $\end{document}. Our hypotheses allow us to consider examples of nonlinearities which do not verify the Ambrosetti-Rabinowitz condition, neither monotonicity conditions for the function \begin{document}$ \frac{f(x, s)}{s} $\end{document}. Our argument requires new estimates in order to prove the boundedness of a Cerami sequence.

Analysis of a spatial memory model with nonlocal maturation delay and hostile boundary condition
Qi An, Chuncheng Wang and Hao Wang
2020, 40(10): 5845-5868 doi: 10.3934/dcds.2020249 +[Abstract](90) +[HTML](23) +[PDF](4683.93KB)

In this paper, we propose and investigate a memory-based reaction-diffusion equation with nonlocal maturation delay and homogeneous Dirichlet boundary condition. We first study the existence of the spatially inhomogeneous steady state. By analyzing the associated characteristic equation, we obtain sufficient conditions for local stability and Hopf bifurcation of this inhomogeneous steady state, respectively. For the Hopf bifurcation analysis, a geometric method and prior estimation techniques are combined to find all bifurcation values because the characteristic equation includes a non-self-adjoint operator and two time delays. In addition, we provide an explicit formula to determine the crossing direction of the purely imaginary eigenvalues. The bifurcation analysis reveals that the diffusion with memory effect could induce spatiotemporal patterns which were never possessed by an equation without memory-based diffusion. Furthermore, these patterns are different from the ones of a spatial memory equation with Neumann boundary condition.

Extended symmetry groups of multidimensional subshifts with hierarchical structure
Álvaro Bustos
2020, 40(10): 5869-5895 doi: 10.3934/dcds.2020250 +[Abstract](106) +[HTML](24) +[PDF](925.6KB)

The centralizer (automorphism group) and normalizer (extended symmetry group) of the shift action inside the group of self-homeomorphisms are studied, in the context of certain \begin{document}$ \mathbb{Z}^d $\end{document} subshifts with a hierarchical supertile structure, such as bijective substitutive subshifts and the Robinson tiling. Restrictions on these groups via geometrical considerations are used to characterize explicitly their structure: nontrivial extended symmetries can always be described via relabeling maps and rigid transformations of the Euclidean plane permuting the coordinate axes. The techniques used also carry over to the well-known Robinson tiling, both in its minimal and non-minimal versions.

Continuous orbit equivalence of topological Markov shifts and KMS states on Cuntz–Krieger algebras
Kengo Matsumoto
2020, 40(10): 5897-5909 doi: 10.3934/dcds.2020251 +[Abstract](82) +[HTML](11) +[PDF](360.25KB)

We study KMS states for gauge actions with potential functions on Cuntz–Krieger algebras whose underlying one-sided topological Markov shifts are continuously orbit equivalent. As a result, we have a certain relationship between topological entropy of continuously orbit equivalent one-sided topological Markov shifts.

A degenerate KAM theorem for partial differential equations with periodic boundary conditions
Meina Gao and Jianjun Liu
2020, 40(10): 5911-5928 doi: 10.3934/dcds.2020252 +[Abstract](48) +[HTML](15) +[PDF](388.36KB)

In this paper, an infinite dimensional KAM theorem with double normal frequencies is established under qualitative non-degenerate conditions. This is an extension of the degenerate KAM theorem with simple normal frequencies in [3] by Bambusi, Berti and Magistrelli. As applications, for nonlinear wave equation and nonlinear Schr\begin{document}$ \ddot{\mbox{o}} $\end{document}dinger equation with periodic boundary conditions, quasi-periodic solutions of small amplitude and quasi-periodic solutions around plane wave are obtained respectively.

The two-component $ \mu $-Camassa–Holm system with peaked solutions
Yingying Li, Ying Fu and Changzheng Qu
2020, 40(10): 5929-5954 doi: 10.3934/dcds.2020253 +[Abstract](82) +[HTML](21) +[PDF](598.66KB)

This paper is mainly concerned with the classification of the general two-component \begin{document}$ \mu $\end{document}-Camassa-Holm systems with quadratic nonlinearities. As a conclusion of such classification, a two-component \begin{document}$ \mu $\end{document}-Camassa-Holm system admitting multi-peaked solutions and \begin{document}$ H^1 $\end{document}-norm conservation law is found, which is a \begin{document}$ \mu $\end{document}-version of the two-component modified Camassa-Holm system and can be derived from the semidirect-product Euler-Poincaré equations corresponding to a Lagrangian. The local well-posedness for solutions to the initial value problem associated with the two-component \begin{document}$ \mu $\end{document}-Camassa-Holm system is established. And the precise blow-up scenario, wave breaking phenomena and blow-up rate for solutions of this problem are also investigated.

Gradient regularity for a singular parabolic equation in non-divergence form
Amal Attouchi and Eero Ruosteenoja
2020, 40(10): 5955-5972 doi: 10.3934/dcds.2020254 +[Abstract](80) +[HTML](11) +[PDF](368.8KB)

In this paper we consider viscosity solutions of a class of non-homogeneous singular parabolic equations

where \begin{document}$ -1<\gamma<0 $\end{document}, \begin{document}$ 1<p<\infty $\end{document}, and \begin{document}$ f $\end{document} is a given bounded function. We establish interior Hölder regularity of the gradient by studying two alternatives: The first alternative uses an iteration which is based on an approximation lemma. In the second alternative we use a small perturbation argument.

Filtering the $ L^2- $critical focusing Schrödinger equation
Ruoci Sun
2020, 40(10): 5973-5990 doi: 10.3934/dcds.2020255 +[Abstract](96) +[HTML](23) +[PDF](419.24KB)

We study the influence of Szegő projector on the \begin{document}$ L^2- $\end{document}critical non linear focusing Schrödinger equation, leading to the quintic focusing NLS–Szegő equation on the line

It has no Galilean invariance but the momentum \begin{document}$ P(u) = \langle -i\partial_x u, u\rangle_{L^2} $\end{document} becomes the \begin{document}$ \dot{H}^{\frac{1}{2}}- $\end{document}norm. Thus this equation is globally well-posed in \begin{document}$ H^1_+ = \Pi(H^1(\mathbb{R})) $\end{document}, for every initial datum \begin{document}$ u_0 $\end{document}. The solution \begin{document}$ L^2- $\end{document}scatters both forward and backward in time if \begin{document}$ u_0 $\end{document} has sufficiently small mass. By using the concentration–compactness principle, we prove the orbital stability of some weak type of the traveling wave : \begin{document}$ u_{\omega, c}(t, x) = e^{i\omega t}Q(x+ct) $\end{document}, for some \begin{document}$ \omega, c>0 $\end{document}, where \begin{document}$ Q $\end{document} is a ground state associated to Gagliardo–Nirenberg type functional

for some \begin{document}$ \gamma\geq 0 $\end{document}. Its Euler–Lagrange equation is a non local elliptic equation. The ground states are completely classified in the case \begin{document}$ \gamma = 2 $\end{document}, leading to the actual orbital stability for appropriate traveling waves. As a consequence, the scattering mass threshold of the focusing quintic NLS–Szegő equation is strictly below the mass of ground state associated to the functional \begin{document}$ I^{(0)} $\end{document}, unlike the recent result by Dodson [8] on the usual quintic focusing non linear Schrödinger equation.

Asymptotic behavior of global solutions to a class of heat equations with gradient nonlinearity
Caihong Chang, Qiangchang Ju and Zhengce Zhang
2020, 40(10): 5991-6014 doi: 10.3934/dcds.2020256 +[Abstract](190) +[HTML](27) +[PDF](423.8KB)

The paper is devoted to investigating a semilinear parabolic equation with a nonlinear gradient source term:

where \begin{document}$ p>m+2 $\end{document}, \begin{document}$ m\geq0 $\end{document}. Zhang and Hu [Discrete Contin. Dyn. Syst. 26 (2010) 767-779] showed that finite time gradient blowup occurs at the boundary and the accurate blowup rate is also obtained for super-critical boundary value. Throughout this paper, we present a complete large time behavior of a classical solution \begin{document}$ u $\end{document}: \begin{document}$ u $\end{document} is global and converges to the unique stationary solution in \begin{document}$ C^1 $\end{document} norm for subcritical boundary value, and \begin{document}$ u_x $\end{document} blows up in infinite time for critical boundary value. Gradient growup rate is also established by the method of matched asymptotic expansions. In addition, gradient estimate of solutions is obtained by the Bernstein-type arguments.

Nonuniformly hyperbolic systems arising from coupling of chaotic and gradient-like systems
Matteo Tanzi and Lai-Sang Young
2020, 40(10): 6015-6041 doi: 10.3934/dcds.2020257 +[Abstract](67) +[HTML](16) +[PDF](913.98KB)

We investigate dynamical systems obtained by coupling two maps, one of which is chaotic and is exemplified by an Anosov diffeomorphism, and the other is of gradient type and is exemplified by a N-pole-to-S-pole map of the circle. Leveraging techniques from the geometric and ergodic theories of hyperbolic systems, we analyze three different ways of coupling together the two maps above. For weak coupling, we offer an addendum to existing theory showing that almost always the attractor has fractal-like geometry when it is not normally hyperbolic. Our main results are for stronger couplings in which the action of the Anosov diffeomorphism on the circle map has certain monotonicity properties. Under these conditions, we show that the coupled systems have invariant cones and possess SRB measures even though there are genuine obstructions to uniform hyperbolicity.

No-shadowing for singular hyperbolic sets with a singularity
Xiao Wen and Lan Wen
2020, 40(10): 6043-6059 doi: 10.3934/dcds.2020258 +[Abstract](71) +[HTML](13) +[PDF](341.76KB)

We prove that every singular hyperbolic chain transitive set with a singularity does not admit the shadowing property. Using this result we show that if a star flow has the shadowing property on its chain recurrent set then it satisfies Axiom A and the no-cycle conditions; and that if a multisingular hyperbolic set has the shadowing property then it is hyperbolic.

Integrability of moduli and regularity of denjoy counterexamples
Sang-hyun Kim and Thomas Koberda
2020, 40(10): 6061-6088 doi: 10.3934/dcds.2020259 +[Abstract](75) +[HTML](19) +[PDF](449.9KB)

We study the regularity of exceptional actions of groups by \begin{document}$ C^{1, \alpha} $\end{document} diffeomorphisms on the circle, i.e. ones which admit exceptional minimal sets, and whose elements have first derivatives that are continuous with concave modulus of continuity \begin{document}$ \alpha $\end{document}. Let \begin{document}$ G $\end{document} be a finitely generated group admitting a \begin{document}$ C^{1, \alpha} $\end{document} action \begin{document}$ \rho $\end{document} with a free orbit on the circle, and such that the logarithms of derivatives of group elements are uniformly bounded at some point of the circle. We prove that if \begin{document}$ G $\end{document} has spherical growth bounded by \begin{document}$ c n^{d-1} $\end{document} and if the function \begin{document}$ 1/\alpha^d $\end{document} is integrable near zero, then under some mild technical assumptions on \begin{document}$ \alpha $\end{document}, there is a sequence of exceptional \begin{document}$ C^{1, \alpha} $\end{document} actions of \begin{document}$ G $\end{document} which converge to \begin{document}$ \rho $\end{document} in the \begin{document}$ C^1 $\end{document} topology. As a consequence for a single diffeomorphism, we obtain that if the function \begin{document}$ 1/\alpha $\end{document} is integrable near zero, then there exists a \begin{document}$ C^{1, \alpha} $\end{document} exceptional diffeomorphism of the circle. This corollary accounts for all previously known moduli of continuity for derivatives of exceptional diffeomorphisms. We also obtain a partial converse to our main result. For finitely generated free abelian groups, the existence of an exceptional action, together with some natural hypotheses on the derivatives of group elements, puts integrability restrictions on the modulus \begin{document}$ \alpha $\end{document}. These results are related to a long-standing question of D. McDuff concerning the length spectrum of exceptional \begin{document}$ C^1 $\end{document} diffeomorphisms of the circle.

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