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Discrete & Continuous Dynamical Systems - A

August 2020 , Volume 40 , Issue 8

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Boundary layer for 3D plane parallel channel flows of nonhomogeneous incompressible Navier-Stokes equations
Shijin Ding, Zhilin Lin and Dongjuan Niu
2020, 40(8): 4579-4596 doi: 10.3934/dcds.2020193 +[Abstract](288) +[HTML](73) +[PDF](371.51KB)

In this paper, we establish the mathematical validity of the Prandtl boundary layer theory for a class of nonlinear plane parallel flows of nonhomogeneous incompressible Navier-Stokes equations. The convergence is shown under various Sobolev norms, including the physically important space-time uniform norm, as well as the \begin{document}$ L^\infty(H^1) $\end{document} norm. It is mentioned that the mathematical validity of the Prandtl boundary layer theory for nonlinear plane parallel flow is generalized to the nonhomogeneous case.

Magnus-type integrator for non-autonomous SPDEs driven by multiplicative noise
Antoine Tambue and Jean Daniel Mukam
2020, 40(8): 4597-4624 doi: 10.3934/dcds.2020194 +[Abstract](194) +[HTML](65) +[PDF](492.57KB)

This paper aims to investigate numerical approximation of a general second order non-autonomous semilinear parabolic stochastic partial differential equation (SPDE) driven by multiplicative noise. Numerical approximations of autonomous SPDEs are thoroughly investigated in the literature, while the non-autonomous case is not yet understood. We discretize the non-autonomous SPDE by the finite element method in space and the Magnus-type integrator in time. We provide a strong convergence proof of the fully discrete scheme toward the mild solution in the root-mean-square \begin{document}$ L^2 $\end{document} norm. The result reveals how the convergence orders in both space and time depend on the regularity of the noise and the initial data. In particular, for multiplicative trace class noise we achieve convergence order \begin{document}$ \mathcal{O}\left(h^2\left(1+\max(0, \ln\left(t_m/h^2\right)\right)\right. \left.+\Delta t^{\frac{1}{2}}\right) $\end{document}. Numerical simulations to illustrate our theoretical finding are provided.

Ruelle operator for continuous potentials and DLR-Gibbs measures
Leandro Cioletti, Artur O. Lopes and Manuel Stadlbauer
2020, 40(8): 4625-4652 doi: 10.3934/dcds.2020195 +[Abstract](211) +[HTML](57) +[PDF](446.59KB)

In this work we study the Ruelle Operator associated to a continuous potential defined on a countable product of a compact metric space. We prove a generalization of Bowen's criterion for the uniqueness of the eigenmeasures and that one-sided one-dimensional DLR-Gibbs measures associated to a continuous translation invariant specifications are eigenmeasures of the transpose of the Ruelle operator. From the last claim one gets that for a continuous potential the concept of eigenprobability for the transpose of the Ruelle operator is equivalent to the concept of DLR probability.

Bounded extensions of the Ruelle operator to the Lebesgue space of integrable functions, with respect to the eigenmeasures, are studied and the problem of existence of maximal positive eigenfunctions for them is considered. One of our main results in this direction is the existence of such positive eigenfunctions for Bowen's potential in the setting of a compact and metric alphabet. We also present a version of Dobrushin's Theorem in the setting of Thermodynamic Formalism.

Structure of accessibility classes
Jana Rodriguez Hertz and Carlos H. Vásquez
2020, 40(8): 4653-4664 doi: 10.3934/dcds.2020196 +[Abstract](204) +[HTML](57) +[PDF](309.35KB)

In this work we deal with dynamically coherent partially hyperbolic diffeomorphisms whose central direction is two dimensional. We prove that in general the accessibility classes are topologically immersed manifolds. If, furthermore, the diffeomorphism satisfies certain bunching condition, then the accessibility classes are immersed \begin{document}$ C^{1} $\end{document}-manifolds.

Fluctuations of ergodic sums on periodic orbits under specification
Manfred Denker, Samuel Senti and Xuan Zhang
2020, 40(8): 4665-4687 doi: 10.3934/dcds.2020197 +[Abstract](339) +[HTML](69) +[PDF](407.62KB)

We study the fluctuations of ergodic sums using global and local specifications on periodic points. We obtain Lindeberg-type central limit theorems in both situations. As an application, when the system possesses a unique measure of maximal entropy, we show weak convergence of ergodic sums to a mixture of normal distributions. Our results suggest decomposing the variances of ergodic sums according to global and local sources.

Existence of periodic waves for a perturbed quintic BBM equation
Lina Guo and Yulin Zhao
2020, 40(8): 4689-4703 doi: 10.3934/dcds.2020198 +[Abstract](118) +[HTML](56) +[PDF](254.25KB)

This paper dealt with the existence of periodic waves for a perturbed quintic BBM equation by using geometric singular perturbation theory. By analyzing the perturbations of the Hamiltonian vector field with a hyperelliptic Hamiltonian of degree six, we proved that periodic wave solutions persist for sufficiently small perturbation parameter. It is also proved that the wave speed \begin{document}$ c_0(h) $\end{document} is decreasing on \begin{document}$ h $\end{document} by analyzing the ratio of Abelian integrals, where \begin{document}$ h $\end{document} is the energy level value. Moreover, the upper and lower bounds of the limit wave speed are given.

Representation formula for symmetrical symplectic capacity and applications
Rongrong Jin and Guangcun Lu
2020, 40(8): 4705-4765 doi: 10.3934/dcds.2020199 +[Abstract](98) +[HTML](51) +[PDF](670.45KB)

This is the second installment in a series of papers aimed at generalizing symplectic capacities and homologies. We study symmetric versions of symplectic capacities for real symplectic manifolds, and obtain corresponding results for them to those of the first [19] of this series (such as representation formula, a theorem by Evgeni Neduv, Brunn-Minkowski type inequality and Minkowski billiard trajectories proposed by Artstein-Avidan-Ostrover).

Mean dimension of shifts of finite type and of generalized inverse limits
Kazuhiro Kawamura
2020, 40(8): 4767-4775 doi: 10.3934/dcds.2020200 +[Abstract](108) +[HTML](46) +[PDF](302.54KB)

We study mean dimension of shifts of finite type defined on compact metric spaces and give its lower bound when the shift possesses a certain "periodic block" of arbitrarily large length. The result is applied to shift maps on generalized inverse limits with upper semi-continuous closed set-valued functions. In particular we obtain a refinement of some results due to Banič [1] and Erceg and Kennedy [5] on the dimension of the inverse limit spaces and topological entropy of their shifts.

Exponential upper bounds on the spectral gaps and homogeneous spectrum for the non-critical extended Harper's model
Xu Xu and Xin Zhao
2020, 40(8): 4777-4800 doi: 10.3934/dcds.2020201 +[Abstract](117) +[HTML](58) +[PDF](534.62KB)

For non-critical extended Harper's model with Diophantine frequency, we establish the exponential decay of the upper bounds on the spectral gaps and prove the spectrum is homogeneous. Especially we give a relationship between the decaying rate and Lyapunov exponent in non-self-dual region.

Asymptotic behavior for a Schrödinger equation with nonlinear subcritical dissipation
Thierry Cazenave and Zheng Han
2020, 40(8): 4801-4819 doi: 10.3934/dcds.2020202 +[Abstract](132) +[HTML](47) +[PDF](393.49KB)

We study the time-asymptotic behavior of solutions of the Schrödinger equation with nonlinear dissipation

in \begin{document}$ {\mathbb R}^N $\end{document}, \begin{document}$ N\geq1 $\end{document}, where \begin{document}$ \lambda\in {\mathbb C} $\end{document}, \begin{document}$ \Re \lambda <0 $\end{document} and \begin{document}$ 0<\alpha<\frac2N $\end{document}. We give a precise description of the behavior of the solutions (including decay rates in \begin{document}$ L^2 $\end{document} and \begin{document}$ L^\infty $\end{document}, and asymptotic profile), for a class of arbitrarily large initial data, under the additional assumption that \begin{document}$ \alpha $\end{document} is sufficiently close to \begin{document}$ \frac2N $\end{document}.

Classification to the positive radial solutions with weighted biharmonic equation
Xia Huang and Liping Wang
2020, 40(8): 4821-4837 doi: 10.3934/dcds.2020203 +[Abstract](137) +[HTML](54) +[PDF](402.71KB)

In this paper, we consider the weighted problem

where \begin{document}$ n\ge 5, -n<\alpha<n-4 $\end{document} and \begin{document}$ (p, \alpha,\beta, n), p>1 $\end{document} belongs to the critical hyperbola

We give two type-homoclinic functions \begin{document}$ v(t): = |x|^{\frac{n-4-\alpha}{2}}u(|x|), t = -\ln |x| $\end{document}. On the other hand, for radial solution \begin{document}$ u $\end{document} with non-removable singularity at origin, \begin{document}$ v(t) $\end{document} is periodic and classification for all periodic functions are obtained with \begin{document}$ -2<\alpha<n-4 $\end{document}; while for \begin{document}$ -n<\alpha \le -2, $\end{document} there always exists a solution \begin{document}$ u(|x|) $\end{document} with non-removable singularity and the corresponding function \begin{document}$ v(t) $\end{document} is not periodic. It is also closely related to the Caffarelli-Kohn-Nirenberg inequality, and we get some results such as the best embedding constants and the existence in radial case. In particular, for \begin{document}$ \alpha = \beta = 0 $\end{document}, it is related to the \begin{document}$ Q $\end{document}-curvature problem in conformal geometry.

Critically finite random maps of an interval
Jason Atnip and Mariusz Urbański
2020, 40(8): 4839-4906 doi: 10.3934/dcds.2020204 +[Abstract](107) +[HTML](55) +[PDF](630.73KB)

Given a finite collection \begin{document}$ {\mathcal{G}} $\end{document} of closed subintervals of the unit interval \begin{document}$ [0,1] $\end{document} with mutually empty interiors, we consider random multimodal \begin{document}$ C^3 $\end{document} maps with negative Schwarzian derivative, mapping each interval of \begin{document}$ {\mathcal{G}} $\end{document} onto the unit interval \begin{document}$ [0,1] $\end{document}. The randomness is governed by an invertible ergodic map \begin{document}$ {\theta}:{\Omega}\to{\Omega} $\end{document} preserving a probability measure \begin{document}$ m $\end{document} on some probability space \begin{document}$ {\Omega} $\end{document}. We denote the corresponding skew product map by \begin{document}$ T $\end{document} and call it a critically finite random map of an interval. We prove that there exists a subset \begin{document}$ AA(T) $\end{document}, defined in Definition 9.1, of \begin{document}$ [0,1] $\end{document} with the following properties:

1. For each \begin{document}$ t\in AA(T) $\end{document} a \begin{document}$ t $\end{document}–conformal random measure \begin{document}$ \nu_t $\end{document} exists. We denote by \begin{document}$ {\lambda}_{t,\nu_t,{\omega}} $\end{document} the corresponding generalized eigenvalues of the corresponding dual operators \begin{document}$ {\mathcal{L}}_{t,{\omega}}^* $\end{document}, \begin{document}$ {\omega}\in{\Omega} $\end{document}.

2. Given \begin{document}$ t\ge 0 $\end{document} any two \begin{document}$ t $\end{document}–conformal random measures are equivalent.

3. The expected topological pressure of the parameter \begin{document}$ t $\end{document}:

is independent of the choice of a \begin{document}$ t $\end{document}–conformal random measure \begin{document}$ \nu $\end{document}.

4. The function

is monotone decreasing and Lipschitz continuous.

5. With \begin{document}$ b_T $\end{document} being defined as the supremum of such parameters \begin{document}$ t\in AA(T) $\end{document} that \begin{document}$ { {\mathcal{E}}{{\rm{P}}}}(t)\ge 0 $\end{document}, it holds that

6. \begin{document}$ {\rm{HD}}( {\mathcal{J}}_{\omega}(T)) = b_T $\end{document} for \begin{document}$ m $\end{document}–a.e \begin{document}$ {\omega}\in{\Omega} $\end{document}, where \begin{document}$ {\mathcal{J}}_{\omega}(T) $\end{document}, \begin{document}$ {\omega}\in{\Omega} $\end{document}, form the random closed set generated by the skew product map \begin{document}$ T $\end{document}.

7. \begin{document}$ b_T = 1 $\end{document} if and only if \begin{document}$ {\bigcup}_{ {\Delta}\in {\mathcal{G}}} {\Delta} = [0,1] $\end{document}, and then \begin{document}$ {\mathcal{J}}_{\omega}(T) = [0,1] $\end{document} for all \begin{document}$ {\omega}\in{\Omega} $\end{document}.

Representation formulas of solutions and bifurcation sheets to a nonlocal Allen-Cahn equation
Tatsuki Mori, Kousuke Kuto, Tohru Tsujikawa and Shoji Yotsutani
2020, 40(8): 4907-4925 doi: 10.3934/dcds.2020205 +[Abstract](115) +[HTML](61) +[PDF](815.51KB)

We are interested in the Neumann problem of a 1D stationary Allen-Cahn equation with a nonlocal term. In our previous papers [4] and [5], we obtained a global bifurcation branch, and showed the existence and uniqueness of secondary bifurcation point. At this point, asymmetric solutions bifurcate from a branch of odd-symmetric solutions. In this paper, we give representation formulas of all solutions on the secondary bifurcation branch, and a bifurcation sheet which consists of bifurcation curves with heights.

A structure-preserving scheme for the Allen–Cahn equation with a dynamic boundary condition
Makoto Okumura and Daisuke Furihata
2020, 40(8): 4927-4960 doi: 10.3934/dcds.2020206 +[Abstract](163) +[HTML](48) +[PDF](699.91KB)

We propose a structure-preserving finite difference scheme for the Allen–Cahn equation with a dynamic boundary condition using the discrete variational derivative method [9]. In this method, how to discretize the energy which characterizes the equation is essential. Modifying the conventional manner and using an appropriate summation-by-parts formula, we can use a central difference operator as an approximation of an outward normal derivative on the boundary condition in the scheme. We show the stability and the existence and uniqueness of the solution for the proposed scheme. Also, we give the error estimate for the scheme. Numerical experiments demonstrate the effectiveness of the proposed scheme. Besides, through numerical experiments, we confirm that the long-time behavior of the solution under a dynamic boundary condition may differ from that under the Neumann boundary condition.

Positivity, monotonicity, and convexity for convolution operators
Christopher Goodrich and Carlos Lizama
2020, 40(8): 4961-4983 doi: 10.3934/dcds.2020207 +[Abstract](128) +[HTML](69) +[PDF](388.15KB)

We consider the convolution inequality

for given functions \begin{document}$ a $\end{document} and \begin{document}$ v $\end{document}, and we then investigate conditions on \begin{document}$ a $\end{document} and \begin{document}$ v $\end{document} that force the unknown function \begin{document}$ u $\end{document} to be positive or monotone or convex. We demonstrate that these results for abstract convolution equations can be specialized to yield new insights into the qualitative properties of fractional difference and differential operators. Finally, we apply our results to finite difference methods for fractional differential equations, and we show that our results yield insights into the qualitative behavior of these types of numerical approximations.

Lifespan of solutions to the Strauss type wave system on asymptotically flat space-times
Wei Dai, Daoyuan Fang and Chengbo Wang
2020, 40(8): 4985-4999 doi: 10.3934/dcds.2020208 +[Abstract](86) +[HTML](50) +[PDF](439.15KB)

By assuming certain local energy estimates on \begin{document}$ (1+3) $\end{document}-dimensional asymptotically flat space-time, we study the existence portion of the Strauss type wave system. Firstly we give a kind of space-time estimates which are related to the local energy norm that appeared in [13]. These estimates can be used to prove a series of weighted Strichartz and KSS type estimates, for wave equations on asymptotically flat space-time. Then we apply the space-time estimates to obtain the lower bound of the lifespan when the nonlinear exponents \begin{document}$ p $\end{document} and \begin{document}$ q\ge 2 $\end{document}. In particular, our bound for the subcritical case is sharp in general and we extend the known region of \begin{document}$ (p, q) $\end{document} to admit global solutions. In addition, the initial data are not required to be compactly supported, when \begin{document}$ p, q>2 $\end{document}.

Blowup results and concentration in focusing Schrödinger-Hartree equation
Yingying Xie, Jian Su and Liquan Mei
2020, 40(8): 5001-5017 doi: 10.3934/dcds.2020209 +[Abstract](125) +[HTML](46) +[PDF](417.11KB)

This paper is concerned with the Cauchy problem of the Schrödinger-Hartree equation. Applying the profile decomposition of bounded sequence in \begin{document}$ \dot{H}^1(\mathbb{R}^N)\cap\dot{H}^{S_c}(\mathbb{R}^N) $\end{document} and corresponding variational structure, a refined Gagliardo-Nirenberg inequality is established and the sharp constant for this inequality is deduced. Secondly, via construction and analysis of some invariant manifolds, we derive a different criterion of global existence and blowup results. Under the discussion of Bootstrap argument, we additionally obtain other sufficient condition for global existence. Finally, A compactness result is applied to show that the blowup solutions with bounded \begin{document}$ \dot{H}^{S_c} $\end{document} norm definitely have concentration properties related to a fixed \begin{document}$ \dot{H}^{S_c} $\end{document} norm of certain standing waves.

$ L^p $ Neumann problems in homogenization of general elliptic operators
Li Wang, Qiang Xu and Shulin Zhou
2020, 40(8): 5019-5045 doi: 10.3934/dcds.2020210 +[Abstract](139) +[HTML](44) +[PDF](463.0KB)

In this paper, we extend the nontangential maximal function estimates in \begin{document}$ L^p $\end{document}-norm obtained by C. Kenig, F. Lin and Z. Shen [13] to the nonhomogeneous elliptic operators with rapidly oscillating periodic coefficients. The result relies on a local Lipschitz boundary estimate, which has not been established in [29]. The present argument develops some new techniques to make the Campanato iteration and real methods workable for general elliptic operators. The result is new even for effective operators, as well as general elliptic equations of scalar.

Asymptotic behavior of sign-changing radial solutions of a semilinear elliptic equation in $ \mathbb{R}^2 $ when exponent approaches $ +\infty $
Weiwei Ao and Chao Liu
2020, 40(8): 5047-5077 doi: 10.3934/dcds.2020211 +[Abstract](180) +[HTML](44) +[PDF](476.57KB)

In this paper, we consider a semi-linear elliptic equation in \begin{document}$ \mathbb{R}^2 $\end{document} with the nonlinear exponent approaching infinity. We study asymptotic behavior of sign-changing once radial solutions obtained by Bartsch-Willem in [3] and [16]. Assuming \begin{document}$ u_{p}(0)>0 $\end{document}, we prove that a suitable rescaling of the positive part \begin{document}$ u^+_{p} $\end{document} converges to the unique regular solution of Liouville equation in \begin{document}$ \mathbb{R}^2 $\end{document}, while a suitable rescaling of the negative part \begin{document}$ u^-_{p} $\end{document} converges to a solution of a singular Liouville equation in \begin{document}$ \mathbb{R}^2 $\end{document}. We also obtain the asymptotic value of the \begin{document}$ L^\infty $\end{document}-norms of \begin{document}$ u^-_{p} $\end{document} and \begin{document}$ u^+_{p} $\end{document}. Moreover, we show that \begin{document}$ pu_p $\end{document} blow up at the origin and \begin{document}$ pu_p $\end{document} convergence to the fundamental solution of \begin{document}$ -\Delta +1 $\end{document} in \begin{document}$ \mathbb{R}^2 $\end{document} (up to a multiplier).

On a discrete self-organized-criticality finite time result
Umberto Mosco and Maria Agostina Vivaldi
2020, 40(8): 5079-5103 doi: 10.3934/dcds.2020212 +[Abstract](96) +[HTML](48) +[PDF](386.69KB)

In this paper we deal with theoretical and numerical aspects of some nonlinear problems related to sandpile models. We introduce a purely discrete model for infinitely many particles interacting according to a toppling process on a uniform two-dimensional grid and prove the convergence of the solutions to a differential initial value problem.

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