Electronic Research Archive

September 2020 , Volume 28 , Issue 3

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The longtime behavior of the model with nonlocal diffusion and free boundaries in online social networks
Meng Zhao
2020, 28(3): 1143-1160 doi: 10.3934/era.2020063 +[Abstract](173) +[HTML](60) +[PDF](346.33KB)

In this paper we consider a free boundary problem with nonlocal diffusion describing information diffusion in online social networks. This model can be viewed as a nonlocal version of the free boundary problem studied by Ren et al. (Spreading-vanishing dichotomy in information diffusion in online social networks with intervention, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019) 1843–1865). We first show that this problem has a unique solution for all \begin{document}$ t>0 $\end{document}, and then we show that its longtime behaviour is determined by a spreading-vanishing dichotomy. We also obtain sharp criteria for spreading and vanishing, and show that the spreading always happen if the diffusion rate of any one of the information is small, which is very different from the local diffusion model.

The Jacobi spectral collocation method for fractional integro-differential equations with non-smooth solutions Special Issues
Yin Yang, Sujuan Kang and Vasiliy I. Vasil'ev
2020, 28(3): 1161-1189 doi: 10.3934/era.2020064 +[Abstract](192) +[HTML](69) +[PDF](640.45KB)

In recent years, many numerical methods have been extended to fractional integro-differential equations. But most of them ignore an important problem. Even if the input function is smooth, the solutions of these equations would exhibit some weak singularity, which leads to non-smooth solutions, and a deteriorate order of convergence. To overcome this problem, we first study in detail the singularity of the fractional integro-differential equation, and then eliminate the singularity by introducing some smoothing transformation. We can maximize the convergence rate by adjusting the parameters in the auxiliary transformation. We use the Jacobi spectral-collocation method with global and high precision characteristics to solve the transformed equation. A comprehensive and rigorous error estimation under the \begin{document}$ L^{\infty} $\end{document}- and \begin{document}$ L^{2}_{\omega^{\alpha, \beta}} $\end{document}-norms is derived. Finally, we give specific numerical examples to show the accuracy of the theoretical estimation and the feasibility and effectiveness of the proposed method.

Numerical analysis of modular grad-div stability methods for the time-dependent Navier-Stokes/Darcy model Special Issues
Jiangshan Wang, Lingxiong Meng and Hongen Jia
2020, 28(3): 1191-1205 doi: 10.3934/era.2020065 +[Abstract](125) +[HTML](54) +[PDF](361.28KB)

In this paper, we construct a modular grad-div stabilization method for the Navier-Stokes/Darcy model, which is based on the first order Backward Euler scheme. This method does not enlarge the accuracy of numerical solution, but also can improve mass conservation and relax the influence of parameters. Herein, we give stability analysis and error estimations. Finally, by some numerical experiment, the scheme our proposed is shown to be valid.

A multiple-relaxation-time lattice Boltzmann method with Beam-Warming scheme for a coupled chemotaxis-fluid model Special Issues
Zhonghua Qiao and Xuguang Yang
2020, 28(3): 1207-1225 doi: 10.3934/era.2020066 +[Abstract](165) +[HTML](50) +[PDF](1610.81KB)

In this work, a multiple-relaxation-time (MRT) lattice Boltzmann method (LBM) is proposed to solve a coupled chemotaxis-fluid model. In the evolution equation of the proposed LBM, Beam-Warming (B-W) scheme is used to enhance the numerical stability. In numerical experiments, at first, the comparison between the classical LBM and the present LBM with B-W scheme is carried out by simulating blow up phenomenon of the Keller-Segel (K-S) model. Numerical results show that the stability of the present LBM with B-W scheme is better than the classical one. Then, the second order convergence rate of the proposed B-W scheme is verified in the numerical study of the coupled Navier-Stokes (N-S) K-S model. Finally, through solving the coupled chemotaxis-fluid model, the formation of falling bacterial plumes is numerically investigated. Numerical results agree well with existing ones in the literature.

Colimits of crossed modules in modified categories of interest Special Issues
Ali Aytekin and Kadir Emir
2020, 28(3): 1227-1238 doi: 10.3934/era.2020067 +[Abstract](131) +[HTML](44) +[PDF](324.17KB)

In this paper, we give the constructions of the coequalizer and coproduct objects for the category of crossed modules, in a modified category of interest (MCI). In other words, we prove that the corresponding category is finitely cocomplete.

On simultaneous recovery of sources/obstacles and surrounding mediums by boundary measurements Special Issues
Xiaoping Fang, Youjun Deng, Wing-Yan Tsui and Zaiyun Zhang
2020, 28(3): 1239-1255 doi: 10.3934/era.2020068 +[Abstract](117) +[HTML](52) +[PDF](367.52KB)

We consider a particular type of inverse problems where an unknown source embedded in an inhomogeneous medium, and one intends to recover the source and/or the medium by knowledge of the wave field (generated by the unknown source) outside the medium. This type of inverse problems arises in many applications of practical importance, including photoacoustic and thermoacoustic tomography, brain imaging and geomagnetic anomaly detections. We survey the recent mathematical developments on this type of inverse problems. We discuss the mathematical tools developed for effectively tackling this type of inverse problems. We also discuss a related inverse problem of recovering an embedded obstacle and its surrounding medium by active measurements.

Recent progress on the mathematical study of anomalous localized resonance in elasticity Special Issues
Hongjie Li
2020, 28(3): 1257-1272 doi: 10.3934/era.2020069 +[Abstract](113) +[HTML](48) +[PDF](361.94KB)

We consider the anomalous localized resonance induced by negative elastic metamaterials and its application in invisibility cloaking. We survey the recent mathematical developments in the literature and discuss two mathematical strategies that have been developed for tackling this peculiar resonance phenomenon. The first one is the spectral method, which explores the anomalous localized resonance through investigating the spectral system of the associated Neumann-Poincaré operator. The other one is the variational method, which considers the anomalous localized resonance via calculating the nontrivial kernels of a non-elliptic partial differential operator. The advantages and the relationship between the two methods are discussed. Finally, we propose some open problems for the future study.

New series for powers of $ \pi $ and related congruences Special Issues
Zhi-Wei Sun
2020, 28(3): 1273-1342 doi: 10.3934/era.2020070 +[Abstract](147) +[HTML](101) +[PDF](571.91KB)

Via symbolic computation we deduce 97 new type series for powers of \begin{document}$ \pi $\end{document} related to Ramanujan-type series. Here are three typical examples:



where the generalized central trinomial coefficient \begin{document}$ T_k(b,c) $\end{document} denotes the coefficient of \begin{document}$ x^k $\end{document} in the expansion of \begin{document}$ (x^2+bx+c)^k $\end{document}. We also formulate a general characterization of rational Ramanujan-type series for \begin{document}$ 1/\pi $\end{document} via congruences, and pose 117 new conjectural series for powers of \begin{document}$ \pi $\end{document} via looking for corresponding congruences. For example, we conjecture that

Eighteen of the new series in this paper involve some imaginary quadratic fields with class number \begin{document}$ 8 $\end{document}.

The boundedness and upper semicontinuity of the pullback attractors for a 2D micropolar fluid flows with delay Special Issues
Wenlong Sun
2020, 28(3): 1343-1356 doi: 10.3934/era.2020071 +[Abstract](118) +[HTML](55) +[PDF](352.36KB)

In this paper, two properties of the pullback attractor for a 2D non-autonomous micropolar fluid flows with delay on unbounded domains are investigated. First, we establish the \begin{document}$ H^1 $\end{document}-boundedness of the pullback attractor. Further, with an additional regularity limit on the force and moment with respect to time t, we remark the \begin{document}$ H^2 $\end{document}-boundedness of the pullback attractor. Then, we verify the upper semicontinuity of the pullback attractor with respect to the domains.

Strong $ (L^2,L^\gamma\cap H_0^1) $-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension Special Issues
Hongyong Cui, Peter E. Kloeden and Wenqiang Zhao
2020, 28(3): 1357-1374 doi: 10.3934/era.2020072 +[Abstract](132) +[HTML](48) +[PDF](384.95KB)

In this paper we study the continuity in initial data of a classical reaction-diffusion equation with arbitrary \begin{document}$ p>2 $\end{document} order nonlinearity and in any space dimension \begin{document}$ N \geqslant 1 $\end{document}. It is proved that the weak solutions can be \begin{document}$ (L^2, L^\gamma\cap H_0^1) $\end{document}-continuous in initial data for arbitrarily large \begin{document}$ \gamma \geqslant 2 $\end{document} (independent of the physical parameters of the system), i.e., can converge in the norm of any \begin{document}$ L^\gamma\cap H_0^1 $\end{document} as the corresponding initial values converge in \begin{document}$ L^2 $\end{document}. In fact, the system is shown to be \begin{document}$ (L^2, L^\gamma\cap H_0^1) $\end{document}-smoothing in a H\begin{document}$ \ddot{\rm o} $\end{document}lder way. Applying this to the global attractor we find that, with external forcing only in \begin{document}$ L^2 $\end{document}, the attractor \begin{document}$ \mathscr{A} $\end{document} attracts bounded subsets of \begin{document}$ L^2 $\end{document} in the norm of any \begin{document}$ L^\gamma\cap H_0^1 $\end{document}, and that every translation set \begin{document}$ \mathscr{A}-z_0 $\end{document} of \begin{document}$ \mathscr{A} $\end{document} for any \begin{document}$ z_0\in \mathscr{A} $\end{document} is a finite dimensional compact subset of \begin{document}$ L^\gamma\cap H_0^1 $\end{document}. The main technique we employ is a combination of a Moser iteration and a decomposition of the nonlinearity, by which the interpolation inequalities are avoided and the new continuity result is obtained without any restrictions on the order \begin{document}$ p>2 $\end{document} of the nonlinearity and the space dimension \begin{document}$ N \geqslant 1 $\end{document}.

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