# American Institute of Mathematical Sciences

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2688-1594

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## Electronic Research Archive

June 2021 , Volume 29 , Issue 2

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2021, 29(2): 1925-1944 doi: 10.3934/era.2020098 +[Abstract](1285) +[HTML](631) +[PDF](398.24KB)
Abstract:

Flajolet and Françon [European. J. Combin. 10 (1989) 235-241] gave a combinatorial interpretation for the Taylor coefficients of the Jacobian elliptic functions in terms of doubled permutations. We show that a multivariable counting of the doubled permutations has also an explicit continued fraction expansion generalizing the continued fraction expansions of Rogers and Stieltjes. The second goal of this paper is to study the expansion of the Taylor coefficients of the generalized Jacobian elliptic functions, which implies the symmetric and unimodal property of the Taylor coefficients of the generalized Jacobian elliptic functions. The main tools are the combinatorial theory of continued fractions due to Flajolet and bijections due to Françon-Viennot, Foata-Zeilberger and Clarke-Steingrímsson-Zeng.

2021, 29(2): 1945-1967 doi: 10.3934/era.2020099 +[Abstract](1054) +[HTML](423) +[PDF](461.93KB)
Abstract:

We investigate the sharp time decay rates of the solution \begin{document}$U$\end{document} for the compressible Navier-Stokes system (1.1) in \begin{document}${\mathop{\mathbb R\kern 0pt}\nolimits}^3$\end{document} to the constant equilibrium \begin{document}$(\bar\rho>0, 0)$\end{document} when the initial data is a small smooth perturbation of \begin{document}$(\bar\rho,0)$\end{document}. Let \begin{document}$\widetilde U$\end{document} be the solution to the corresponding linearized equations with the same initial data. Under a mild non-degenerate condition on initial perturbations, we show that \begin{document}$\|U-\widetilde U\|_{L^2}$\end{document} decays at least at the rate of \begin{document}$(1+t)^{-\frac54}$\end{document}, which is faster than the rate \begin{document}$(1+t)^{-\frac34}$\end{document} for the \begin{document}$\widetilde U$\end{document} to its equilibrium \begin{document}$(\bar\rho ,0)$\end{document}. Our method is based on a combination of the linear sharp decay rate obtained from the spectral analysis and the energy estimates.

2021, 29(2): 1969-1990 doi: 10.3934/era.2020100 +[Abstract](951) +[HTML](421) +[PDF](462.41KB)
Abstract:

This paper deals with the limiting dynamical behavior of non-autonomous stochastic reaction-diffusion equations on thin domains. Firstly, we prove the existence and uniqueness of the regular random attractor. Then we prove the upper semicontinuity of the regular random attractors for the equations on a family of \begin{document}$(n+1)$\end{document}-dimensional thin domains collapses onto an \begin{document}$n$\end{document}-dimensional domain.

2021, 29(2): 1991-2006 doi: 10.3934/era.2020101 +[Abstract](1009) +[HTML](458) +[PDF](1523.36KB)
Abstract:

We investigate the matrix structure of the discrete system of the multiscale discontinuous Galerkin method (MDG) for general second order partial differential equations [10]. The MDG solution is obtained by composition of DG and the inter-scale operator. We show that the MDG matrix is given by the product of the DG matrix and the inter-scale matrix of the local problem. We apply an ILU preconditioned GMRES to solve the matrix equation effectively. Numerical examples are presented for convection dominated problems.

2021, 29(2): 2007-2028 doi: 10.3934/era.2020102 +[Abstract](911) +[HTML](427) +[PDF](3144.54KB)
Abstract:

We study decoupled numerical methods for multi-domain, multi-physics applications. By investigating various stages of numerical approximation and decoupling and tracking how the information is transmitted across the interface for a typical multi-modeling model problem, we derive an approximate intrinsic or inertial type Robin condition for its semi-discrete model. This new interface condition is justified both mathematically and physically in contrast to the classical Robin interface condition conventionally introduced for decoupling multi-modeling problems. Based on the intrinsic or inertial Robin condition, an equivalent semi-discrete model is introduced, which provides a general framework for devising effective decoupled numerical methods. Numerical experiments also confirm the effectiveness of this new decoupling approach.

2021, 29(2): 2029-2045 doi: 10.3934/era.2020103 +[Abstract](838) +[HTML](402) +[PDF](435.38KB)
Abstract:

The main aim of this paper is to study the \begin{document}$P_1$\end{document} nonconforming finite element approximations of the variational inequality arisen from the Signorini problem. We describe the finite dimensional closed convex cone approximation in a meanvalue-oriented sense. In this way, the optimal convergence rate \begin{document}$O(h)$\end{document} can be obtained by a refined analysis when the exact solution belongs to \begin{document}$H^{2}(\Omega)$\end{document} without any assumption. Furthermore, we also study the optimal convergence for the case \begin{document}$u\in H^{1+\nu}(\Omega)$\end{document} with \begin{document}$\frac{1}{2}<\nu<1$\end{document}.

2021, 29(2): 2047-2061 doi: 10.3934/era.2020104 +[Abstract](1146) +[HTML](484) +[PDF](378.87KB)
Abstract:

In this paper, a unified theoretical method is presented to implement the finite/fixed-time synchronization control for complex networks with uncertain inner coupling. The quantized controller and the quantized adaptive controller are designed to reduce the control cost and save the channel resources, respectively. By means of the linear matrix inequalities technique, two sufficient conditions are proposed to guarantee that the synchronization error system of the complex networks is finite/fixed-time stable in virtue of the Lyapunov stability theory. Moreover, two types of setting time, which are dependent and independent on the initial values, are given respectively. Finally, the effectiveness of the control strategy is verified by a simulation example.

2021, 29(2): 2063-2075 doi: 10.3934/era.2020105 +[Abstract](925) +[HTML](450) +[PDF](358.63KB)
Abstract:

In this paper, we study the large time behavior of the solution for one-dimensional compressible micropolar fluid model with large initial data. This model describes micro-rotational motions and spin inertia which is commonly used in the suspensions, animal blood, and liquid crystal. We get the uniform positive lower and upper bounds of the density and temperature independent of both space and time. In particular, we also obtain the asymptotic behavior of the micro-rotation velocity.

2021, 29(2): 2077-2099 doi: 10.3934/era.2020106 +[Abstract](955) +[HTML](413) +[PDF](791.6KB)
Abstract:

This paper concerns the motion of the supersonic potential flow in a two-dimensional expanding duct. In the case that two Riemann invariants are both monotonically increasing along the inlet, which means the gases are spread at the inlet, we obtain the global solution by solving the problem in those inner and border regions divided by two characteristics in \begin{document}$(x, y)$\end{document}-plane, and the vacuum will appear in some finite place adjacent to the boundary of the duct. In addition, we point out that the vacuum here is not the so-called physical vacuum. On the other hand, for the case that at least one Riemann invariant is strictly monotonic decreasing along some part of the inlet, which means the gases have some local squeezed properties at the inlet, we show that the \begin{document}$C^1$\end{document} solution to the problem will blow up at some finite location in the non-convex duct.

2021, 29(2): 2101-2127 doi: 10.3934/era.2020107 +[Abstract](1112) +[HTML](362) +[PDF](456.23KB)
Abstract:

We study second order linear differential equations with analytic coefficients. One important case is when the equation admits a so called regular singular point. In this case we address some untouched and some new aspects of Frobenius methods. For instance, we address the problem of finding formal solutions and studying their convergence. A characterization of regular singularities is given in terms of the space of solutions. An analytic-geometric classification of such linear polynomial homogeneous ODEs is obtained by the use of techniques from geometric theory of foliations means. This is done by associating to such an ODE a rational Riccati differential equation and therefore a global holonomy group. This group is a computable group of Moebius maps. These techniques apply to classical equations as Bessel and Legendre equations. We also address the problem of deciding which such polynomial equations admit a Liouvillian solution. A normal form for such a solution is then obtained. Our results are concrete and (computationally) constructive and are aimed to shed a new light in this important subject.

2021, 29(2): 2129-2140 doi: 10.3934/era.2020108 +[Abstract](1099) +[HTML](405) +[PDF](410.23KB)
Abstract:

A model of COVID-19 in an interconnected network of communities is studied. This model considers the dynamics of susceptible, asymptomatic and symptomatic individuals, deceased but not yet buried people, as well as the dynamics of the virus or pathogen at connected nodes or communities. People can move between communities carrying the virus to any node in the region of \begin{document}$n$\end{document} communities (or patches). This model considers both virus direct (person to person) and indirect (contaminated environment to person) transmissions. Using either matrix and graph-theoretic methods and some combinatorial identities, appropriate Lyapunov functions are constructed to study global stability properties of both the disease-free and the endemic equilibrium of the corresponding system of \begin{document}$5n$\end{document} differential equations.

2021, 29(2): 2141-2147 doi: 10.3934/era.2020109 +[Abstract](721) +[HTML](357) +[PDF](281.76KB)
Abstract:

We outline the construction of special functions in terms of Fredholm determinants to solve boundary value problems of the string spectral problem. Our motivation is that the string spectral problem is related to the spectral equations in Lax pairs of at least three nonlinear evolution equations from mathematical physics.

2021, 29(2): 2149-2165 doi: 10.3934/era.2020110 +[Abstract](1054) +[HTML](374) +[PDF](335.94KB)
Abstract:

This article is an overview on some recent advances in the inverse scattering problems with phaseless data. Based upon our previous studies on the uniqueness issues in phaseless inverse acoustic scattering theory, this survey aims to briefly summarize the relevant rudiments comprising prototypical model problems, major results therein, as well as the rationale behind the basic techniques. We hope to sort out the essential ideas and shed further lights on this intriguing field.

2021, 29(2): 2167-2185 doi: 10.3934/era.2020111 +[Abstract](1004) +[HTML](404) +[PDF](435.77KB)
Abstract:

The Euler number \begin{document}$E_n$\end{document} (resp. Entringer number \begin{document}$E_{n,k}$\end{document}) enumerates the alternating (down-up) permutations of \begin{document}$\{1,\dots,n\}$\end{document} (resp. starting with \begin{document}$k$\end{document}). The Springer number \begin{document}$S_n$\end{document} (resp. Arnold number \begin{document}$S_{n,k}$\end{document}) enumerates the type \begin{document}$B$\end{document} alternating permutations (resp. starting with \begin{document}$k$\end{document}). In this paper, using bijections we first derive the counterparts in André permutations and Simsun permutations for the Entringer numbers \begin{document}$(E_{n,k})$\end{document}, and then the counterparts in signed André permutations and type \begin{document}$B$\end{document} increasing 1-2 trees for the Arnold numbers \begin{document}$(S_{n,k})$\end{document}.

2021, 29(2): 2187-2221 doi: 10.3934/era.2020112 +[Abstract](971) +[HTML](371) +[PDF](511.36KB)
Abstract:

In this paper, we investigate a class of stochastic recurrent neural networks with discrete and distributed delays for both biological and mathematical interests. We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth conditions so that the uniqueness of the Cauchy problem fails to be true. Moreover, the existence of pullback attractors with or without periodicity is presented for the multi-valued noncompact random dynamical system. In particular, a new method for checking the asymptotical compactness of solutions to the class of nonautonomous stochastic lattice systems with infinite delay is used.

2020 Impact Factor: 1.833
5 Year Impact Factor: 1.833