Electronic Research Archive

March 2021 , Volume 29 , Issue 1

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Regularity criteria for weak solutions of the Magneto-micropolar equations
Jens Lorenz, Wilberclay G. Melo and Suelen C. P. de Souza
2021, 29(1): 1625-1639 doi: 10.3934/era.2020083 +[Abstract](481) +[HTML](210) +[PDF](357.62KB)

In this paper, we show that a weak solution \begin{document}$ (\mathbf{u},\mathbf{w},\mathbf{b})(\cdot,t) $\end{document} of the magneto-micropolar equations, defined in \begin{document}$ [0,T) $\end{document}, which satisfies \begin{document}$ \nabla u_3, \nabla_{h} \mathbf{w}, \nabla_{h} \mathbf{b} $\end{document} \begin{document}$ \in L^{\frac{32}{7}}(0,T; $\end{document} \begin{document}$ L^2(\mathbb{R}^3)) $\end{document} or \begin{document}$ \partial_3 u_3, \partial_3 \mathbf{w}, \partial_3 \mathbf{b} \in L^{\infty}(0,T;L^2(\mathbb{R}^3)) $\end{document}, is regular in \begin{document}$ \mathbb{R}^3\times(0,T) $\end{document} and can be extended as a \begin{document}$ C^\infty $\end{document} solution beyond \begin{document}$ T $\end{document}.

Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting
Kerioui Nadjah and Abdelouahab Mohammed Salah
2021, 29(1): 1641-1660 doi: 10.3934/era.2020084 +[Abstract](489) +[HTML](189) +[PDF](819.15KB)

The objective of the current paper is to investigate the dynamics of a new bioeconomic predator prey system with only predator's harvesting and Holling type Ⅲ response function. The system is equipped with an algebraic equation because of the economic revenue. We offer a detailed mathematical analysis of the proposed model to illustrate some of the significant results. The boundedness and positivity of solutions for the model are examined. Coexistence equilibria of the bioeconomic system have been thoroughly investigated and the behaviours of the model around them are described by means of qualitative theory of dynamical systems (such as local stability and Hopf bifurcation). The obtained results provide a useful platform to understand the role of the economic revenue \begin{document}$ v $\end{document}. We show that a positive equilibrium point is locally asymptotically stable when the profit \begin{document}$ v $\end{document} is less than a certain critical value \begin{document}$ v^{*}_1 $\end{document}, while a loss of stability by Hopf bifurcation can occur as the profit increases. It is evident from our study that the economic revenue has the capability of making the system stable (survival of all species). Finally, some numerical simulations have been carried out to substantiate the analytical findings.

Threshold dynamics of stochastic models with time delays: A case study for Yunnan, China
Zhimin Li, Tailei Zhang and Xiuqing Li
2021, 29(1): 1661-1679 doi: 10.3934/era.2020085 +[Abstract](439) +[HTML](217) +[PDF](387.44KB)

In this paper, we provide an effective method for estimating the thresholds of the stochastic models with time delays by using of the nonnegative semimartingale convergence theorem. Firstly, we establish the stochastic delay differential equation models for two diseases, and obtain two thresholds of two diseases and the sufficient conditions for the persistence and extinction of two diseases. Then, numerical simulations for co-infection of HIV/AIDS and Gonorrhea in Yunnan Province, China, are carried out. Finally, we discuss some biological implications and focus on the impact of some key model parameters. One of the most interesting findings is that the stochastic fluctuation and time delays introduced into the deterministic models can suppress the outbreak of the diseases, which can provide some useful control strategies to regulate the dynamics of the diseases, and the numerical simulations verify this phenomenon.

Some properties for almost cellular algebras
Yongjie Wang and Nan Gao
2021, 29(1): 1681-1689 doi: 10.3934/era.2020086 +[Abstract](412) +[HTML](209) +[PDF](297.34KB)

In this paper, we will investigate some properties for almost cellular algebras. We compare the almost cellular algebras with quasi-hereditary algebras, which are known to carry any homological and categorical structures. We prove that any almost cellular algebra is the iterated inflation and obtain some sufficient and necessary conditions for an almost cellular algebra \begin{document}$ \mathrm{A} $\end{document} to be quasi-hereditary.

Global conservative solutions for a modified periodic coupled Camassa-Holm system
Rong Chen, Shihang Pan and Baoshuai Zhang
2021, 29(1): 1691-1708 doi: 10.3934/era.2020087 +[Abstract](406) +[HTML](190) +[PDF](415.27KB)

In present paper, we deal with the behavior of a solution beyond the occurrence of wave breaking for a modified periodic Coupled Camassa-Holm system. By introducing a new set of independent and dependent variables, which resolve all singularities due to possible wave breaking, this evolution system is rewritten as a closed semilinear system. The local existence of the semilinear system is obtained as fixed points of a contractive transformation. Moreover, this formulation allows us to continue the solution after wave breaking, and gives a global conservative solution where the energy is conserved for almost all times. Returning to the original variables. We finally obtain a semigroup of global conservative solutions, which depend continuously on the initial data. Additionally, our results repair some gaps in the pervious work.

On a final value problem for a nonlinear fractional pseudo-parabolic equation
Vo Van Au, Hossein Jafari, Zakia Hammouch and Nguyen Huy Tuan
2021, 29(1): 1709-1734 doi: 10.3934/era.2020088 +[Abstract](691) +[HTML](171) +[PDF](484.76KB)

In this paper, we investigate a final boundary value problem for a class of fractional with parameter \begin{document}$ \beta $\end{document} pseudo-parabolic partial differential equations with nonlinear reaction term. For \begin{document}$ 0<\beta < 1, $\end{document} the solution is regularity-loss, we establish the well-posedness of solutions. In the case that \begin{document}$ \beta >1 $\end{document}, it has a feature of regularity-gain. Then, the instability of a mild solution is proved. We introduce two methods to regularize the problem. With the help of the modified Lavrentiev regularization method and Fourier truncated regularization method, we propose the regularized solutions in the cases of globally or locally Lipschitzian source term. Moreover, the error estimates is established.

Error estimates for second-order SAV finite element method to phase field crystal model
Liupeng Wang and Yunqing Huang
2021, 29(1): 1735-1752 doi: 10.3934/era.2020089 +[Abstract](520) +[HTML](218) +[PDF](883.1KB)

In this paper, the second-order scalar auxiliary variable approach in time and linear finite element method in space are employed for solving the Cahn-Hilliard type equation of the phase field crystal model. The energy stability of the fully discrete scheme and the boundedness of numerical solution are studied. The rigorous error estimates of order \begin{document}$ O(\tau^2+h^2) $\end{document} in the sense of \begin{document}$ L^2 $\end{document}-norm is derived. Finally, some numerical results are given to demonstrate the theoretical analysis.

Some recent progress on inverse scattering problems within general polyhedral geometry
Xinlin Cao, Huaian Diao and Jinhong Li
2021, 29(1): 1753-1782 doi: 10.3934/era.2020090 +[Abstract](423) +[HTML](246) +[PDF](492.02KB)

Unique identifiability by finitely many far-field measurements in the inverse scattering theory is a highly challenging fundamental mathematical topic. In this paper, we survey some recent progress on the inverse obstacle scattering problems and the inverse medium scattering problems associated with time-harmonic waves within a certain polyhedral geometry, where one can establish the unique identifiability results by finitely many measurements. Some unique identifiability issues on the inverse diffraction grating problems are also considered. Furthermore, the geometrical structures of Laplacian and transmission eigenfunctions are reviewed, which have important applications in the unique determination for inverse obstacle and medium scattering problems with finitely many measurements. We discuss the mathematical techniques and methods developed in the literature. Finally, we raise some intriguing open problems for the future investigation.

Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations
José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello and Marko A. Rojas-Medar
2021, 29(1): 1783-1801 doi: 10.3934/era.2020091 +[Abstract](400) +[HTML](162) +[PDF](424.01KB)

We prove some results on the stability of slow stationary solutions of the MHD equations in two- and three-dimensional bounded domains for external force fields that are asymptotically autonomous. Our results show that weak solutions are asymptotically stable in time in the \begin{document}$ L^2 $\end{document}-norm. Further, assuming certain regularity hypotheses on the problem data, strong solutions are asymptotically stable in the \begin{document}$ H^1 $\end{document} and \begin{document}$ H^2 $\end{document}-norms.

Combinatorics of some fifth and sixth order mock theta functions
Meenakshi Rana and Shruti Sharma
2021, 29(1): 1803-1818 doi: 10.3934/era.2020092 +[Abstract](332) +[HTML](181) +[PDF](346.04KB)

The goal of this paper is to provide a new combinatorial meaning to two fifth order and four sixth order mock theta functions. Lattice paths of Agarwal and Bressoud with certain modifications are used as a tool to study these functions.

A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations
Guoliang Zhang, Shaoqin Zheng and Tao Xiong
2021, 29(1): 1819-1839 doi: 10.3934/era.2020093 +[Abstract](473) +[HTML](285) +[PDF](522.1KB)

In this paper, we propose a conservative semi-Lagrangian finite difference (SLFD) weighted essentially non-oscillatory (WENO) scheme, based on Runge-Kutta exponential integrator (RKEI) method, to solve one-dimensional scalar nonlinear hyperbolic equations. Conservative semi-Lagrangian schemes, under the finite difference framework, usually are designed only for linear or quasilinear conservative hyperbolic equations. Here we combine a conservative SLFD scheme developed in [21], with a high order RKEI method [7], to design conservative SLFD schemes, which can be applied to nonlinear hyperbolic equations. Our new approach will enjoy several good properties as the scheme for the linear or quasilinear case, such as, conservation, high order and large time steps. The new ingredient is that it can be applied to nonlinear hyperbolic equations, e.g., the Burgers' equation. Numerical tests will be performed to illustrate the effectiveness of our proposed schemes.

Skew doubled shifted plane partitions: Calculus and asymptotics
Guo-Niu Han and Huan Xiong
2021, 29(1): 1841-1857 doi: 10.3934/era.2020094 +[Abstract](316) +[HTML](149) +[PDF](368.99KB)

In this paper, we establish a new summation formula for Schur processes, called the complete summation formula. As an application, we obtain the generating function and the asymptotic formula for the number of doubled shifted plane partitions, which can be viewed as plane partitions "shifted at the two sides". We prove that the order of the asymptotic formula depends only on the diagonal width of the doubled shifted plane partition, not on the profile (the skew zone) itself. By using similar methods, the generating function and the asymptotic formula for the number of symmetric cylindric partitions are also derived.

Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method
Ying Liu, Yanping Chen, Yunqing Huang and Yang Wang
2021, 29(1): 1859-1880 doi: 10.3934/era.2020095 +[Abstract](409) +[HTML](172) +[PDF](2296.84KB)

The mathematical model of a semiconductor device is described by a coupled system of three quasilinear partial differential equations. The mixed finite element method is presented for the approximation of the electrostatic potential equation, and the characteristics finite element method is used for the concentration equations. First, we estimate the mixed finite element and the characteristics finite element method solution in the sense of the \begin{document}$ L^q $\end{document} norm. To linearize the full discrete scheme of the problem, we present an efficient two-grid method based on the idea of Newton iteration. The two-grid algorithm is to solve the nonlinear coupled equations on the coarse grid and then solve the linear equations on the fine grid. Moreover, we obtain the \begin{document}$ L^{q} $\end{document} error estimates for this algorithm. It is shown that a mesh size satisfies \begin{document}$ H = O(h^{1/2}) $\end{document} and the two-grid method still achieves asymptotically optimal approximations. Finally, the numerical experiment is given to illustrate the theoretical results.

Viscosity robust weak Galerkin finite element methods for Stokes problems
Bin Wang and Lin Mu
2021, 29(1): 1881-1895 doi: 10.3934/era.2020096 +[Abstract](397) +[HTML](179) +[PDF](1246.21KB)

In this paper, we develop a viscosity robust weak Galerkin finite element scheme for Stokes equations. The major idea for achieving pressure-independent energy-error estimate is to use a divergence preserving velocity reconstruction operator in the discretization of the right hand side body force. The optimal convergence results for velocity and pressure have been established in this paper. Finally, numerical examples are presented for validating the theoretical conclusions.

A weak Galerkin finite element method for nonlinear conservation laws
Xiu Ye, Shangyou Zhang and Peng Zhu
2021, 29(1): 1897-1923 doi: 10.3934/era.2020097 +[Abstract](382) +[HTML](164) +[PDF](430.32KB)

A weak Galerkin (WG) finite element method is presented for nonlinear conservation laws. There are two built-in parameters in this WG framework. Different choices of the parameters will lead to different approaches for solving hyperbolic conservation laws. The convergence analysis is obtained for the forward Euler time discrete and the third order explicit TVDRK time discrete WG schemes respectively. The theoretical results are verified by numerical experiments.

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