Electronic Research Archive

December 2020 , Volume 28 , Issue 4

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The regularized Boussinesq equations with partial dissipations in dimension two
Hua Qiu and Zheng-An Yao
2020, 28(4): 1375-1393 doi: 10.3934/era.2020073 +[Abstract](514) +[HTML](215) +[PDF](372.62KB)

The incompressible Boussinesq system plays an important role in modelling geophysical fluids and studying the Raleigh-Bernard convection. We consider the regularized model (also named as Boussinesq-\begin{document}$ \alpha $\end{document} model) to the Boussinesq equations. We consider the Cauchy problem of a two-dimensional regularized Boussinesq model with vertical dissipation in the horizontal regularized velocity equation and horizontal dissipation in the vertical regularized velocity equation and prove that this system has a unique global classical solution. Next, we consider a two-dimensional Boussinesq-\begin{document}$ \alpha $\end{document} model with only vertical thermal diffusion and establish a Beale-Kato-Majda type regularity condition of smooth solution for this system.

The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay
Xin-Guang Yang, Lu Li, Xingjie Yan and Ling Ding
2020, 28(4): 1395-1418 doi: 10.3934/era.2020074 +[Abstract](513) +[HTML](202) +[PDF](410.68KB)

This paper concerns the stability of pullback attractors for 3D Brinkman-Forchheimer equation with delays. By some regular estimates and the variable index to deal with the delay term, we get the sufficient conditions for asymptotic stability of trajectories inside the pullback attractors for a fluid flow model in porous medium by generalized Grashof numbers.

A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior
Yichen Zhang and Meiqiang Feng
2020, 28(4): 1419-1438 doi: 10.3934/era.2020075 +[Abstract](563) +[HTML](204) +[PDF](390.49KB)

We prove uniqueness, existence and asymptotic behavior of positive solutions to the system coupled by \begin{document}$ p $\end{document}-Laplacian elliptic equations

where \begin{document}$ \Delta_p u = \text{div}({|\nabla u|}^{p-2}\nabla u),\ 1<p<\infty $\end{document}, \begin{document}$ \lambda_1 $\end{document} and \begin{document}$ \lambda_2 $\end{document} are positive parameters, \begin{document}$ \Omega $\end{document} is the open unit ball in \begin{document}$ \mathbb{R}^N,\ N\geq 2 $\end{document}.

A robust adaptive grid method for singularly perturbed Burger-Huxley equations Special Issues
Li-Bin Liu, Ying Liang, Jian Zhang and Xiaobing Bao
2020, 28(4): 1439-1457 doi: 10.3934/era.2020076 +[Abstract](526) +[HTML](322) +[PDF](466.79KB)

In this paper, an adaptive grid method is proposed to solve one-dimensional unsteady singularly perturbed Burger-Huxley equation with appropriate initial and boundary conditions. Firstly, we use the classical backward-Euler scheme on a uniform mesh to approximate time derivative. The resulting nonlinear singularly perturbed semi-discrete problem is linearized by using Newton-Raphson-Kantorovich approximation method which is quadratically convergent. Then, an upwind finite difference scheme on an adaptive nonuniform grid is used for space derivative. The nonuniform grid is generated by equidistribution of a positive monitor function, which is similar to the arc-length function. It is shown that the presented adaptive grid method is first order uniform convergent in the time and spatial directions, respectively. Finally, numerical results are given to validate the theoretical results.

Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems Special Issues
Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li and Shang Liu
2020, 28(4): 1459-1486 doi: 10.3934/era.2020077 +[Abstract](568) +[HTML](216) +[PDF](2201.39KB)

This paper aims at investigating the convergence and quasi- optimality of an adaptive finite element method for control constrained nonlinear elliptic optimal control problems. We derive a posteriori error estimation for both the control, the state and adjoint state variables under controlling by \begin{document}$ L^2 $\end{document}-norms where bubble function is a wonderful tool to deal with the global lower error bound. Then a contraction is proved before the convergence is proposed. Furthermore, we find that if keeping the grids sufficiently mildly graded, we can prove the optimal convergence and the quasi-optimality for the adaptive finite element method. In addition, some numerical results are presented to verify our theoretical analysis.

A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction Special Issues
Zexuan Liu, Zhiyuan Sun and Jerry Zhijian Yang
2020, 28(4): 1487-1501 doi: 10.3934/era.2020078 +[Abstract](549) +[HTML](242) +[PDF](5508.22KB)

We numerically investigate the superconvergence property of the discontinuous Galerkin method by patch reconstruction. The convergence rate \begin{document}$ 2m+1 $\end{document} can be observed at the grid points and barycenters in one dimensional case with uniform partitions. The convergence rate \begin{document}$ m + 2 $\end{document} is achieved at the center of the element faces in two and three dimensions. The meshes are uniformly partitioned into triangles/tetrahedrons or squares/hexahedrons. We also demonstrate the details of the implementation of the proposed method. The numerical results for all three dimensional cases are presented to illustrate the propositions.

Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium Special Issues
Xuefei He, Kun Wang and Liwei Xu
2020, 28(4): 1503-1528 doi: 10.3934/era.2020079 +[Abstract](543) +[HTML](222) +[PDF](1371.93KB)

In this paper, we consider a kind of efficient finite difference methods (FDMs) for solving the nonlinear Helmholtz equation in the Kerr medium. Firstly, by applying several iteration methods, we linearize the nonlinear Helmholtz equation in several different ways. Then, based on the resulted linearized problem at each iterative step, by rearranging the Taylor expansion and using the ADI method, we deduce a kind of new FDMs, which also provide a route to deal with the problem with discontinuous coefficients.Finally, some numerical results are presented to validate the efficiency of the proposed schemes, and to show that our schemes perform with much higher accuracy and better convergence compared with the classical ones.

Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation Special Issues
Yangrong Li, Shuang Yang and Qiangheng Zhang
2020, 28(4): 1529-1544 doi: 10.3934/era.2020080 +[Abstract](500) +[HTML](207) +[PDF](356.71KB)

We study the random dynamics for the stochastic non-autonomous Kuramoto-Sivashinsky equation in the possibly non-dissipative case. We first prove the existence of a pullback attractor in the Lebesgue space of odd functions, then show that the fiber of the odd pullback attractor semi-converges to a nonempty compact set as the time-parameter goes to minus infinity and finally prove the measurability of the attractor. In a word, we obtain a longtime stable random attractor formed from odd functions. A key tool is the existence of a bridge function between Lebesgue and Sobolev spaces of odd functions.

Global weak solutions for the two-component Novikov equation Special Issues
Cheng He and Changzheng Qu
2020, 28(4): 1545-1562 doi: 10.3934/era.2020081 +[Abstract](584) +[HTML](236) +[PDF](411.13KB)

The two-component Novikov equation is an integrable generalization of the Novikov equation, which has the peaked solitons in the sense of distribution as the Novikov and Camassa-Holm equations. In this paper, we prove the existence of the \begin{document}$ H^1 $\end{document}-weak solution for the two-component Novikov equation by the regular approximation method due to the existence of three conserved densities. The key elements in our approach are some a priori estimates on the approximation solutions.

Gorenstein global dimensions relative to balanced pairs Special Issues
Haiyu Liu, Rongmin Zhu and Yuxian Geng
2020, 28(4): 1563-1571 doi: 10.3934/era.2020082 +[Abstract](500) +[HTML](209) +[PDF](294.33KB)

Let \begin{document}$ \mathcal{G}(\mathcal{X}) $\end{document} and \begin{document}$ \mathcal{G}(\mathcal{Y}) $\end{document} be Gorenstein subcategories induced by an admissible balanced pair \begin{document}$ (\mathcal{X}, \mathcal{Y}) $\end{document} in an abelian category \begin{document}$ \mathcal{A} $\end{document}. In this paper, we establish Gorenstein homological dimensions in terms of these two subcategories and investigate the Gorenstein global dimensions of \begin{document}$ \mathcal{A} $\end{document} induced by the balanced pair \begin{document}$ (\mathcal{X}, \mathcal{Y}) $\end{document}. As a consequence, we give some new characterizations of pure global dimensions and Gorenstein global dimensions of a ring \begin{document}$ R $\end{document}.

A survey of gradient methods for solving nonlinear optimization
Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma and Dijana Mosić
2020, 28(4): 1573-1624 doi: 10.3934/era.2020115 +[Abstract](461) +[HTML](303) +[PDF](716.69KB)

The paper surveys, classifies and investigates theoretically and numerically main classes of line search methods for unconstrained optimization. Quasi-Newton (QN) and conjugate gradient (CG) methods are considered as representative classes of effective numerical methods for solving large-scale unconstrained optimization problems. In this paper, we investigate, classify and compare main QN and CG methods to present a global overview of scientific advances in this field. Some of the most recent trends in this field are presented. A number of numerical experiments is performed with the aim to give an experimental and natural answer regarding the numerical one another comparison of different QN and CG methods.

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