# American Institute of Mathematical Sciences

eISSN:
2688-1594

All Issues

## Electronic Research Archive

### Special issue on PDEs in fluid flow problems

Xiaoming He, Department of Mathematics and Statistics,Missouri University of Science and Technology,400 W. 12th St. Rolla, MO 65401, USA hex@mst.edu

Chunpeng Wang, School of Mathematiccs, Jilin University,Changchun 130012, China wangcp@jlu.edu.cn

Ran Zhang, School of Mathematics, Jilin University,Changchun 130012, China zhangran@jlu.edu.cn

2020, 28(4): 1545-1562 doi: 10.3934/era.2020081 +[Abstract](15)+[HTML](14) +[PDF](411.13KB)
Abstract:

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.

Bin Wang and
2021, 29(1): 1881-1895 doi: 10.3934/era.2020096 +[Abstract](699)+[HTML](298) +[PDF](1246.21KB)
Abstract:

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.

2021, 29(1): 1897-1923 doi: 10.3934/era.2020097 +[Abstract](720)+[HTML](283) +[PDF](430.32KB)
Abstract:

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.

2021, 29(2): 1991-2006 doi: 10.3934/era.2020101 +[Abstract](692)+[HTML](338) +[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): 2063-2075 doi: 10.3934/era.2020105 +[Abstract](588)+[HTML](309) +[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(3): 2223-2247 doi: 10.3934/era.2020113 +[Abstract](699)+[HTML](335) +[PDF](5380.94KB)
Abstract:

We study a continuous data assimilation (CDA) algorithm for a velocity-vorticity formulation of the 2D Navier-Stokes equations in two cases: nudging applied to the velocity and vorticity, and nudging applied to the velocity only. We prove that under a typical finite element spatial discretization and backward Euler temporal discretization, application of CDA preserves the unconditional long-time stability property of the velocity-vorticity method and provides optimal long-time accuracy. These properties hold if nudging is applied only to the velocity, and if nudging is also applied to the vorticity then the optimal long-time accuracy is achieved more rapidly in time. Numerical tests illustrate the theory, and show its effectiveness on an application problem of channel flow past a flat plate.

2021, 29(3): 2417-2444 doi: 10.3934/era.2020122 +[Abstract](519)+[HTML](262) +[PDF](403.32KB)
Abstract:

This paper focuses on two-dimensional continuous subsonic-sonic potential flows in a semi-infinitely long nozzle with a straight lower wall and an upper wall which is convergent at the outlet while straight at the far fields. It is proved that if the variation rate of the cross section of the nozzle is suitably small, there exists a unique continuous subsonic-sonic flows in the nozzle such that the sonic curve intersects the upper wall at a fixed point and the velocity of the flow is along the normal direction at the sonic curve. Furthermore, the sonic curve is free, where the flow is singular in the sense that the flow speed is only Hölder continuous and the flow acceleration blows up. Additionally, the asymptotic behaviors of the flow speed at the far fields is shown.

2021, 29(3): 2489-2516 doi: 10.3934/era.2020126 +[Abstract](536)+[HTML](249) +[PDF](455.77KB)
Abstract:

This paper presents a hybridized weak Galerkin (HWG) finite element method for solving the Brinkman equations. Mathematically, Brinkman equations can model the Stokes and Darcy flows in a unified framework so as to describe the fluid motion in porous media with fractures. Numerical schemes for Brinkman equations, therefore, must be designed to tackle Stokes and Darcy flows at the same time. We demonstrate that HWG is capable of providing very accurate and stable numerical approximations for both Darcy and Stokes. The main features of HWG is that it approximates the differential operators by their weak forms as distributions and it introduces the Lagrange multipliers to relax certain constraints. We establish the optimal order error estimates for HWG solutions of Brinkman equations. We also present a Schur complement formulation of HWG, which reduces the systems' computational complexity significantly. A number of numerical experiments are provided to confirm the theoretical developments.

2021, 29(3): 2517-2532 doi: 10.3934/era.2020127 +[Abstract](490)+[HTML](227) +[PDF](400.6KB)
Abstract:

This article presents a four-field mixed finite element method for Biot's consolidation problems, where the four fields include the displacement, total stress, flux and pressure for the porous medium component of the modeling system. The mixed finite element method involving Raviart-Thomas element is used for the fluid flow equation, while the Crank-Nicolson scheme is employed for the time discretization. The main contribution of this work is the derivation of the optimal order error estimates for semi-discrete and fully-discrete schemes for the unknowns in energy norm or \begin{document}$L^2$\end{document} norm. Numerical experiments are presented to validate the theoretical results.

2021, 29(3): 2533-2552 doi: 10.3934/era.2020128 +[Abstract](487)+[HTML](252) +[PDF](6781.76KB)
Abstract:

An efficient computing method for a target velocity tracking problem of fluid flows is considered. We first adopts the Lagrange multipliers method to obtain the optimality system, and then designs a simple and effective feedback control law based on the relationship between the control \begin{document}${{\boldsymbol f}}$\end{document} and the adjoint variable \begin{document}${{\boldsymbol w}}$\end{document} in the optimality system. We consider a reduced order modeling (ROM) of this problem for real-time computing. In order to improve the existing ROM method, the deep learning technique, which is currently being actively researched, is applied. We review previous research results and some computational results are presented.

2020 Impact Factor: 1.833
5 Year Impact Factor: 1.833