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August 2021 , Volume 29 , Issue 3

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Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations
Matthew Gardner, Adam Larios, Leo G. Rebholz, Duygu Vargun and Camille Zerfas
2021, 29(3): 2223-2247 doi: 10.3934/era.2020113 +[Abstract](648) +[HTML](320) +[PDF](5380.94KB)

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.

Solvability of the matrix equation $ AX^{2} = B $ with semi-tensor product
Jin Wang, Jun-E Feng and Hua-Lin Huang
2021, 29(3): 2249-2267 doi: 10.3934/era.2020114 +[Abstract](593) +[HTML](306) +[PDF](418.13KB)

We investigate the solvability of the matrix equation \begin{document}$ AX^{2} = B $\end{document} in which the multiplication is the semi-tensor product. Then compatible conditions on the matrices \begin{document}$ A $\end{document} and \begin{document}$ B $\end{document} are established in each case and necessary and sufficient condition for the solvability is discussed. In addition, concrete methods of solving the equation are provided.

Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity
Shao-Xia Qiao and Li-Jun Du
2021, 29(3): 2269-2291 doi: 10.3934/era.2020116 +[Abstract](501) +[HTML](226) +[PDF](367.09KB)

This paper is concerned with the nonlocal dispersal equations with inhomogeneous bistable nonlinearity in one dimension. The varying nonlinearity consists of two spatially independent bistable nonlinearities, which are connected by a compact transition region. We establish the existence of a unique entire solution connecting two traveling wave solutions pertaining to the different nonlinearities. In particular, we use a "squeezing" technique to show that the traveling wave of the equation with one nonlinearity approaching from infinity, after going through the transition region, converges to the other traveling wave prescribed by the nonlinearity on the other side. Furthermore, we also prove that such an entire solution is Lyapunov stable.

On projective threefolds of general type with small positive geometric genus
Meng Chen, Yong Hu and Matteo Penegini
2021, 29(3): 2293-2323 doi: 10.3934/era.2020117 +[Abstract](489) +[HTML](266) +[PDF](483.45KB)

In this paper we study the pluricanonical maps of minimal projective 3-folds of general type with geometric genus \begin{document}$ 1 $\end{document}, \begin{document}$ 2 $\end{document} and \begin{document}$ 3 $\end{document}. We go in the direction pioneered by Enriques and Bombieri, and other authors, pinning down, for low projective genus, a finite list of exceptions to the birationality of some pluricanonical map. In particular, apart from a finite list of weighted baskets, we prove the birationality of \begin{document}$ \varphi_{16} $\end{document}, \begin{document}$ \varphi_{6} $\end{document} and \begin{document}$ \varphi_{5} $\end{document} respectively.

Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment
Hai-Feng Huo, Shi-Ke Hu and Hong Xiang
2021, 29(3): 2325-2358 doi: 10.3934/era.2020118 +[Abstract](577) +[HTML](291) +[PDF](726.08KB)

A reaction-diffusion SEIR model, including the self-protection for susceptible individuals, treatments for infectious individuals and constant recruitment, is introduced. The existence of traveling wave solution, which is determined by the basic reproduction number \begin{document}$ R_0 $\end{document} and wave speed \begin{document}$ c, $\end{document} is firstly proved as \begin{document}$ R_0>1 $\end{document} and \begin{document}$ c\geq c^* $\end{document} via the Schauder fixed point theorem, where \begin{document}$ c^* $\end{document} is minimal wave speed. Asymptotic behavior of traveling wave solution at infinity is also proved by applying the Lyapunov functional. Furthermore, when \begin{document}$ R_0\leq1 $\end{document} or \begin{document}$ R_0>1 $\end{document} with \begin{document}$ c\in(0,\ c^*), $\end{document} we derive the non-existence of traveling wave solution with utilizing two-sides Laplace transform. We take advantage of numerical simulations to indicate the existence of traveling wave, and show that self-protection and treatment can reduce the spread speed at last.

Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case
Yongxiu Shi and Haitao Wan
2021, 29(3): 2359-2373 doi: 10.3934/era.2020119 +[Abstract](509) +[HTML](239) +[PDF](370.32KB)

This paper is considered with the quasilinear elliptic equation \begin{document}$ \Delta_{p}u = b(x)f(u),\,u(x)>0,\,x\in\Omega, $\end{document} where \begin{document}$ \Omega $\end{document} is an exterior domain with compact smooth boundary, \begin{document}$ b\in \rm C(\Omega) $\end{document} is non-negative in \begin{document}$ \Omega $\end{document} and may be singular or vanish on \begin{document}$ \partial\Omega $\end{document}, \begin{document}$ f\in C[0, \infty) $\end{document} is positive and increasing on \begin{document}$ (0, \infty) $\end{document} which satisfies a generalized Keller-Osserman condition and is regularly varying at infinity with critical index \begin{document}$ p-1 $\end{document}. By structuring a new comparison function, we establish the new asymptotic behavior of large solutions to the above equation in the exterior domain. We find that the lower term of \begin{document}$ f $\end{document} has an important influence on the asymptotic behavior of large solutions. And then we further establish the uniqueness of such solutions.

A conforming discontinuous Galerkin finite element method on rectangular partitions
Yue Feng, Yujie Liu, Ruishu Wang and Shangyou Zhang
2021, 29(3): 2375-2389 doi: 10.3934/era.2020120 +[Abstract](536) +[HTML](251) +[PDF](316.24KB)

This article presents a conforming discontinuous Galerkin (conforming DG) scheme for second order elliptic equations on rectangular partitions. The new method is based on DG finite element space and uses a weak gradient arising from local Raviart Thomas space for gradient approximations. By using the weak gradient and enforcing inter-element continuity strongly, the scheme maintains the simple formulation of conforming finite element method while have the flexibility of using discontinuous approximations. Hence, the programming complexity of this new conforming DG scheme is significantly reduced compared to other existing DG methods. Error estimates of optimal order are established for the corresponding conforming DG approximations in various discrete Sobolev norms. Numerical results are presented to confirm the developed convergence theory.

On recent progress of single-realization recoveries of random Schrödinger systems
Shiqi Ma
2021, 29(3): 2391-2415 doi: 10.3934/era.2020121 +[Abstract](500) +[HTML](249) +[PDF](535.93KB)

We consider the recovery of some statistical quantities by using the near-field or far-field data in quantum scattering generated under a single realization of the randomness. We survey the recent main progress in the literature and point out the similarity among the existing results. The methodologies in the reformulation of the forward problems are also investigated. We consider two separate cases of using the near-field and far-field data, and discuss the key ideas of obtaining some crucial asymptotic estimates. We pay special attention on the use of the theory of pseudodifferential operators and microlocal analysis needed in the proofs.

Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle
Mingjun Zhou and Jingxue Yin
2021, 29(3): 2417-2444 doi: 10.3934/era.2020122 +[Abstract](478) +[HTML](245) +[PDF](403.32KB)

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.

Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $
Wenjun Liu, Yukun Xiao and Xiaoqing Yue
2021, 29(3): 2445-2456 doi: 10.3934/era.2020123 +[Abstract](558) +[HTML](253) +[PDF](330.24KB)

We study a family of non-simple Lie conformal algebras \begin{document}$ \mathcal{W}(a,b,r) $\end{document} (\begin{document}$ a,b,r\in {\mathbb{C}} $\end{document}) of rank three with free \begin{document}$ {\mathbb{C}}[{\partial}] $\end{document}-basis \begin{document}$ \{L, W,Y\} $\end{document} and relations \begin{document}$ [L_{\lambda} L] = ({\partial}+2{\lambda})L,\ [L_{\lambda} W] = ({\partial}+ a{\lambda} +b)W,\ [L_{\lambda} Y] = ({\partial}+{\lambda})Y,\ [Y_{\lambda} W] = rW $\end{document} and \begin{document}$ [Y_{\lambda} Y] = [W_{\lambda} W] = 0 $\end{document}. In this paper, we investigate the irreducibility of all free nontrivial \begin{document}$ \mathcal{W}(a,b,r) $\end{document}-modules of rank one over \begin{document}$ {\mathbb{C}}[{\partial}] $\end{document} and classify all finite irreducible conformal modules over \begin{document}$ \mathcal{W}(a,b,r) $\end{document}.

A generalization on derivations of Lie algebras
Hongliang Chang, Yin Chen and Runxuan Zhang
2021, 29(3): 2457-2473 doi: 10.3934/era.2020124 +[Abstract](587) +[HTML](224) +[PDF](414.98KB)

We initiate a study on a range of new generalized derivations of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero. This new generalization of derivations has an analogue in the theory of associative prime rings and unites many well-known generalized derivations that have already appeared extensively in the study of Lie algebras and other nonassociative algebras. After exploiting fundamental properties, we introduce and analyze their interiors, especially focusing on the rationality of the corresponding Hilbert series. Applying techniques in computational ideal theory we develop an approach to explicitly compute these new generalized derivations for the three-dimensional special linear Lie algebra over the complex field.

The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations
Li Cai and Fubao Zhang
2021, 29(3): 2475-2488 doi: 10.3934/era.2020125 +[Abstract](472) +[HTML](213) +[PDF](350.26KB)

In this paper, we study the following Schrödinger-Poisson equations with double critical exponents:

where \begin{document}$ \Omega $\end{document} is a bounded domain in \begin{document}$ \mathbb{R}^3 $\end{document} with Lipschitz boundary, \begin{document}$ \lambda $\end{document} is a real parameter satisfying suitable conditions. Using variational methods, we show the existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson equations.

Hybridized weak Galerkin finite element methods for Brinkman equations
Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang and Zhongshu Zhao
2021, 29(3): 2489-2516 doi: 10.3934/era.2020126 +[Abstract](496) +[HTML](235) +[PDF](455.77KB)

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.

A four-field mixed finite element method for Biot's consolidation problems
Wenya Qi, Padmanabhan Seshaiyer and Junping Wang
2021, 29(3): 2517-2532 doi: 10.3934/era.2020127 +[Abstract](459) +[HTML](215) +[PDF](400.6KB)

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.

Efficient computations for linear feedback control problems for target velocity matching of Navier-Stokes flows via POD and LSTM-ROM
Hyung-Chun Lee
2021, 29(3): 2533-2552 doi: 10.3934/era.2020128 +[Abstract](454) +[HTML](236) +[PDF](6781.76KB)

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.

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