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2688-1594

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August 2021 , Volume 29 , Issue 3

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**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.

*+*[Abstract](593)

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**Abstract:**

We investigate the solvability of the matrix equation

*+*[Abstract](501)

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**Abstract:**

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.

*+*[Abstract](489)

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**Abstract:**

In this paper we study the pluricanonical maps of minimal projective 3-folds of general type with geometric genus

*+*[Abstract](577)

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**Abstract:**

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

*+*[Abstract](509)

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**Abstract:**

This paper is considered with the quasilinear elliptic equation

*+*[Abstract](536)

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**Abstract:**

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.

*+*[Abstract](500)

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**Abstract:**

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.

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**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.

*+*[Abstract](558)

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**Abstract:**

We study a family of non-simple Lie conformal algebras

*+*[Abstract](587)

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**Abstract:**

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.

*+*[Abstract](472)

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**Abstract:**

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

where

*+*[Abstract](496)

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**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.

*+*[Abstract](459)

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**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

*+*[Abstract](454)

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**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

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