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December 2021 , Volume 29 , Issue 6

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Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients
Vo Van Au, Jagdev Singh and Anh Tuan Nguyen
2021, 29(6): 3581-3607 doi: 10.3934/era.2021052 +[Abstract](700) +[HTML](192) +[PDF](489.64KB)
Abstract:

The semi-linear problem of a fractional diffusion equation with the Caputo-like counterpart of a hyper-Bessel differential is considered. The results on existence, uniqueness and regularity estimates (local well-posedness) of the solutions are established in the case of linear source and the source functions that satisfy the globally Lipschitz conditions. Moreover, we prove that the problem exists a unique positive solution. In addition, the unique continuation of solutions and a finite-time blow-up are proposed with the reaction terms are logarithmic functions.

A stabilizer free WG method for the Stokes equations with order two superconvergence on polytopal mesh Special Issues
Xiu Ye and Shangyou Zhang
2021, 29(6): 3609-3627 doi: 10.3934/era.2021053 +[Abstract](381) +[HTML](205) +[PDF](5398.53KB)
Abstract:

A stabilizer free WG method is introduced for the Stokes equations with superconvergence on polytopal mesh in primary velocity-pressure formulation. Convergence rates two order higher than the optimal-order for velocity of the WG approximation is proved in both an energy norm and the \begin{document}$ L^2 $\end{document} norm. Optimal order error estimate for pressure in the \begin{document}$ L^2 $\end{document} norm is also established. The numerical examples cover low and high order approximations, and 2D and 3D cases.

A simple virtual element-based flux recovery on quadtree Special Issues
Shuhao Cao
2021, 29(6): 3629-3647 doi: 10.3934/era.2021054 +[Abstract](397) +[HTML](221) +[PDF](825.67KB)
Abstract:

In this paper, we introduce a simple local flux recovery for \begin{document}$ \mathcal{Q}_k $\end{document} finite element of a scalar coefficient diffusion equation on quadtree meshes, with no restriction on the irregularities of hanging nodes. The construction requires no specific ad hoc tweaking for hanging nodes on \begin{document}$ l $\end{document}-irregular (\begin{document}$ l\geq 2 $\end{document}) meshes thanks to the adoption of virtual element families. The rectangular elements with hanging nodes are treated as polygons as in the flux recovery context. An efficient a posteriori error estimator is then constructed based on the recovered flux, and its reliability is proved under common assumptions, both of which are further verified in numerics.

On blowup of secant varieties of curves Special Issues
Lawrence Ein, Wenbo Niu and Jinhyung Park
2021, 29(6): 3649-3654 doi: 10.3934/era.2021055 +[Abstract](421) +[HTML](237) +[PDF](311.11KB)
Abstract:

In this paper, we show that for a nonsingular projective curve and a positive integer \begin{document}$ k $\end{document}, the \begin{document}$ k $\end{document}-th secant bundle is the blowup of the \begin{document}$ k $\end{document}-th secant variety along the \begin{document}$ (k-1) $\end{document}-th secant variety. This answers a question raised in the recent paper of the authors on secant varieties of curves.

Regularity of Wong-Zakai approximation for non-autonomous stochastic quasi-linear parabolic equation on $ {\mathbb{R}}^N $
Guifen Liu and Wenqiang Zhao
2021, 29(6): 3655-3686 doi: 10.3934/era.2021056 +[Abstract](377) +[HTML](176) +[PDF](463.03KB)
Abstract:

In this paper, we investigate a non-autonomous stochastic quasi-linear parabolic equation driven by multiplicative white noise by a Wong-Zakai approximation technique. The convergence of the solutions of quasi-linear parabolic equations driven by a family of processes with stationary increment to that of stochastic differential equation with white noise is obtained in the topology of \begin{document}$ L^2( {\mathbb{R}}^N) $\end{document} space. We establish the Wong-Zakai approximations of solutions in \begin{document}$ L^l( {\mathbb{R}}^N) $\end{document} for arbitrary \begin{document}$ l\geq q $\end{document} in the sense of upper semi-continuity of their random attractors, where \begin{document}$ q $\end{document} is the growth exponent of the nonlinearity. The \begin{document}$ L^l $\end{document}-pre-compactness of attractors is proved by using the truncation estimate in \begin{document}$ L^q $\end{document} and the higher-order bound of solutions.

Semilinear pseudo-parabolic equations on manifolds with conical singularities
Yitian Wang, Xiaoping Liu and Yuxuan Chen
2021, 29(6): 3687-3720 doi: 10.3934/era.2021057 +[Abstract](421) +[HTML](180) +[PDF](511.79KB)
Abstract:

This paper studies the well-posedness of the semilinear pseudo-parabolic equations on manifolds with conical degeneration. By employing the Galerkin method and performing energy estimates, we first establish the local-in-time well-posedness of the solution. Moreover, to reveal the relationship between the initial datum and the global-in-time well-posedness of the solution we divide the initial datum into three classes by the potential well depth, i.e., the sub-critical initial energy level, the critical initial energy level and the sup-critical initial energy level (included in the arbitrary high initial energy case), and finally we give an affirmative answer to the question whether the solution exists globally or not. For the sub-critical and critical initial energy, thanks to the potential well theory, we not only obtain the invariant manifolds, global existence and asymptotic behavior of solutions, but also prove the finite time blow up of solutions and estimate the lower bound the of blowup time. For the sup-critical case, we show the assumptions for initial datum which cause the finite time blowup of the solution, realized by introducing a new auxiliary function. Additionally, we also provide some results concerning the estimates of the upper bound of the blowup time in the sup-critical initial energy.

Multidimensional stability of pyramidal traveling fronts in degenerate Fisher-KPP monostable and combustion equations
Denghui Wu and Zhen-Hui Bu
2021, 29(6): 3721-3740 doi: 10.3934/era.2021058 +[Abstract](421) +[HTML](170) +[PDF](436.37KB)
Abstract:

In this paper, multidimensional stability of pyramidal traveling fronts are studied to the reaction-diffusion equations with degenerate Fisher-KPP monostable and combustion nonlinearities. By constructing supersolutions and subsolutions coupled with the comparison principle, we firstly prove that under any initial perturbation (possibly large) decaying at space infinity, the three-dimensional pyramidal traveling fronts are asymptotically stable in weighted \begin{document}$ L^{\infty} $\end{document} spaces on \begin{document}$ \mathbb{R}^{n}\; (n\geq4) $\end{document}. Secondly, we show that under general bounded perturbations (even very small), the pyramidal traveling fronts are not asymptotically stable by constructing a solution which oscillates permanently between two three-dimensional pyramidal traveling fronts on \begin{document}$ \mathbb{R}^{4} $\end{document}.

Constructions of three kinds of Bihom-superalgebras Special Issues
Ying Hou and Liangyun Chen
2021, 29(6): 3741-3760 doi: 10.3934/era.2021059 +[Abstract](319) +[HTML](174) +[PDF](401.3KB)
Abstract:

The purpose of this paper is to study the constructions between Bihom-alternative superalgebras and Bihom-Malcev superalgebras and Bihom-Jordan superalgebras. First, we explain in detail that every regular Bihom-alternative superalgebra could be Bihom-Malcev-admissible superalgebra or Bihom-Jordan-admissible superalgebra. Next, the bimodules and \begin{document}$ T^*_\theta $\end{document}-extensions of Bihom-alternative superalgebras are also discussed as properties of Bihom-alternative superalgebras.

Uniqueness and nondegeneracy of positive solutions to an elliptic system in ecology Special Issues
Zaizheng Li and Zhitao Zhang
2021, 29(6): 3761-3774 doi: 10.3934/era.2021060 +[Abstract](349) +[HTML](178) +[PDF](382.84KB)
Abstract:

In this paper, we study the follwing important elliptic system which arises from the Lotka-Volterra ecological model in \begin{document}$ \mathbb{R}^N $\end{document}

where \begin{document}$ N\leq 5, $\end{document} \begin{document}$ \lambda, \mu_1, \mu_2 $\end{document} are positive constants, \begin{document}$ \beta\geq 0 $\end{document} is a coupling constant. Firstly, we prove the uniqueness of positive solutions under general conditions, then we show the nondegeneracy of the positive solution and the degeneracy of semi-trivial solutions. Finally, we give a complete classification of positive solutions when \begin{document}$ \mu_1 = \mu_2 = \beta. $\end{document}

Weighted fourth order elliptic problems in the unit ball Special Issues
Zongming Guo and Fangshu Wan
2021, 29(6): 3775-3803 doi: 10.3934/era.2021061 +[Abstract](341) +[HTML](179) +[PDF](438.28KB)
Abstract:

Existence and uniqueness of positive radial solutions of some weighted fourth order elliptic Navier and Dirichlet problems in the unit ball \begin{document}$ B $\end{document} are studied. The weights can be singular at \begin{document}$ x = 0 \in B $\end{document}. Existence of positive radial solutions of the problems is obtained via variational methods in the weighted Sobolev spaces. To obtain the uniqueness results, we need to know exactly the asymptotic behavior of the solutions at the singular point \begin{document}$ x = 0 $\end{document}.

Recent progress on stable and finite Morse index solutions of semilinear elliptic equations Special Issues
Kelei Wang
2021, 29(6): 3805-3816 doi: 10.3934/era.2021062 +[Abstract](444) +[HTML](246) +[PDF](332.06KB)
Abstract:

We discuss some recent results (mostly from the last decade) on stable and finite Morse index solutions of semilinear elliptic equations, where Norman Dancer has made many important contributions. Some open questions in this direction are also discussed.

On mathematical analysis of complex fluids in active hydrodynamics Special Issues
Yazhou Chen, Dehua Wang and Rongfang Zhang
2021, 29(6): 3817-3832 doi: 10.3934/era.2021063 +[Abstract](366) +[HTML](184) +[PDF](353.68KB)
Abstract:

This is a survey article for this special issue providing a review of the recent results in the mathematical analysis of active hydrodynamics. Both the incompressible and compressible models are discussed for the active liquid crystals in the Landau-de Gennes Q-tensor framework. The mathematical results on the weak solutions, regularity, and weak-strong uniqueness are presented for the incompressible flows. The global existence of weak solution to the compressible flows is recalled. Other related results on the inhomogeneous flows, incompressible limits, and stochastic analysis are also reviewed.

Initial boundary value problem of a class of mixed pseudo-parabolic Kirchhoff equations
Yang Cao and Qiuting Zhao
2021, 29(6): 3833-3851 doi: 10.3934/era.2021064 +[Abstract](276) +[HTML](144) +[PDF](378.74KB)
Abstract:

In this paper, we consider the initial boundary value problem for a mixed pseudo-parabolic Kirchhoff equation. Due to the comparison principle being invalid, we use the potential well method to give a threshold result of global existence and non-existence for the sign-changing weak solutions with initial energy \begin{document}$ J(u_0)\leq d $\end{document}. When the initial energy \begin{document}$ J(u_0)>d $\end{document}, we find another criterion for the vanishing solution and blow-up solution. Our interest also lies in the discussion of the exponential decay rate of the global solution and life span of the blow-up solution.

A computable formula for the class number of the imaginary quadratic field $ \mathbb Q(\sqrt{-p}), \ p = 4n-1 $
Jorge Garcia Villeda
2021, 29(6): 3853-3865 doi: 10.3934/era.2021065 +[Abstract](377) +[HTML](140) +[PDF](321.17KB)
Abstract:

Using elementary methods, we count the quadratic residues of a prime number of the form \begin{document}$ p = 4n-1 $\end{document} in a manner that has not been explored before. The simplicity of the pattern found leads to a novel formula for the class number \begin{document}$ h $\end{document} of the imaginary quadratic field \begin{document}$ \mathbb Q(\sqrt{-p}). $\end{document} Such formula is computable and does not rely on the Dirichlet character or the Kronecker symbol at all. Examples are provided and formulas for the sum of the quadratic residues are also found.

A class of fourth-order hyperbolic equations with strongly damped and nonlinear logarithmic terms
Yi Cheng and Ying Chu
2021, 29(6): 3867-3887 doi: 10.3934/era.2021066 +[Abstract](376) +[HTML](147) +[PDF](413.81KB)
Abstract:

In this paper, we study a class of hyperbolic equations of the fourth order with strong damping and logarithmic source terms. Firstly, we prove the local existence of the weak solution by using the contraction mapping principle. Secondly, in the potential well framework, the global existence of weak solutions and the energy decay estimate are obtained. Finally, we give the blow up result of the solution at a finite time under the subcritical initial energy.

Optimal time-decay rates of the compressible Navier–Stokes–Poisson system in $ \mathbb R^3 $
Guochun Wu, Han Wang and Yinghui Zhang
2021, 29(6): 3889-3908 doi: 10.3934/era.2021067 +[Abstract](337) +[HTML](166) +[PDF](388.28KB)
Abstract:

We are concerned with the Cauchy problem of the 3D compressible Navier–Stokes–Poisson system. Compared to the previous related works, the main purpose of this paper is two–fold: First, we prove the optimal decay rates of the higher spatial derivatives of the solution. Second, we investigate the influences of the electric field on the qualitative behaviors of solution. More precisely, we show that the density and high frequency part of the momentum of the compressible Navier–Stokes–Poisson system have the same \begin{document}$ L^2 $\end{document} decay rates as the compressible Navier–Stokes equation and heat equation, but the \begin{document}$ L^2 $\end{document} decay rate of the momentum is slower due to the effect of the electric field.

The algebraic classification of nilpotent commutative algebras Special Issues
Doston Jumaniyozov, Ivan Kaygorodov and Abror Khudoyberdiyev
2021, 29(6): 3909-3993 doi: 10.3934/era.2021068 +[Abstract](323) +[HTML](138) +[PDF](665.92KB)
Abstract:

This paper is devoted to the complete algebraic classification of complex \begin{document}$ 5 $\end{document}-dimensional nilpotent commutative algebras. Our method of classification is based on the standard method of classification of central extensions of smaller nilpotent commutative algebras and the recently obtained classification of complex \begin{document}$ 5 $\end{document}-dimensional nilpotent commutative \begin{document}$ \mathfrak{CD} $\end{document}-algebras.

Variations on Lyapunov's stability criterion and periodic prey-predator systems Special Issues
Rafael Ortega
2021, 29(6): 3995-4008 doi: 10.3934/era.2021069 +[Abstract](320) +[HTML](154) +[PDF](334.88KB)
Abstract:

A classical stability criterion for Hill's equation is extended to more general families of periodic two-dimensional linear systems. The results are motivated by the study of mechanical vibrations with friction and periodic prey-predator systems.

Local well-posedness of solutions to the boundary layer equations for compressible two-fluid flow Special Issues
Long Fan, Cheng-Jie Liu and Lizhi Ruan
2021, 29(6): 4009-4050 doi: 10.3934/era.2021070 +[Abstract](317) +[HTML](139) +[PDF](501.24KB)
Abstract:

In this paper, we consider the two-dimensional (2D) two-fluid boundary layer system, which is a hyperbolic-degenerate parabolic-elliptic coupling system derived from the compressible isentropic two-fluid flow equations with nonslip boundary condition for the velocity. The local existence and uniqueness is established in weighted Sobolev spaces under the monotonicity assumption on tangential velocity along normal direction based on a nonlinear energy method by employing a nonlinear cancelation technic introduced in [R. Alexandre, Y.-G. Wang, C.-J. Xu and T. Yang, J. Amer. Math. Soc., 28 (2015), 745-784; N. Masmoudi and T.K. Wong, Comm. Pure Appl. Math., 68(2015), 1683-1741] and developed in [C.-J. Liu, F. Xie and T. Yang, Comm. Pure Appl. Math., 72(2019), 63-121].

Stabilizing effect of elasticity on the motion of viscoelastic/elastic fluids Special Issues
Fei Jiang
2021, 29(6): 4051-4074 doi: 10.3934/era.2021071 +[Abstract](311) +[HTML](136) +[PDF](414.7KB)
Abstract:

It is well-known that viscoelasticity is a material property that exhibits both viscous and elastic characteristics with deformation. In particular, an elastic fluid strains when it is stretched and quickly returns to its original state once the stress is removed. In this review, we first introduce some mathematical results, which exhibit the stabilizing effect of elasticity on the motion of viscoelastic fluids. Then we further briefly introduce similar stabilizing effect in the elastic fluids.

Nonexistence of entire positive solutions for conformal Hessian quotient inequalities
Feida Jiang, Xi Chen and Juhua Shi
2021, 29(6): 4075-4086 doi: 10.3934/era.2021072 +[Abstract](250) +[HTML](110) +[PDF](328.43KB)
Abstract:

In this paper, we consider the nonexistence problem for conformal Hessian quotient inequalities in \begin{document}$ \mathbb{R}^n $\end{document}. We prove the nonexistence results of entire positive \begin{document}$ k $\end{document}-admissible solution to a conformal Hessian quotient inequality, and entire \begin{document}$ (k, k') $\end{document}-admissible solution pair to a system of Hessian quotient inequalities, respectively. We use the contradiction method combining with the integration by parts, suitable choices of test functions, Taylor's expansion and Maclaurin's inequality for Hessian quotient operators.

Non-global solution for visco-elastic dynamical system with nonlinear source term in control problem
Xiaoqiang Dai and Wenke Li
2021, 29(6): 4087-4098 doi: 10.3934/era.2021073 +[Abstract](219) +[HTML](105) +[PDF](320.73KB)
Abstract:

In this paper, we study the initial boundary value problem of the visco-elastic dynamical system with the nonlinear source term in control system. By variational arguments and an improved convexity method, we prove the global nonexistence of solution, and we also give a sharp condition for global existence and nonexistence.

Some estimates of virtual element methods for fourth order problems Special Issues
Qingguang Guan
2021, 29(6): 4099-4118 doi: 10.3934/era.2021074 +[Abstract](247) +[HTML](113) +[PDF](354.25KB)
Abstract:

In this paper, we employ the techniques developed for second order operators to obtain the new estimates of Virtual Element Method for fourth order operators. The analysis bases on elements with proper shape regularity. Estimates for projection and interpolation operators are derived. Also, the biharmonic problem is solved by Virtual Element Method, optimal error estimates were obtained. Our choice of the discrete form for the right hand side function relaxes the regularity requirement in previous work and the error estimates between exact solutions and the computable numerical solutions were proved.

Accelerating reinforcement learning with a Directional-Gaussian-Smoothing evolution strategy Special Issues
Jiaxin Zhang, Hoang Tran and Guannan Zhang
2021, 29(6): 4119-4135 doi: 10.3934/era.2021075 +[Abstract](295) +[HTML](113) +[PDF](2281.58KB)
Abstract:

The objective of reinforcement learning (RL) is to find an optimal strategy for solving a dynamical control problem. Evolution strategy (ES) has been shown great promise in many challenging reinforcement learning (RL) tasks, where the underlying dynamical system is only accessible as a black box such that adjoint methods cannot be used. However, existing ES methods have two limitations that hinder its applicability in RL. First, most existing methods rely on Monte Carlo based gradient estimators to generate search directions. Due to low accuracy of Monte Carlo estimators, the RL training suffers from slow convergence and requires more iterations to reach the optimal solution. Second, the landscape of the reward function can be deceptive and may contain many local maxima, causing ES algorithms to prematurely converge and be unable to explore other parts of the parameter space with potentially greater rewards. In this work, we employ a Directional Gaussian Smoothing Evolutionary Strategy (DGS-ES) to accelerate RL training, which is well-suited to address these two challenges with its ability to (i) provide gradient estimates with high accuracy, and (ii) find nonlocal search direction which lays stress on large-scale variation of the reward function and disregards local fluctuation. Through several benchmark RL tasks demonstrated herein, we show that the DGS-ES method is highly scalable, possesses superior wall-clock time, and achieves competitive reward scores to other popular policy gradient and ES approaches.

Pullback dynamics of a 3D modified Navier-Stokes equations with double delays
Pan Zhang, Lan Huang, Rui Lu and Xin-Guang Yang
2021, 29(6): 4137-4157 doi: 10.3934/era.2021076 +[Abstract](185) +[HTML](86) +[PDF](392.68KB)
Abstract:

This paper is concerned with the tempered pullback dynamics for a 3D modified Navier-Stokes equations with double time-delays, which includes delays on external force and convective terms respectively. Based on the property of monotone operator and some suitable hypotheses on the external forces, the existence and uniqueness of weak solutions can be shown in an appropriate functional Banach space. By using the energy equation technique and weak convergence method to achieve asymptotic compactness for the process, the existence of minimal family of pullback attractors has also been derived.

Global dynamics of some system of second-order difference equations
Tran Hong Thai, Nguyen Anh Dai and Pham Tuan Anh
2021, 29(6): 4159-4175 doi: 10.3934/era.2021077 +[Abstract](209) +[HTML](107) +[PDF](664.58KB)
Abstract:

In this paper, we study the boundedness and persistence of positive solution, existence of invariant rectangle, local and global behavior, and rate of convergence of positive solutions of the following systems of exponential difference equations

where the parameters \begin{document}$ \alpha_i,\ \beta_i,\ \gamma_i $\end{document} for \begin{document}$ i \in \{1,2\} $\end{document} and the initial conditions \begin{document}$ x_{-1}, x_0, y_{-1}, y_0 $\end{document} are positive real numbers. Some numerical example are given to illustrate our theoretical results.

On the existence of solutions for the Frenkel-Kontorova models on quasi-crystals Special Issues
Jianxing Du and Xifeng Su
2021, 29(6): 4177-4198 doi: 10.3934/era.2021078 +[Abstract](188) +[HTML](86) +[PDF](685.78KB)
Abstract:

This article focuses on recent investigations on equilibria of the Frenkel-Kontorova models subjected to potentials generated by quasi-crystals.

We present a specific one-dimensional model with an explicit potential driven by the Fibonacci quasi-crystal. For a given positive number \begin{document}$ \theta $\end{document}, we show that there are multiple equilibria with rotation number \begin{document}$ \theta $\end{document}, e.g., a minimal configuration and a non-minimal equilibrium configuration. Some numerical experiments verifying the existence of such equilibria are provided.

Path-connectedness in global bifurcation theory Special Issues
J. F. Toland
2021, 29(6): 4199-4213 doi: 10.3934/era.2021079 +[Abstract](186) +[HTML](77) +[PDF](359.03KB)
Abstract:

A celebrated result in bifurcation theory is that, when the operators involved are compact, global connected sets of non-trivial solutions bifurcate from trivial solutions at non-zero eigenvalues of odd algebraic multiplicity of the linearized problem. This paper presents a simple example in which the hypotheses of the global bifurcation theorem are satisfied, yet all the path-connected components of the connected sets that bifurcate are singletons. Another example shows that even when the operators are everywhere infinitely differentiable and classical bifurcation occurs locally at a simple eigenvalue, the global continua may not be path-connected away from the bifurcation point. A third example shows that the non-trivial solutions which bifurcate at non-zero eigenvalues, irrespective of multiplicity when the problem has gradient structure, may not be connected and may not contain any paths except singletons.

On the number of critical points of solutions of semilinear elliptic equations Special Issues
Massimo Grossi
2021, 29(6): 4215-4228 doi: 10.3934/era.2021080 +[Abstract](207) +[HTML](110) +[PDF](918.14KB)
Abstract:

In this survey we discuss old and new results on the number of critical points of solutions of the problem

where \begin{document}$ \Omega\subset \mathbb{R}^N $\end{document} with \begin{document}$ N\ge2 $\end{document} is a smooth bounded domain. Both cases where \begin{document}$ u $\end{document} is a positive or nodal solution will be considered.

Planar vortices in a bounded domain with a hole Special Issues
Shusen Yan and Weilin Yu
2021, 29(6): 4229-4241 doi: 10.3934/era.2021081 +[Abstract](188) +[HTML](97) +[PDF](386.58KB)
Abstract:

In this paper, we consider the inviscid, incompressible planar flows in a bounded domain with a hole and construct stationary classical solutions with single vortex core, which is closed to the hole. This is carried out by constructing solutions to the following semilinear elliptic problem

where \begin{document}$ p>1 $\end{document}, \begin{document}$ \kappa $\end{document} is a positive constant, \begin{document}$ \rho_\lambda $\end{document} is a constant, depending on \begin{document}$ \lambda $\end{document}, \begin{document}$ \Omega = \Omega_0\setminus \bar{O}_0 $\end{document} and \begin{document}$ \Omega_0 $\end{document}, \begin{document}$ O_0 $\end{document} are two planar bounded simply-connected domains. We show that under the assumption \begin{document}$ (\ln\lambda)^\sigma\leq\rho_\lambda\leq (\ln\lambda)^{1-\sigma} $\end{document} for some \begin{document}$ \sigma>0 $\end{document} small, (1) has a solution \begin{document}$ \psi_\lambda $\end{document}, whose vorticity set \begin{document}$ \{y\in \Omega:\, \psi(y)-\kappa+\rho_\lambda\eta(y)>0\} $\end{document} shrinks to the boundary of the hole as \begin{document}$ \lambda\to +\infty $\end{document}.

Well-posedness in Sobolev spaces of the two-dimensional MHD boundary layer equations without viscosity Special Issues
Wei-Xi Li and Rui Xu
2021, 29(6): 4243-4255 doi: 10.3934/era.2021082 +[Abstract](202) +[HTML](98) +[PDF](368.62KB)
Abstract:

We consider the two-dimensional MHD Boundary layer system without hydrodynamic viscosity, and establish the existence and uniqueness of solutions in Sobolev spaces under the assumption that the tangential component of magnetic fields dominates. This gives a complement to the previous works of Liu-Xie-Yang [Comm. Pure Appl. Math. 72 (2019)] and Liu-Wang-Xie-Yang [J. Funct. Anal. 279 (2020)], where the well-posedness theory was established for the MHD boundary layer systems with both viscosity and resistivity and with viscosity only, respectively. We use the pseudo-differential calculation, to overcome a new difficulty arising from the treatment of boundary integrals due to the absence of the diffusion property for the velocity.

Congruences for sixth order mock theta functions $ \lambda(q) $ and $ \rho(q) $
Harman Kaur and Meenakshi Rana
2021, 29(6): 4257-4268 doi: 10.3934/era.2021084 +[Abstract](167) +[HTML](46) +[PDF](301.81KB)
Abstract:

Ramanujan introduced sixth order mock theta functions \begin{document}$ \lambda(q) $\end{document} and \begin{document}$ \rho(q) $\end{document} defined as:

listed in the Lost Notebook. In this paper, we present some Ramanujan-like congruences and also find their infinite families modulo 12 for the coefficients of mock theta functions mentioned above.

Optimal control for the coupled chemotaxis-fluid models in two space dimensions
Yunfei Yuan and Changchun Liu
2021, 29(6): 4269-4296 doi: 10.3934/era.2021085 +[Abstract](206) +[HTML](73) +[PDF](389.03KB)
Abstract:

This paper deals with a distributed optimal control problem to the coupled chemotaxis-fluid models. We first explore the global-in-time existence and uniqueness of a strong solution. Then, we define the cost functional and establish the existence of Lagrange multipliers. Finally, we derive some extra regularity for the Lagrange multiplier.

Global dynamics in a competitive two-species and two-stimuli chemotaxis system with chemical signalling loop
Rong Zhang and Liangchen Wang
2021, 29(6): 4297-4314 doi: 10.3934/era.2021086 +[Abstract](139) +[HTML](87) +[PDF](425.7KB)
Abstract:

This paper deals with the following competitive two-species and two-stimuli chemotaxis system with chemical signalling loop

under homogeneous Neumann boundary conditions in a bounded domain \begin{document}$ \Omega\subset \mathbb{R}^n $\end{document} with \begin{document}$ n\geq1 $\end{document}, where the parameters \begin{document}$ a_1,a_2 $\end{document}, \begin{document}$ \chi_1, \chi_2, \chi_3 $\end{document}, \begin{document}$ \mu_1, \mu_2 $\end{document} are positive constants. We first showed some conditions between \begin{document}$ \frac{\chi_1}{\mu_1} $\end{document}, \begin{document}$ \frac{\chi_2}{\mu_2} $\end{document}, \begin{document}$ \frac{\chi_3}{\mu_2} $\end{document} and other ingredients to guarantee boundedness. Moreover, the large time behavior and rates of convergence have also been investigated under some explicit conditions.

Canonical maps of general hypersurfaces in Abelian varieties Special Issues
Fabrizio Catanese and Luca Cesarano
2021, 29(6): 4315-4325 doi: 10.3934/era.2021087 +[Abstract](169) +[HTML](51) +[PDF](342.07KB)
Abstract:

The main theorem of this paper is that, for a general pair \begin{document}$ (A,X) $\end{document} of an (ample) hypersurface \begin{document}$ X $\end{document} in an Abelian Variety \begin{document}$ A $\end{document}, the canonical map \begin{document}$ \Phi_X $\end{document} of \begin{document}$ X $\end{document} is birational onto its image if the polarization given by \begin{document}$ X $\end{document} is not principal (i.e., its Pfaffian \begin{document}$ d $\end{document} is not equal to \begin{document}$ 1 $\end{document}).

We also easily show that, setting \begin{document}$ g = dim (A) $\end{document}, and letting \begin{document}$ d $\end{document} be the Pfaffian of the polarization given by \begin{document}$ X $\end{document}, then if \begin{document}$ X $\end{document} is smooth and

is an embedding, then necessarily we have the inequality \begin{document}$ d \geq g + 1 $\end{document}, equivalent to \begin{document}$ N : = g+d-2 \geq 2 \ dim(X) + 1. $\end{document}

Hence we formulate the following interesting conjecture, motivated by work of the second author: if \begin{document}$ d \geq g + 1, $\end{document} then, for a general pair \begin{document}$ (A,X) $\end{document}, \begin{document}$ \Phi_X $\end{document} is an embedding.

Yamabe systems and optimal partitions on manifolds with symmetries Special Issues
Mónica Clapp and Angela Pistoia
2021, 29(6): 4327-4338 doi: 10.3934/era.2021088 +[Abstract](152) +[HTML](40) +[PDF](393.98KB)
Abstract:

We prove the existence of regular optimal \begin{document}$ G $\end{document}-invariant partitions, with an arbitrary number \begin{document}$ \ell\geq 2 $\end{document} of components, for the Yamabe equation on a closed Riemannian manifold \begin{document}$ (M,g) $\end{document} when \begin{document}$ G $\end{document} is a compact group of isometries of \begin{document}$ M $\end{document} with infinite orbits. To this aim, we study a weakly coupled competitive elliptic system of \begin{document}$ \ell $\end{document} equations, related to the Yamabe equation. We show that this system has a least energy \begin{document}$ G $\end{document}-invariant solution with nontrivial components and we show that the limit profiles of its components separate spatially as the competition parameter goes to \begin{document}$ -\infty $\end{document}, giving rise to an optimal partition. For \begin{document}$ \ell = 2 $\end{document} the optimal partition obtained yields a least energy sign-changing \begin{document}$ G $\end{document}-invariant solution to the Yamabe equation with precisely two nodal domains.

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