Abstract: Publication date: September 2018Source: Acta Mathematica Scientia, Volume 38, Issue 5Author(s): Huanyao WEN, Lei YAO, Changjiang ZHU The two-phase flow models are commonly used in industrial applications, such as nuclear, power, chemical-process, oil-and-gas, cryogenics, bio-medical, micro-technology and so on. This is a survey paper on the study of compressible nonconservative two-fluid model, drift-flux model and viscous liquid-gas two-phase flow model. We give the research developments of these three two-phase flow models, respectively. In the last part, we give some open problems about the above models.

Abstract: Publication date: September 2018Source: Acta Mathematica Scientia, Volume 38, Issue 5Author(s): Tao TAO, Liqun ZHANG We show the existence of dissipative Hölder continuous solutions of the Boussinesq equations. More precise, for any β ∈(0,⅕), a time interval [0,T] and any given smooth energy profile e:[0,T] → (0,∞), there exist a weak solution (v, θ) of the 3d Boussinesq equations such that (v,θ) ∈ Cβ × [0,T] with e(t) = ∫T3 v(x,t) 2 dx for all t ∈ [0,T]. This extend the result of [2] about Onsager's conjecture into Boussinesq equation and improve our previous result in [30].

Abstract: Publication date: September 2018Source: Acta Mathematica Scientia, Volume 38, Issue 5Author(s): Yanyan LI, Bo WANG In this paper, we obtain the strong comparison principle and Hopf Lemma for locally Lipschitz viscosity solutions to a class of nonlinear degenerate elliptic operators of the form ∇2 ψ + L(x, ∇ ψ), including the conformal hessian operator.

Abstract: Publication date: September 2018Source: Acta Mathematica Scientia, Volume 38, Issue 5Author(s): Congming LI, Zhigang WU In this paper, we consider systems of fractional Laplacian equations in ℝn with nonlinear terms satisfying some quite general structural conditions. These systems were categorized critical and subcritical cases. We show that there is no positive solution in the subcritical cases, and we classify all positive solutions ui in the critical cases by using a direct method of moving planes introduced in Chen-Li-Li [11] and some new maximum principles in Li-Wu-Xu [27].

Abstract: Publication date: September 2018Source: Acta Mathematica Scientia, Volume 38, Issue 5Author(s): Feimin HUANG, Yong WANG The regularity of solutions to the Boltzmann equation is a fundamental problem in the kinetic theory. In this paper, the case with angular cut-off is investigated. It is shown that the macroscopic parts of solutions to the Boltzmann equation, i.e., the density, momentum and total energy are continuous functions of (x,t) in the region ℝ3 × (0, + ∞). More precisely, these macroscopic quantities immediately become continuous in any positive time even though they are initially discontinuous and the discontinuities of solutions propagate only in the microscopic level. It should be noted that such kind of phenomenon can not happen for the compressible Navier-Stokes equations in which the initial discontinuities of the density never vanish in any finite time, see [22]. This hints that the Boltzmann equation has better regularity effect in the macroscopic level than compressible Navier-Stokes equations.

Abstract: Publication date: September 2018Source: Acta Mathematica Scientia, Volume 38, Issue 5Author(s): Lin HE, Yongkai LIAO, Tao WANG, Huijiang ZHAO This paper is concerned with the construction of global, large amplitude solutions to the Cauchy problem of the one-dimensional compressible Navier-Stokes system for a viscous radiative gas when the viscosity and heat conductivity coefficients depend on both specific volume and absolute temperature. The data are assumed to be without vacuum, mass concentrations, or vanishing temperatures, and the same is shown to be hold for the global solution constructed. The proof is based on some detailed analysis on uniform positive lower and upper bounds of the specific volume and absolute temperature.

Abstract: Publication date: September 2018Source: Acta Mathematica Scientia, Volume 38, Issue 5Author(s): Gui-Qiang G. CHEN, Matthew RIGBY We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem (initial value problem) in the flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than $1$, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.

Abstract: Publication date: September 2018Source: Acta Mathematica Scientia, Volume 38, Issue 5Author(s): Binglong CHEN, Xiping ZHU The well-known Yau's uniformization conjecture states that any complete noncompact Kähler manifold with positive bisectional curvature is bi-holomorphic to the Euclidean space. The conjecture for the case of maximal volume growth has been recently confirmed by G. Liu in [23]. In the first part, we will give a survey on the progress. In the second part, we will consider Yau's conjecture for manifolds with non-maximal volume growth. We will show that the finiteness of the first Chern number C1n is an essential condition to solve Yau's conjecture by using algebraic embedding method. Moreover, we prove that, under bounded curvature conditions, C1n is automatically finite provided that there exists a positive line bundle with finite Chern number. In particular, we obtain a partial answer to Yau's uniformization conjecture on Kähler manifolds with minimal volume growth.

Abstract: Publication date: September 2018Source: Acta Mathematica Scientia, Volume 38, Issue 5Author(s): Daomin CAO, Shuangjie PENG, Shusen YAN In this paper, we continue to construct stationary classical solutions for the incompressible planar flows approximating singular stationary solutions of this problem. This procedure is carried out by constructing solutions for the following elliptic equations {-Δu=λ∑j=1k1Bδ(xO,j)(u-kj)+p, in Ω,u=0,on ∂Ωwhere 0 < p < 1, ⊂ ℝ2 is a bounded simply-connected smooth domain, κ (i, 1, …, k) is prescribed positive constant. The result we prove is that for any given non-degenerate critical point x0 = (x0,1, … x0, k) of the Kirchhoff-Routh function defined on Ωk corresponding to (κ1, …, κk), there exists a stationary classical solution approximating stationary k points vortex solution. Moreover, as λ → + ∞, the vorticity set {y: uλ> κj} ∩ Bδ(x0, j) shrinks to {x0, j}, and the local vorticity strength near each x0, j approaches κj, j = 1, …, k. This result makes the study of the above problem with p ≥ 0 complete since the cases p 1, p = 1, p = 0 have already been studied in [11, 12] and [13] respectively.

Abstract: Publication date: July 2018Source: Acta Mathematica Scientia, Volume 38, Issue 4Author(s): Maofa WANG, Xingxing YAO, Fangwen DENG In this paper, we study weighted composition operators on the Hilbert space of Dirichlet series with square summable coefficients. The Hermitianness, Fredholmness and invertibility of such operators are characterized, and the spectra of compact and invertible weighted composition operators are also described.

Abstract: Publication date: July 2018Source: Acta Mathematica Scientia, Volume 38, Issue 4Author(s): Kamel MEZLINI, Néji BETTAIBI In this paper, we present an explicit realization of q-deformed Calogero-Vasiliev algebra whose generators are first-order q-difference operators related to the generalized discrete q-Hermite I polynomials recently introduced in [14]. Furthermore, we construct the wave functions and we determine the q-coherent states.

Abstract: Publication date: July 2018Source: Acta Mathematica Scientia, Volume 38, Issue 4Author(s): Bochra NEFZI, Kamel BRAHIM, Ahmed FITOUHI The aim of the present paper is to introduce and study a new type of q-Mellin transform [11], that will be called q-finite Mellin transform. In particular, we prove for this new transform an inversion formula and q-convolution product. The application of this transform is also earlier proposed in solving procedure for a new equation with a new fractional differential operator of a variational type.

Abstract: Publication date: July 2018Source: Acta Mathematica Scientia, Volume 38, Issue 4Author(s): Wenlong SUN, Yeping LI This paper investigates the pullback asymptotic behaviors for the non-autonomous micropolar fluid flows in 2D bounded domains. We use the energy method, combining with some important properties of the generated processes, to prove the existence of pullback exponential attractors and global pullback attractors and show that they both with finite fractal dimension. Further, we give the relationship between global pullback attractors and pullback exponential attractors.

Abstract: Publication date: July 2018Source: Acta Mathematica Scientia, Volume 38, Issue 4Author(s): Yi QIN, Yanren HOU Although the two-grid finite element decoupled scheme for mixed Navier-Stokes/Darcy model in literatures has given the numerical results of optimal convergence order, the theoretical analysis only obtain the optimal error order for the porous media flow and a non-optimal error order for the fluid flow. In this article, we give a more rigorous of the error analysis for the fluid flow and obtain the optimal error estimates of the velocity and the pressure.

Abstract: Publication date: July 2018Source: Acta Mathematica Scientia, Volume 38, Issue 4Author(s): R. ABOULAICH, B. ACHCHAB, A. DAROUICHI In the present work, we investigate the inverse problem of reconstructing the parameter of an integro-differential parabolic equation, which comes from pollution problems in porous media, when the final observation is given. We use the optimal control framework to establish both the existence and necessary condition of the minimizer for the cost functional. Furthermore, we prove the stability and the local uniqueness of the minimizer. Some numerical results will be presented and discussed.

Abstract: Publication date: July 2018Source: Acta Mathematica Scientia, Volume 38, Issue 4Author(s): Mohammad ZAREBNIA, Reza PARVAZ, Amir SABOOR BAGHERZADEH In this paper, we study an efficient asymptotically correction of a-posteriori error estimator for the numerical approximation of Volterra integro-differential equations by piecewise polynomial collocation method. The deviation of the error for Volterra integro-differential equations by using the defect correction principle is defined. Also, it is shown that for m degree piecewise polynomial collocation method, our method provides O(hm+1) as the order of the deviation of the error. The theoretical behavior is tested on examples and it is shown that the numerical results confirm the theoretical part.

Abstract: Publication date: July 2018Source: Acta Mathematica Scientia, Volume 38, Issue 4Author(s): Yue-Loong CHANG, Meng-Rong LI, C. Jack YUE, Yong-Shiuan LEE, Tsung-Jui CHIANG-LIN In this article, we work with the ordinary equation u″-n-q-1u(n)q=0 and learn some interesting phenomena concerning the blow-up and the blow-up rate of solution to the equation.

Abstract: Publication date: July 2018Source: Acta Mathematica Scientia, Volume 38, Issue 4Author(s): Amin ESFAHANI In this article, we establish the existence of a sign-changing solution and two sign-constant solutions for nonlinear nonlocal problem involving the BO-ZK operator on bounded domain. Our main tool is constrained minimization on appropriate Nehari manifolds.

Abstract: Publication date: July 2018Source: Acta Mathematica Scientia, Volume 38, Issue 4Author(s): Fethi BEN BELGACEM In this paper, we introduce and study a method for the numerical solution of the elliptic Monge-Ampère equation with Dirichlet boundary conditions. We formulate the Monge-Ampère equation as an optimization problem. The latter involves a Poisson Problem which is solved by the finite element Galerkin method and the minimum is computed by the conjugate gradient algorithm. We also present some numerical experiments.

Abstract: Publication date: July 2018Source: Acta Mathematica Scientia, Volume 38, Issue 4Author(s): M. PIRHAJI, M. ZANGIABADI, H. MANSOURI In this paper, a corrector-predictor interior-point algorithm is proposed for symmetric optimization. The algorithm approximates the central path by an ellipse, follows the ellipsoidal approximation of the central-path step by step and generates a sequence of iterates in a wide neighborhood of the central-path. Using the machinery of Euclidean Jordan algebra and the commutative class of search directions, the convergence analysis of the algorithm is shown and it is proved that the algorithm has the complexity bound O(rL) for the well-known Nesterov-Todd search direction and O(rL) for the xs and sx search directions.

Abstract: Publication date: July 2018Source: Acta Mathematica Scientia, Volume 38, Issue 4Author(s): Zhiqiang GAO Consider a branching random walk, where the underlying branching mechanism is governed by a Galton-Watson process and the moving law of particles by a discrete random variable on the integer lattice Z. Denote by Z_n(z) the number of particles in the n-th generation in the model for each z ∈ Z. We derive the exact convergence rate in the local limit theorem for Z_n(z) assuming a condition like “ EN(logN)1+λ

Abstract: Publication date: July 2018Source: Acta Mathematica Scientia, Volume 38, Issue 4Author(s): Nihal Yilmaz ÖZGÜR, Nihal TAŞ Recently, the notion of an S-metric space is defined and extensively studied as a generalization of a metric space. In this paper, we define the notion of the S∞-space and prove its completeness. We obtain a new generalization of the classical “Picard Theorem”.

Abstract: Publication date: July 2018Source: Acta Mathematica Scientia, Volume 38, Issue 4Author(s): Salah BOULAARAS, Mohammed Said TOUATI BRAHIM, Smail BOUZENADA, Abderrahmane ZARAI In this paper, a posteriori error estimates for the generalized Schwartz method with Dirichlet boundary conditions on the interfaces for advection-diffusion equation with second order boundary value problems are proved by using the Euler time scheme combined with Galerkin spatial method. Furthermore, an asymptotic behavior in Sobolev norm is deduced using Benssoussan-Lions' algorithm. Finally, the results of some numerical experiments are presented to support the theory.

Abstract: Publication date: July 2018Source: Acta Mathematica Scientia, Volume 38, Issue 4Author(s): Ping LI, Congbian MA, Youliang HOU Let O(PτL) be the oscillation of the Possion semigroup associated with the parabolic Hermite operator L=∂t-Δ+ x 2. We show that O(PτL) is bounded from Lp(Rn+1) into itself for 1 < p < ∞, bounded from L1(Rn+1) into weak- L1(Rn+1) and bounded from Lc∞(Rn+1) into BMO (Rn+1). In the case p =∞ we show that the range of the image of the operator O(PτL) is strictly smaller than the range of a general singular operator.

Abstract: Publication date: July 2018Source: Acta Mathematica Scientia, Volume 38, Issue 4Author(s): Ozge AKCAY The aim of this paper is to construct the integral representation of the solution of Sturm-Liouville equation with eigenparameter-dependent discontinuity conditions at an interior point of the finite interval. Moreover, we examine the properties of the kernel function of this integral representation and obtain the partial differential equation provided by this kernel function.

Abstract: Publication date: July 2018Source: Acta Mathematica Scientia, Volume 38, Issue 4Author(s): Nihed TRABELSI The existence of singular limit solutions are investigated by establishing a new Liouville type theorem for nonlinear elliptic system with sub-quadratic convection term and by using the nonlinear domain decomposition method.

Abstract: Publication date: July 2018Source: Acta Mathematica Scientia, Volume 38, Issue 4Author(s): Mahmood POURGHOLAMHOSSEIN, Mohammad ROUZBEHANI, Massoud AMINI In this paper we define and study chain conditions for Hilbert C*-modules through their C*-algebras of compact operators and discuss their perseverance under Morita equivalence and tensor products. We show that these chain conditions are passed from the C*-algebra to its Hilbert module under certain conditions. We also study chain conditions for Hilbert modules coming from inclusion of C*-algebra with a faithful conditional expectation.

Abstract: Publication date: July 2018Source: Acta Mathematica Scientia, Volume 38, Issue 4Author(s): Peide LIU, Maofa WANG In this article, by extending classical Dellacherie's theorem on stochastic sequences to variable exponent spaces, we prove that the famous Burkholder-Gundy-Davis inequality holds for martingales in variable exponent Hardy spaces. We also obtain the variable exponent analogues of several martingale inequalities in classical theory, including convexity lemma, Chevalier's inequality and the equivalence of two kinds of martingale spaces with predictable control. Moreover, under the regular condition on σ-algebra sequence we prove the equivalence between five kinds of variable exponent martingale Hardy spaces.

Abstract: Publication date: July 2018Source: Acta Mathematica Scientia, Volume 38, Issue 4Author(s): Xiao ZHANG, Guoxing JI In this paper, we present some necessary and sufficient conditions for the existence of solutions, hermitian solutions and positive solutions to the system of operator equations AXB=C=BXA in the setting of bounded linear operators on a Hilbert space. Moreover, we obtain the general forms of solutions, hermitian solutions and positive solutions to the system above.

Abstract: Publication date: July 2018Source: Acta Mathematica Scientia, Volume 38, Issue 4Author(s): Shinji ADACHI, Masataka SHIBATA, Tatsuya WATANABE In this paper, we are concerned with the uniqueness and the non-degeneracy of positive radial solutions for a class of semilinear elliptic equations. Using detailed ODE analysis, we extend previous results to cases where nonlinear terms may have sublinear growth. As an application, we obtain the uniqueness and the non-degeneracy of ground states for modified Schrödinger equations.

Abstract: Publication date: July 2018Source: Acta Mathematica Scientia, Volume 38, Issue 4Author(s): Jasbir Singh MANHAS, Ruhan ZHAO We characterize boundedness and compactness of products of differentiation operators and weighted composition operators between weighted Banach spaces of analytic functions and weighted Zygmund spaces or weighted Bloch spaces with general weights.