数学物理学报(英文版)

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RELAXED INERTIAL METHODS FOR SOLVING SPLIT VARIATIONAL INEQUALITY PROBLEMS WITHOUT PRODUCT SPACE FORMULATION

Grace Nnennaya OGWO, Chinedu IZUCHUKWU, Oluwatosin Temitope MEWOMO

数学物理学报(英文版). 2022 (5): 1701-1733. DOI: 10.1007/s10473-022-0501-5

摘要 ( 91 ) RICH HTML

Many methods have been proposed in the literature for solving the split variational inequality problem. Most of these methods either require that this problem is transformed into an equivalent variational inequality problem in a product space, or that the underlying operators are co-coercive. However, it has been discovered that such product space transformation may cause some potential difficulties during implementation and its approach may not fully exploit the attractive splitting nature of the split variational inequality problem. On the other hand, the co-coercive assumption of the underlying operators would preclude the potential applications of these methods. To avoid these setbacks, we propose two new relaxed inertial methods for solving the split variational inequality problem without any product space transformation, and for which the underlying operators are freed from the restrictive co-coercive assumption. The methods proposed, involve projections onto half-spaces only, and originate from an explicit discretization of a dynamical system, which combines both the inertial and relaxation techniques in order to achieve high convergence speed. Moreover, the sequence generated by these methods is shown to converge strongly to a minimum-norm solution of the problem in real Hilbert spaces. Furthermore, numerical implementations and comparisons are given to support our theoretical findings.

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PHASE PORTRAITS OF THE LESLIE-GOWER SYSTEM

Jaume LLIBRE, Claudia VALLS

数学物理学报(英文版). 2022 (5): 1734-1742. DOI: 10.1007/s10473-022-0502-4

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In this paper we characterize the phase portraits of the Leslie-Gower model for competition among species. We give the complete description of their phase portraits in the Poincaré disc (i.e., in the compactification of $\mathbb{R}^2$ adding the circle $\mathbb{S}^1$ of the infinity) modulo topological equivalence.
It is well-known that the equilibrium point of the Leslie-Gower model in the interior of the positive quadrant is a global attractor in this open quadrant, and in this paper we characterize where the orbits attracted by this equilibrium born.

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A GROUND STATE SOLUTION TO THE CHERN-SIMONS-SCHRÖDINGER SYSTEM

Jin DENG, Benniao LI

数学物理学报(英文版). 2022 (5): 1743-1764. DOI: 10.1007/s10473-022-0503-3

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In this paper, we consider the Chern-Simons-Schrödinger system \begin{equation*}\left\{\begin{array}{lll} - \Delta u+\left[e^{2}|\mathbf{A}|^{2}+\left(V(x)+2e A_{0}\right)+2\left(1+\frac{\kappa q}{2 }\right) N\right] u+ q |u|^{p-2}u=0, \\ -\Delta N+\kappa^{2} q^{2} N+q\left(1+\frac{\kappa q}{2}\right) u^{2}=0, \\ \kappa\left(\partial_{1} A_{2}-\partial_{2} A_{1}\right)= - e u^{2}, \, \, \partial_{1} A_{1}+\partial_{2} A_{2}=0, \\ \kappa \partial_{1} A_{0}= e^{2} A_{2} u^{2}, \, \, \kappa \partial_{2} A_{0}= - e^{2} A_{1} u^{2}, \, \, \end{array} \right.{\rm (P)} \end{equation*} where $u \in H^{1}(\mathbb{R}^{2})$, $p \in (2, 4)$, $A_{\alpha}: \mathbb{R}^{2} \rightarrow \mathbb{R}$ are the components of the gauge potential $(\alpha=0, 1, 2)$, $N: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is a neutral scalar field, $V(x)$ is a potential function, the parameters $ \kappa, q>0$ represent the Chern-Simons coupling constant and the Maxwell coupling constant, respectively, and $ e>0$ is the coupling constant. In this paper, the truncation function is used to deal with a neutral scalar field and a gauge field in the Chern-Simons-Schrödinger problem. The ground state solution of the problem (P) is obtained by using the variational method.

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ITERATIVE METHODS FOR OBTAINING AN INFINITE FAMILY OF STRICT PSEUDO-CONTRACTIONS IN BANACH SPACES

Meng WEN, Haiyang LI, Changsong HU, Jigen PENG

数学物理学报(英文版). 2022 (5): 1765-1778. DOI: 10.1007/s10473-022-0504-2

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In this paper, we introduce a general hybrid iterative method to find an infinite family of strict pseudo-contractions in a q-uniformly smooth and strictly convex Banach space. Moreover, we show that the sequence defined by the iterative method converges strongly to a common element of the set of fixed points, which is the unique solution of the variational inequality $\left\langle(\lambda \varphi-\nu \mathcal{F}) \tilde{z}, j_q(z-\tilde{z})\right\rangle \leq 0$, for $z \in \bigcap_{i=1}^{\infty} \Gamma\left(S_i\right)$. The results introduced in our work extend to some corresponding theorems.

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A NON-LOCAL DIFFUSION EQUATION FOR NOISE REMOVAL

Jingfeng SHAO, Zhichang GUO, Wenjuan YAO, Dong YAN, Boying WU

数学物理学报(英文版). 2022 (5): 1779-1808. DOI: 10.1007/s10473-022-0505-1

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In this paper, we propose a new non-local diffusion equation for noise removal, which is derived from the classical Perona-Malik equation (PM equation) and the regularized PM equation. Using the convolution of the image gradient and the gradient, we propose a new diffusion coefficient. Due to the use of the convolution, the diffusion coefficient is non-local. However, the solution of the new diffusion equation may be discontinuous and belong to the bounded variation space (BV space). By virtue of Young measure method, the existence of a BV solution to the new non-local diffusion equation is established. Experimental results illustrate that the new method has some non-local performance and performs better than the original PM and other methods.

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BLOW-UP IN A FRACTIONAL LAPLACIAN MUTUALISTIC MODEL WITH NEUMANN BOUNDARY CONDITIONS

Chao Jiang, Zuhan Liu, Ling Zhou

数学物理学报(英文版). 2022 (5): 1809-1816. DOI: 10.1007/s10473-022-0506-0

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In this paper, a fractional Laplacian mutualistic system under Neumann boundary conditions is studied. Using the method of upper and lower solutions, it is proven that the solutions of the fractional Laplacian strong mutualistic model with Neumann boundary conditions will blow up when the intrinsic growth rates of species are large.

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POSITIVE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH PERTURBED SOURCE TERMS

Narimane AISSAOUI, Wei LONG

数学物理学报(英文版). 2022 (5): 1817-1830. DOI: 10.1007/s10473-022-0507-z

摘要 ( 31 ) RICH HTML

This paper deals with the existence of positive solutions to the following nonlinear Kirchhoff equation with perturbed external source terms: $$ \left\{ \begin{array}{ll} -\Big(a+b \int_{\mathbb{R}^3} | \nabla u |^2 {\rm d}x\Big) \Delta u+ V(x)u=Q(x)u^p+\varepsilon f(x),\quad &x\in \mathbb{R}^3, \\ u>0,\quad &u\in H^1(\mathbb{R}^3). \end{array} \right.$$ Here $a,b$ are positive constants, $V(x),Q(x)$ are positive radial potentials, 1<p<5, $\varepsilon$ >0 is a small parameter, $f(x)$ is an external source term in $L^2(\mathbb{R}^3)\cap L^{\infty}(\mathbb{R}^3)$.

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EXTREMA OF A GAUSSIAN RANDOM FIELD: BERMAN’S SOJOURN TIME METHOD

Liwen CHEN, Xiaofan PENG

数学物理学报(英文版). 2022 (5): 1831-1842. DOI: 10.1007/s10473-022-0508-y

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In this paper we devote ourselves to extending Berman’s sojourn time method, which is thoroughly described in [1–3], to investigate the tail asymptotics of the extrema of a Gaussian random field over [0,T]^d with T ∈ (0, ∞).

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LARGE TIME BEHAVIOR OF THE 1D ISENTROPIC NAVIER-STOKES-POISSON SYSTEM

Qingyou He, Jiawei Sun

数学物理学报(英文版). 2022 (5): 1843-1874. DOI: 10.1007/s10473-022-0509-x

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The initial value problem (IVP) for the one-dimensional isentropic compressible Navier-Stokes-Poisson (CNSP) system is considered in this paper. For the variables, the electric field and the velocity, under the Lagrange coordinate, we establish the global existence and uniqueness of the classical solutions to this IVP problem. Then we prove by the method of complex analysis, that the solutions to this system converge to those of the corresponding linearized system in the L² norm as time tends to infinity. In addition, we show, using Green’s function, that the solutions to this system are close to a diffusion profile, pointwisely, as time goes to infinity.

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EXISTENCE AND STABILITY OF PERIODIC AND ALMOST PERIODIC SOLUTIONS TO THE BOUSSINESQ SYSTEM IN UNBOUNDED DOMAINS

Thieu Huy NGUYEN, Truong Xuan PHAM, Thi Ngoc Ha VU, The Sac LE

数学物理学报(英文版). 2022 (5): 1875-1901. DOI: 10.1007/s10473-022-0510-4

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In this paper we investigate the existence and stability of periodic solutions (on a half-line $\mathbb{R}_{+}$) and almost periodic solutions on the whole line time-axis $\mathbb{R}$ to the Boussinesq system on several classes of unbounded domains of $\mathbb{R}^n$ in the framework of interpolation spaces. For the linear Boussinesq system we combine the L^p — L^q-smoothing estimates and interpolation functors to prove the existence of bounded mild solutions. Then, we prove the existence of periodic solutions by invoking Massera’s principle. We also prove the existence of almost periodic solutions. Then we use the results of the linear Boussinesq system to establish the existence, uniqueness and stability of the small periodic and almost periodic solutions to the Boussinesq system using fixed point arguments and interpolation spaces. Our results cover and extend the previous ones obtained in [13, 34, 38].

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CHARACTERIZATION OF RESIDUATED LATTICES VIA MULTIPLIERS

Wei WANG, Bin ZHAO

数学物理学报(英文版). 2022 (5): 1902-1920. DOI: 10.1007/s10473-022-0511-3

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In the paper, we introduce some of multipliers on residuated lattices and investigate the relations among them. First, basing on the properties of multipliers, we show that the set of all multiplicative multipliers on a residuated lattice A forms a residuated lattice which is isomorphic to A. Second, we prove that the set of all total multipliers on A is a Boolean subalgebra of the residuated lattice (which is constituted by all multiplicative multipliers on A) and is isomorphic to the Boolean center of A. Moreover, by partial multipliers, we study the maximal residuated lattices of quotients for residuated lattices. Finally, we focus on principal implicative multipliers on residuated lattices and obtain that the set of principal implicative multipliers on A is isomorphic to the set of all multiplicative multipliers on A under the opposite (dual) order.

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LARGE TIME BEHAVIOR OF GLOBAL STRONG SOLUTIONS TO A TWO-PHASE MODEL WITH A MAGNETIC FIELD

Wenjun WANG, Zhen CHENG

数学物理学报(英文版). 2022 (5): 1921-1946. DOI: 10.1007/s10473-022-0512-2

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In this paper, the Cauchy problem for a two-phase model with a magnetic field in three dimensions is considered. Based on a new linearized system with respect to (c - c_∞, P - P_∞, u, H) for constants c_∞ ≥ 0 and P_∞ > 0, the existence theory of global strong solution is established when the initial data is close to its equilibrium in three dimensions for the small H² initial data. We improve the existence results obtained by Wen and Zhu in [40] where an additional assumption that the initial perturbations are bounded in L¹-norm was needed. The energy method combined with the low-frequency and high-frequency decomposition is used to derive the decay of the solution and hence the global existence. As a by-product, the time decay estimates of the solution and its derivatives in the L²-norm are obtained.

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LOCALIZED NODAL SOLUTIONS FOR SCHRÖDINGER-POISSON SYSTEMS

Xing WANG, Rui HE, Xiangqing LIU

数学物理学报(英文版). 2022 (5): 1947-1970. DOI: 10.1007/s10473-022-0513-1

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In this paper, we study the existence of localized nodal solutions for Schrödinger-Poisson systems with critical growth \begin{equation*} \left\{ \begin{aligned} &-\varepsilon^2\Delta v+V(x)v+\lambda \psi v=v^{5}+\mu|v|^{q-2}v, \ \ \ \text{in}\,\,\mathbb{R}^3,\\ &-\varepsilon^2\Delta \psi=v^2, \ \ \ \text{in}\,\,\mathbb{R}^3; \,\,v(x)\rightarrow 0,\,\psi(x)\rightarrow 0\quad\text{as}\,\,|x| \rightarrow\infty. \end{aligned} \right. \end{equation*} We establish, for small $\varepsilon$, the existence of a sequence of localized nodal solutions concentrating near a given local minimum point of the potential function via the perturbation method, and employ some new analytical skills to overcome the obstacles caused by the nonlocal term $\varphi_u(x)=\frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{u^2(y)}{|x-y|}{\rm d}y$. Our results improve and extend related ones in the literature.

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GLEASON’S PROBLEM ON THE SPACE F^p,q,s (B) IN $\mathbb{C}^n$

Pengcheng TANG, Xuejun ZHANG

数学物理学报(英文版). 2022 (5): 1971-1980. DOI: 10.1007/s10473-022-0514-0

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Let $\Omega$ be a domain in $ \mathbb{C}^{n}$ and let $Y$ be a function space on $\Omega$. If $a\in \Omega$ and $g\in Y$ with $g(a)=0$, do there exist functions $f_{1},f_{2},\cdots ,f_{n}\in Y$ such that $$g(z)=\sum_{l=1}^{n}(z_{l}-a_{l})\ f_{l}(z) \ \ \mbox{ for all $z=(z_{1},z_{2},\cdots ,z_{n})\in \Omega$} \ ? $$ This is Gleason's problem. In this paper, we prove that Gleason's problem is solvable on the boundary general function space $F^{p,q,s}(B)$ in the unit ball $B$ of $ \mathbb{C}^{n}$.

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NUMERICAL ANALYSIS OF A BDF2 MODULAR GRAD-DIV STABILITY METHOD FOR THE STOKES/DARCY EQUATIONS

Jiangshan WANG, Lingxiong MENG, Xiaofeng JIA, Hongen JIA

数学物理学报(英文版). 2022 (5): 1981-2000. DOI: 10.1007/s10473-022-0515-z

摘要 ( 24 ) RICH HTML

In this paper, a BDF2 modular grad-div algorithm for the Stokes/Darcy model is constructed. This method not only effectively avoids solver breakdown, but also increases computational efficiency for increasing parameter values. Herein, complete stability and error analysis are provided. Finally, some numerical tests are proposed to justify the theoretical analysis.

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GENERALIZED ORLICZ-TYPE SLICE SPACES, EXTRAPOLATION AND APPLICATIONS

Songbai WANG

数学物理学报(英文版). 2022 (5): 2001-2024. DOI: 10.1007/s10473-022-0516-y

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We introduce a class of generalized Orlicz-type Auscher—Mourgoglou slice space, which is a special case of the Wiener amalgam. We prove versions of the Rubio de Francia extrapolation theorem in this space. As a consequence, we obtain the boundedness results for several classical operators, such as the Calderón—Zygmund operator, the Marcinkiewicz integrals, the Bochner—Riesz means and the Riesz potential, as well as variational inequalities for differential operators and singular integrals. As an application, we obtain global regularity estimates for solutions of non-divergence elliptic equations on generalized Orlicz-type slice spaces if the coefficient matrix is symmetric, uniformly elliptic and has a small (δ, R)-BMO norm for some positive numbers δ and R.

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LIMIT THEOREMS FOR β-LAGUERRE AND β-JACOBI ENSEMBLES

Naqi HUANG, Yutao MA

数学物理学报(英文版). 2022 (5): 2025-2039. DOI: 10.1007/s10473-022-0517-x

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We use tridiagonal models to study the limiting behavior of β-Laguerre and β-Jacobi ensembles, focusing on the limiting behavior of the extremal eigenvalues and the central limit theorem for the two ensembles. For the central limit theorem of β-Laguerre ensembles, we follow the idea in [1] while giving a modified version for the generalized case. Then we use the total variation distance between the two sorts of ensembles to obtain the limiting behavior of β-Jacobi ensembles.

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A SUBSOLUTION THEOREM FOR THE MONGE-AMPÈRE EQUATION OVER AN ALMOST HERMITIAN MANIFOLD

Jiaogen ZHANG

数学物理学报(英文版). 2022 (5): 2040-2062. DOI: 10.1007/s10473-022-0518-9

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Let Ω ⊆M be a bounded domain with a smooth boundary ∂Ω, where (M, J, g) is a compact, almost Hermitian manifold. The main result of this paper is to consider the Dirichlet problem for a complex Monge-Ampère equation on Ω. Under the existence of a C²-smooth strictly J-plurisubharmonic (J-psh for short) subsolution, we can solve this Dirichlet problem. Our method is based on the properties of subsolutions which have been widely used for fully nonlinear elliptic equations over Hermitian manifolds.

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EXISTENCE RESULTS FOR THE KIRCHHOFF TYPE EQUATION WITH A GENERAL NONLINEAR TERM

Huirong PI, Yong ZENG

数学物理学报(英文版). 2022 (5): 2063-2077. DOI: 10.1007/s10473-022-0519-8

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This paper is mainly concerned with existence and nonexistence results for solutions to the Kirchhoff type equation $-\big(a + b \int_{\mathbb{R}^{3}}|\nabla u|^2 \big)\Delta u + V(x) u = f(u)$ in $\mathbb{R}^3$, with the general hypotheses on the nonlinearity f being as introduced by Berestycki and Lions. Our analysis introduces variational techniques to the analysis of the effect of the nonlinearity, especially for those cases when the concentration-compactness principle cannot be applied in terms of obtaining the compactness of the bounded Palais-Smale sequences and a minimizing problem related to the existence of a ground state on the Pohozaev manifold rather than the Nehari manifold associated with the equation.

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GLOBAL STRUCTURE OF A NODAL SOLUTIONS SET OF MEAN CURVATURE EQUATION IN STATIC SPACETIME

Hua LUO, Guowei DAI

数学物理学报(英文版). 2022 (5): 2078-2086. DOI: 10.1007/s10473-022-0520-2

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By bifurcation and topological methods, we study the global structure of a radial nodal solutions set of the mean curvature equation in a standard static spacetime \begin{eqnarray} \text{div} \left(\frac{a\nabla u}{\sqrt{1-a^2\vert \nabla u\vert^2}}\right)+\frac{g(\nabla u, \nabla a)}{\sqrt{1-a^2\vert \nabla u\vert^2}}=\lambda NH,\nonumber \end{eqnarray} with a $0$-Dirichlet boundary condition on the unit ball. According to the behavior of $H$ near $0$, we obtain the global structure of sign-changing radial spacelike graphs for this problem.

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PROBING A STOCHASTIC EPIDEMIC HEPATITIS C VIRUS MODEL WITH A CHRONICALLY INFECTED TREATED POPULATION

S. P. RAJASEKAR, M. PITCHAIMANI, Quanxin ZHU

数学物理学报(英文版). 2022 (5): 2087-2112. DOI: 10.1007/s10473-022-0521-1

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The hepatitis C virus is hitherto a tremendous threat to human beings, but many researchers have analyzed mathematical models for hepatitis C virus transmission dynamics only in the deterministic case. Stochasticity plays an immense role in pathology and epidemiology. Hence, the main theme of this article is to investigate a stochastic epidemic hepatitis C virus model with five states of epidemiological classification: susceptible, acutely infected, chronically infected, recovered or removed and chronically infected, and treated. The stochastic hepatitis C virus model in epidemiology is established based on the environmental influence on individuals, is manifested by stochastic perturbations, and is proportional to each state. We assert that the stochastic HCV model has a unique global positive solution and attains sufficient conditions for the extinction of the hepatotropic RNA virus. Furthermore, by constructing a suitable Lyapunov function, we obtain sufficient conditions for the existence of an ergodic stationary distribution of the solutions to the stochastic HCV model. Moreover, this article confirms that using numerical simulations, the six parameters of the stochastic HCV model can have a high impact over the disease transmission dynamics, specifically the disease transmission rate, the rate of chronically infected population, the rate of progression to chronic infection, the treatment failure rate of chronically infected population, the recovery rate from chronic infection and the treatment rate of the chronically infected population. Eventually, numerical simulations validate the effectiveness of our theoretical conclusions.

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POINTWISE SPACE-TIME BEHAVIOR OF A COMPRESSIBLE NAVIER-STOKES-KORTEWEG SYSTEM IN DIMENSION THREE

Xiaopan JIANG, Zhigang WU

数学物理学报(英文版). 2022 (5): 2113-2130. DOI: 10.1007/s10473-022-0522-0

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The Cauchy problem of compressible Navier-Stokes-Korteweg system in $\mathbb{R}^3$ is considered here. Due to capillarity effect of material, we obtain the pointwise estimates of the solution in an H⁴-framework, which is different from the previous results for the compressible Navier-Stokes system in an H⁶-framework [24, 25]. Our result mainly relies on two different descriptions of the singularity in the short wave of Green’s function for dealing initial propagation and nonlinear coupling respectively. Our pointwise results demonstrate the generalized Huygens’ principle as the compressible Navier-Stokes system. As a corollary, we have an L^p estimate of the solution with p > 1, which is a generalization for p ≥ 2 in [33].

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GLOBAL WELL-POSEDNESS FOR THE FULL COMPRESSIBLE NAVIER-STOKES EQUATIONS

Jinlu LI, Zhaoyang YIN, Xiaoping ZHAI

数学物理学报(英文版). 2022 (5): 2131-2148. DOI: 10.1007/s10473-022-0523-z

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We are concerned with the Cauchy problem regarding the full compressible Navier-Stokes equations in $\mathbb{R}^d$ (d = 2, 3). By exploiting the intrinsic structure of the equations and using harmonic analysis tools (especially the Littlewood-Paley theory), we prove the global solutions to this system with small initial data restricted in the Sobolev spaces. Moreover, the initial temperature may vanish at infinity.

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THE ASYMPTOTIC BEHAVIOR AND SYMMETRY OF POSITIVE SOLUTIONS TO p-LAPLACIAN EQUATIONS IN A HALF-SPACE

Yujuan CHEN, Lei WEI, Yimin ZHANG

数学物理学报(英文版). 2022 (5): 2149-2164. DOI: 10.1007/s10473-022-0524-y

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We study a nonlinear equation in the half-space with a Hardy potential, specifically, \[ \displaystyle - \Delta_p u= \lambda \frac{u ^{p-1}}{x_1^p}-x_1^\theta f(u)\ \ {\rm in}\ T,\] where $\Delta_p$ stands for the $p$-Laplacian operator defined by $\Delta_p u = {\rm div}(|\nabla u|^{p-2}\nabla u)$, $p>1$, $\theta > -p$, and $T$ is a half-space $\{x_1>0\}$. When $\lambda > \Theta$ (where $\Theta$ is the Hardy constant), we show that under suitable conditions on $f$ and $\theta$, the equation has a unique positive solution. Moreover, the exact behavior of the unique positive solution as $x_1\to 0^+$, and the symmetric property of the positive solution are obtained.

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EXPONENTIAL STABILITY OF A MULTI-PARTICLE SYSTEM WITH LOCAL INTERACTION AND DISTRIBUTED DELAY

Yicheng LIU

数学物理学报(英文版). 2022 (5): 2165-2187. DOI: 10.1007/s10473-022-0525-x

摘要 ( 24 ) RICH HTML

For a collective system, the connectedness of the adjacency matrix plays a key role in making the system achieve its emergent feature, such as flocking or multi-clustering. In this paper, we study a nonsymmetric multi-particle system with a constant and local cut-off weight. A distributed communication delay is also introduced into both the velocity adjoint term and the cut-off weight. As a new observation, we show that the desired multi-particle system undergoes both flocking and clustering behaviors when the eigenvalue 1 of the adjacency matrix is semi-simple. In this case, the adjacency matrix may lose the connectedness. In particular, the number of clusters is discussed by using subspace analysis. In terms of results, for both the non-critical and general neighbourhood situations, some criteria of flocking and clustering emergence with an exponential convergent rate are established by the standard matrix analysis for when the delay is free. As a distributed delay is involved, the corresponding criteria are also found, and these small time lags do not change the emergent properties qualitatively, but alter the final value in a nonlinear way. Consequently, some previous works [14] are extended.

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ERRATUM TO: SEEMINGLY INJECTIVE VON NEUMANN ALGEBRAS

Gilles Pisier

数学物理学报(英文版). 2022 (5): 2188-2188. DOI: 10.1007/s10473-022-0526-9

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