The asymptotic behavior of the solutions to a class of pseudoparabolic viscous diffusion equation with periodic initial condition is studied by using the spectral method. The semidiscrete Fourier approximate solution ...The asymptotic behavior of the solutions to a class of pseudoparabolic viscous diffusion equation with periodic initial condition is studied by using the spectral method. The semidiscrete Fourier approximate solution of the problem is constructed and the error estimation between spectral approximate solution and exact solution on large time is also obtained. The existence of the approximate attractor AN and the upper semicontinuity d（AN,A） → 0 are proved.展开更多
A one-dimensional quantum hydrodynamic model （or quantum Euler-Poisson system） for semiconductors with initial boundary conditions is considered for general pressure-density function. The existence and uniqueness of...A one-dimensional quantum hydrodynamic model （or quantum Euler-Poisson system） for semiconductors with initial boundary conditions is considered for general pressure-density function. The existence and uniqueness of the classical solution of the corresponding steady-state quantum hydrodynamic equations is proved. Furthermore, the global existence of classical solution, when the initial datum is a perturbation of t he steadystate solution, is obtained. This solution tends to the corresponding steady-state solution exponentially fast as the time tends to infinity.展开更多
Nonlinear Schrodinger equation (NSE) arises in many physical problems. It is a very important equation. A lot of works studied the wellposed, the existence of solution of NSE etc. And there are many works studied the ...Nonlinear Schrodinger equation (NSE) arises in many physical problems. It is a very important equation. A lot of works studied the wellposed, the existence of solution of NSE etc. And there are many works studied the numerical methods for it. Recently, since the development of infinite dimensional dynamic system the dynamical behavior of NSE has been investigated. The paper [1] studied the long time wellposedness, the existence of universal attractor and the estimate of Lyapunov exponent for NSE with weakly damped. At the same time it was need to study the large time new computational methods and to discuss its convergence error estimate, the existence of approximate attractors etc. In this pape we study the NSE with weakly damped (1.1). We assume,where 0【λ【2 is a constant. If we wish to construct the higher accuracy computational scheme, it will be difficult that staigh from the equation (1.1). Therefore we start with (1. 4) and use fully discrete Fourier spectral method with time difference to展开更多
We are concerned with the large-time behavior of 3D quasilinear hyperbolic equations with nonlinear damping.The main novelty of this paper is two-fold.First,we prove the optimal decay rates of the second and third ord...We are concerned with the large-time behavior of 3D quasilinear hyperbolic equations with nonlinear damping.The main novelty of this paper is two-fold.First,we prove the optimal decay rates of the second and third order spatial derivatives of the solution,which are the same as those of the heat equation,and in particular,are faster than ones of previous related works.Second,for well-chosen initial data,we also show that the lower optimal L^(2) convergence rate of the k(∈[0,3])-order spatial derivatives of the solution is(1+t)^(-(2+2k)/4).Therefore,our decay rates are optimal in this sense.The proofs are based on the Fourier splitting method,low-frequency and high-frequency decomposition,and delicate energy estimates.展开更多
In the present paper, the primitive equations, which can be used to simulate the large-scale motion of the ocean and atmosphere, are considered in the three-dimensional domain bounded below by a fixed solid boundary a...In the present paper, the primitive equations, which can be used to simulate the large-scale motion of the ocean and atmosphere, are considered in the three-dimensional domain bounded below by a fixed solid boundary and above by a free-moving boundary. The global existence and uniqueness of strong solutions are established, and the long-time convergence to the equilibrium state is shown to be either at an exponential rate for the horizontal periodic domain or at an algebraic rate for the horizontal whole space.展开更多
In this paper,we consider 3D compressible magneto-micropolar fluids without resistivity and spin viscosity in a strip domain.The prominent character of the governing model is the presence of the microstructure,a linea...In this paper,we consider 3D compressible magneto-micropolar fluids without resistivity and spin viscosity in a strip domain.The prominent character of the governing model is the presence of the microstructure,a linear coupling structure involving derivatives of the velocity fields,which along with the lack of spin viscosity brings several challenges to the analysis.By exploiting the two-tier energy method developed in Guo and Tice(Arch Ration Mech Anal,2013),we prove the global existence of classical solutions to the governing model around a uniform magnetic field that is non-parallel to the horizontal boundary.Moreover,we show that the solution converges to the steady state at an almost exponential rate as time goes to infinity.One of the main ingredients in our analysis,compared with previous works on micropolar fluids,is that we deal with the microstructure by establishing some delicate estimates based on the analysis of the div-curl decomposition,and the coupling between the fluid velocity and the vorticity of angular velocity.展开更多
This paper is concerned with the large time behavior of the Cauchy problem for Navier-Stokes/AllenCahn system describing the interface motion of immiscible two-phase flow in 3-D. The existence and uniqueness of global...This paper is concerned with the large time behavior of the Cauchy problem for Navier-Stokes/AllenCahn system describing the interface motion of immiscible two-phase flow in 3-D. The existence and uniqueness of global solutions and the stability of the phase separation state are proved under the small initial perturbations. Moreover, the optimal time decay rates are obtained for higher-order spatial derivatives of density,velocity and phase. Our results imply that if the immiscible two-phase flow is initially located near the phase separation state, then under small perturbation conditions, the solution exists globally and decays algebraically to the complete separation state of the two-phase flow, that is, there will be no interface fracture, vacuum, shock wave, mass concentration at any time, and the interface thickness tends to zero as the time t → +∞.展开更多
This paper is concerned with the large-time behavior of solutions to the Cauchy problem of a one-dimensional viscous radiative and reactive gas.Based on the elaborate energy estimates,we develop a new approach to deri...This paper is concerned with the large-time behavior of solutions to the Cauchy problem of a one-dimensional viscous radiative and reactive gas.Based on the elaborate energy estimates,we develop a new approach to derive the upper bound of the absolute temperature by avoiding the use of auxiliary functions Z(t)and W(t)introduced by Liao and Zhao[J.Differential Equations,2018,265(5):2076-2120].Our results also improve upon the results obtained in Liao and Zhao[J.Differential Equations,2018,265(5):2076-2120].展开更多
In this paper, we study the large time asymptotic behavior of solutions to both the Cauchy problem and the exterior problem of the Stokes approximation equations of two dimensional compressible flows.
In this paper the Cauchy problem for the following nonhomogeneous Burgers' equation is considered: (1) ut + uux = μuxx - kx, x∈R, t > 0, where μ and k are positive constants. Since the nonhomogeneous term kx...In this paper the Cauchy problem for the following nonhomogeneous Burgers' equation is considered: (1) ut + uux = μuxx - kx, x∈R, t > 0, where μ and k are positive constants. Since the nonhomogeneous term kx does not belong to any Lp(R) space, this type of equation is beyond usual Sobolev framework in some sense. By Hopf-Cole transformation, (1) takes the form (2) (?)t - (?)xx = - x2(?). With the help of the Hermite polynomials and their properties, (1) and (2) are solved exactly. Moreover, the large time behavior of the solutions is also considered, similar to the discussion in Hopf's paper. Especially, we observe that the nonhomogeneous Burgers' equation (1) is nonlinearly unstable.展开更多
Some basic properties of the small random perturbed dynamical system of Freidlin\|Wentzell type are elicited. A hierarchy structure of attractors is constructed and some further properties of this structure are confir...Some basic properties of the small random perturbed dynamical system of Freidlin\|Wentzell type are elicited. A hierarchy structure of attractors is constructed and some further properties of this structure are confirmed.展开更多
In this note,we present a framework for the large time behavior of general uniformly bounded weak entropy solutions to the Cauchy problem of Euler-Poisson system of semiconductor devices.It is shown that the solutions...In this note,we present a framework for the large time behavior of general uniformly bounded weak entropy solutions to the Cauchy problem of Euler-Poisson system of semiconductor devices.It is shown that the solutions converges to the stationary solutions exponentially in time.No smallness and regularity conditions are assumed.展开更多
We dedicate to the 2D density-dependent nonhomogeneous incompressible Boussinesq equations with vacuum on . At infinity, if the attenuation of initial density and temperature is not very slow. And it is gained that th...We dedicate to the 2D density-dependent nonhomogeneous incompressible Boussinesq equations with vacuum on . At infinity, if the attenuation of initial density and temperature is not very slow. And it is gained that there is a global strong solution and is unique for the 2D Cauchy problem with the initial density which can allow vacuum conditions and even have compact support. Besides, the large time decay rates of the gradients of velocity, temperature and pressure can also be obtained which are also the same as those of the homogeneous case.展开更多
This paper concerns large time behavior of a regular weak solution for non-Newtonian flow equations. It is shown that the decay of the solution is of exponential type when the force term is equal to zero and the domai...This paper concerns large time behavior of a regular weak solution for non-Newtonian flow equations. It is shown that the decay of the solution is of exponential type when the force term is equal to zero and the domain is bounded. Moreover, the ratio of the enstrophy over the energy has a limit as time tends to infinity, which is an eigenvaiue of the Stokes operator.展开更多
基金This work was supported by the National Science Foundation of China(10271034)
文摘The asymptotic behavior of the solutions to a class of pseudoparabolic viscous diffusion equation with periodic initial condition is studied by using the spectral method. The semidiscrete Fourier approximate solution of the problem is constructed and the error estimation between spectral approximate solution and exact solution on large time is also obtained. The existence of the approximate attractor AN and the upper semicontinuity d（AN,A） → 0 are proved.
基金The first author was supported by the China Postdoctoral Science Foundation(2005037318)The second author acknowledges partial support from the Austrian-Chinese Scientific-Technical Collaboration Agreement, the CTS of Taiwanthe Wittgenstein Award 2000 of P.A. Markowich, funded by the Austrian FWF, the Grants-in-Aid of JSPS No.14-02036the NSFC(10431060)the Project-sponsored by SRF for ROCS, SEM
文摘A one-dimensional quantum hydrodynamic model （or quantum Euler-Poisson system） for semiconductors with initial boundary conditions is considered for general pressure-density function. The existence and uniqueness of the classical solution of the corresponding steady-state quantum hydrodynamic equations is proved. Furthermore, the global existence of classical solution, when the initial datum is a perturbation of t he steadystate solution, is obtained. This solution tends to the corresponding steady-state solution exponentially fast as the time tends to infinity.
文摘Nonlinear Schrodinger equation (NSE) arises in many physical problems. It is a very important equation. A lot of works studied the wellposed, the existence of solution of NSE etc. And there are many works studied the numerical methods for it. Recently, since the development of infinite dimensional dynamic system the dynamical behavior of NSE has been investigated. The paper [1] studied the long time wellposedness, the existence of universal attractor and the estimate of Lyapunov exponent for NSE with weakly damped. At the same time it was need to study the large time new computational methods and to discuss its convergence error estimate, the existence of approximate attractors etc. In this pape we study the NSE with weakly damped (1.1). We assume,where 0【λ【2 is a constant. If we wish to construct the higher accuracy computational scheme, it will be difficult that staigh from the equation (1.1). Therefore we start with (1. 4) and use fully discrete Fourier spectral method with time difference to
基金partially supported by the National Nature Science Foundation of China(12271114)the Guangxi Natural Science Foundation(2023JJD110009,2019JJG110003,2019AC20214)+2 种基金the Innovation Project of Guangxi Graduate Education(JGY2023061)the Key Laboratory of Mathematical Model and Application(Guangxi Normal University)the Education Department of Guangxi Zhuang Autonomous Region。
文摘We are concerned with the large-time behavior of 3D quasilinear hyperbolic equations with nonlinear damping.The main novelty of this paper is two-fold.First,we prove the optimal decay rates of the second and third order spatial derivatives of the solution,which are the same as those of the heat equation,and in particular,are faster than ones of previous related works.Second,for well-chosen initial data,we also show that the lower optimal L^(2) convergence rate of the k(∈[0,3])-order spatial derivatives of the solution is(1+t)^(-(2+2k)/4).Therefore,our decay rates are optimal in this sense.The proofs are based on the Fourier splitting method,low-frequency and high-frequency decomposition,and delicate energy estimates.
基金supported by National Natural Science Foundation of China (Grant Nos. 11931010, 12226326, and 12226327)supported by National Natural Science Foundation of China (Grant No. 11701053)+2 种基金the Key Research Project of Academy for Multidisciplinary Studies, Capital Normal Universitythe Capacity Building for Sci-Tech Innovation—Fundamental Scientific Research Funds (Grant No. 007/20530290068)the Fundamental Research Funds for the Central Universities (Grant Nos. 0903005203477, 2020CDJQY-A040, and 2020CDJQY-Z001)。
文摘In the present paper, the primitive equations, which can be used to simulate the large-scale motion of the ocean and atmosphere, are considered in the three-dimensional domain bounded below by a fixed solid boundary and above by a free-moving boundary. The global existence and uniqueness of strong solutions are established, and the long-time convergence to the equilibrium state is shown to be either at an exponential rate for the horizontal periodic domain or at an algebraic rate for the horizontal whole space.
基金supported by National Natural Science Foundation of China(Grant No.12101095)the Science and Technology Research Program of Chongqing Municipal Education Commission(Grant No.KJQN202100517)+6 种基金the Research Project of Chongqing Education Commission(Grant No.CXQT21014)the Natural Science Foundation of Chongqing(Grant No.cstc2021jcyj-msxm X0224)the Grant of Chongqing Young Experts’Workshopsupported by National Natural Science Foundation of China(Grant No.12201221)the Guangdong Basic and Applied Basic Research Foundation(Grant No.2021A1515111038)Changjiang Zhu was supported by National Natural Science Foundation of China(Grant Nos.12171160 and 11831003)the Guangdong Provincial Key Laboratory of Human Digital Twin(Grant No.2022B1212010004)。
文摘In this paper,we consider 3D compressible magneto-micropolar fluids without resistivity and spin viscosity in a strip domain.The prominent character of the governing model is the presence of the microstructure,a linear coupling structure involving derivatives of the velocity fields,which along with the lack of spin viscosity brings several challenges to the analysis.By exploiting the two-tier energy method developed in Guo and Tice(Arch Ration Mech Anal,2013),we prove the global existence of classical solutions to the governing model around a uniform magnetic field that is non-parallel to the horizontal boundary.Moreover,we show that the solution converges to the steady state at an almost exponential rate as time goes to infinity.One of the main ingredients in our analysis,compared with previous works on micropolar fluids,is that we deal with the microstructure by establishing some delicate estimates based on the analysis of the div-curl decomposition,and the coupling between the fluid velocity and the vorticity of angular velocity.
基金supported by the National Natural Science Foundation of China (Nos. 12171024, 11901025,11971217, 11971020)Academic and Technical Leaders Training Plan of Jiangxi Province (No. 20212BCJ23027)。
文摘This paper is concerned with the large time behavior of the Cauchy problem for Navier-Stokes/AllenCahn system describing the interface motion of immiscible two-phase flow in 3-D. The existence and uniqueness of global solutions and the stability of the phase separation state are proved under the small initial perturbations. Moreover, the optimal time decay rates are obtained for higher-order spatial derivatives of density,velocity and phase. Our results imply that if the immiscible two-phase flow is initially located near the phase separation state, then under small perturbation conditions, the solution exists globally and decays algebraically to the complete separation state of the two-phase flow, that is, there will be no interface fracture, vacuum, shock wave, mass concentration at any time, and the interface thickness tends to zero as the time t → +∞.
基金National Postdoctoral Program for Innovative Talents of China(BX20180054).
文摘This paper is concerned with the large-time behavior of solutions to the Cauchy problem of a one-dimensional viscous radiative and reactive gas.Based on the elaborate energy estimates,we develop a new approach to derive the upper bound of the absolute temperature by avoiding the use of auxiliary functions Z(t)and W(t)introduced by Liao and Zhao[J.Differential Equations,2018,265(5):2076-2120].Our results also improve upon the results obtained in Liao and Zhao[J.Differential Equations,2018,265(5):2076-2120].
基金A The research is supported in part by the National Natural Science Foundation of China （Grant No. 10401012） and The Project Sponsored by the Scientific Research Foundation for the Returned 0verseas Chinese Scholars, State Education Ministry.Acknowledgment This is a part of my Ph.D thesis at The Institute of Mathematical Sciences, The Chinese University of Hong Kong. I express my deep gratitude to my graduate advisor, Professor Zhouping Xin, for his guidance and encouragement.
文摘In this paper, we study the large time asymptotic behavior of solutions to both the Cauchy problem and the exterior problem of the Stokes approximation equations of two dimensional compressible flows.
基金Beijing Natural Sciences Foundation (Grant Nos. 1992002 and 1002004) Beijing Education Committee Foundation, and partially supported by the National Youth Foundation of China.
文摘In this paper the Cauchy problem for the following nonhomogeneous Burgers' equation is considered: (1) ut + uux = μuxx - kx, x∈R, t > 0, where μ and k are positive constants. Since the nonhomogeneous term kx does not belong to any Lp(R) space, this type of equation is beyond usual Sobolev framework in some sense. By Hopf-Cole transformation, (1) takes the form (2) (?)t - (?)xx = - x2(?). With the help of the Hermite polynomials and their properties, (1) and (2) are solved exactly. Moreover, the large time behavior of the solutions is also considered, similar to the discussion in Hopf's paper. Especially, we observe that the nonhomogeneous Burgers' equation (1) is nonlinearly unstable.
基金ProjectsupportedbytheNationalNaturalScienceFoundationofChinaandtheDoctoralProgramFoundationofIHE 86 3Program China
文摘Some basic properties of the small random perturbed dynamical system of Freidlin\|Wentzell type are elicited. A hierarchy structure of attractors is constructed and some further properties of this structure are confirmed.
基金the National Natural Science Foundation of China (Grant No.10471138),NSFC-NSAFG (Grant No.10676037) the Major State Basic Research Development Program of China (Grant No.2006CB805902)partially supported by NSF (Grant No.DMS-0505515)
文摘In this note,we present a framework for the large time behavior of general uniformly bounded weak entropy solutions to the Cauchy problem of Euler-Poisson system of semiconductor devices.It is shown that the solutions converges to the stationary solutions exponentially in time.No smallness and regularity conditions are assumed.
文摘We dedicate to the 2D density-dependent nonhomogeneous incompressible Boussinesq equations with vacuum on . At infinity, if the attenuation of initial density and temperature is not very slow. And it is gained that there is a global strong solution and is unique for the 2D Cauchy problem with the initial density which can allow vacuum conditions and even have compact support. Besides, the large time decay rates of the gradients of velocity, temperature and pressure can also be obtained which are also the same as those of the homogeneous case.
文摘This paper concerns large time behavior of a regular weak solution for non-Newtonian flow equations. It is shown that the decay of the solution is of exponential type when the force term is equal to zero and the domain is bounded. Moreover, the ratio of the enstrophy over the energy has a limit as time tends to infinity, which is an eigenvaiue of the Stokes operator.
