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
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
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
In this paper we mainly deal with the global well-posedness and large-time behavior of the 2D tropical climate model with small initial data. We first establish the global well-posedness of solution in the Besov space...In this paper we mainly deal with the global well-posedness and large-time behavior of the 2D tropical climate model with small initial data. We first establish the global well-posedness of solution in the Besov space, then we obtain the optimal decay rates of solutions by virtue of the frequency decomposition method. Specifically, for the low frequency part, we use the Fourier splitting method of Schonbek and the spectrum analysis method, and for the high frequency part, we use the global energy estimate and the behavior of exponentially decay operator.展开更多
In this paper we deal with the initial-boundary value problem for the coupled Keller-Segel-Stokes system with rotational flux, which is corresponding to the case that the chemical is produced instead of consumed,■sub...In this paper we deal with the initial-boundary value problem for the coupled Keller-Segel-Stokes system with rotational flux, which is corresponding to the case that the chemical is produced instead of consumed,■subject to the boundary conditions( n-n S(x, n, c) c) · ν = c · ν = 0 and u = 0, and suitably regular initial data(n0(x), c0(x), u0(x)), where ? ? R3is a bounded domain with smooth boundary ??. Here S is a chemotactic sensitivity satisfying |S(x, n, c)| ≤ CS(1 + n)-αwith some CS> 0 and α > 0. The greatest contribution of this paper is to consider the large time behavior of solutions for the system(KSS), which is still open even in the 2D case. We can prove that the corresponding solution of the system(KSS) decays to(■) exponentially, if the coefficient of chemotactic sensitivity is appropriately small. As a precondition to consider the asymptotic behavior, we also show the global existence and boundedness of the corresponding initial-boundary problem KSS with a simplified method. We find a new phenomenon that the suitably small coefficient CSof chemotactic sensitivity could benefit the global existence and boundedness of solutions to the model KSS.展开更多
This paper is concerned with a diffuse interface model called Navier-Stokes/CahnHilliard system.This model is usually used to describe the motion of immiscible two-phase flows with a diffusion interface.For the period...This paper is concerned with a diffuse interface model called Navier-Stokes/CahnHilliard system.This model is usually used to describe the motion of immiscible two-phase flows with a diffusion interface.For the periodic boundary value problem of this system in torus T3,we prove that there exists a global unique strong solution near the phase separation state,which means that no vacuum,shock wave,mass concentration,interface collision or rupture will be developed in finite time.Furthermore,we establish the large time behavior of the global strong solution of this system.In particular,we find that the phase field decays algebraically to the phase separation state.展开更多
In this paper,we consider the Cauchy problem for the Camassa-Holm-Novikov equation.First,we establish the local well-posedness and the blow-up scenario.Second,infinite propagation speed is obtained as the nontrivial s...In this paper,we consider the Cauchy problem for the Camassa-Holm-Novikov equation.First,we establish the local well-posedness and the blow-up scenario.Second,infinite propagation speed is obtained as the nontrivial solution u(x,t)does not have compact x-support for any t>0 in its lifespan,although the corresponding u0(x)is compactly supported.Then,the global existence and large time behavior for the support of the momentum density are considered.Finally,we study the persistence property of the solution in weighted Sobolev spaces.展开更多
The one-dimensional compressible Navier-Stokes-Vlasov-Fokker-Planck system with density-dependent viscosity and drag force coefficients is investigated in the present paper.The existence,uniqueness,and regularity of g...The one-dimensional compressible Navier-Stokes-Vlasov-Fokker-Planck system with density-dependent viscosity and drag force coefficients is investigated in the present paper.The existence,uniqueness,and regularity of global weak solution to the initial value problem for general initial data are established in spatial periodic domain.Moreover,the iong time behavior of the weak solution is analyzed.It is shown that as the time grows,the distri-bution function of the particles converges to the global Maxwellian,and both the fluid velocity and the macroscopic velocity of the particles converge to the same speed.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 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
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.
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
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 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 key research project of National Natural Science Foundation of China (Grant No.11931010)。
文摘In this paper we mainly deal with the global well-posedness and large-time behavior of the 2D tropical climate model with small initial data. We first establish the global well-posedness of solution in the Besov space, then we obtain the optimal decay rates of solutions by virtue of the frequency decomposition method. Specifically, for the low frequency part, we use the Fourier splitting method of Schonbek and the spectrum analysis method, and for the high frequency part, we use the global energy estimate and the behavior of exponentially decay operator.
基金supported by the Shandong Provincial Natural Science Foundation (No.ZR2022JQ06)the National Natural Science Foundation of China (No.11601215)Beijing Natural Science Foundation (No.Z210002)。
文摘In this paper we deal with the initial-boundary value problem for the coupled Keller-Segel-Stokes system with rotational flux, which is corresponding to the case that the chemical is produced instead of consumed,■subject to the boundary conditions( n-n S(x, n, c) c) · ν = c · ν = 0 and u = 0, and suitably regular initial data(n0(x), c0(x), u0(x)), where ? ? R3is a bounded domain with smooth boundary ??. Here S is a chemotactic sensitivity satisfying |S(x, n, c)| ≤ CS(1 + n)-αwith some CS> 0 and α > 0. The greatest contribution of this paper is to consider the large time behavior of solutions for the system(KSS), which is still open even in the 2D case. We can prove that the corresponding solution of the system(KSS) decays to(■) exponentially, if the coefficient of chemotactic sensitivity is appropriately small. As a precondition to consider the asymptotic behavior, we also show the global existence and boundedness of the corresponding initial-boundary problem KSS with a simplified method. We find a new phenomenon that the suitably small coefficient CSof chemotactic sensitivity could benefit the global existence and boundedness of solutions to the model KSS.
基金partially supported by the NationalNatural Science Foundation of China(12171024,11901025,11971217,11971020)the Academic and Technical Leaders Training Plan of Jiangxi Province(20212BCJ23027)。
文摘This paper is concerned with a diffuse interface model called Navier-Stokes/CahnHilliard system.This model is usually used to describe the motion of immiscible two-phase flows with a diffusion interface.For the periodic boundary value problem of this system in torus T3,we prove that there exists a global unique strong solution near the phase separation state,which means that no vacuum,shock wave,mass concentration,interface collision or rupture will be developed in finite time.Furthermore,we establish the large time behavior of the global strong solution of this system.In particular,we find that the phase field decays algebraically to the phase separation state.
基金partially supported by the National Natural Science Foundation of China(12071439)the Zhejiang Provincial Natural Science Foundation of China(LY19A010016)the Natural Science Foundation of Jiangxi Province(20212BAB201016)。
文摘In this paper,we consider the Cauchy problem for the Camassa-Holm-Novikov equation.First,we establish the local well-posedness and the blow-up scenario.Second,infinite propagation speed is obtained as the nontrivial solution u(x,t)does not have compact x-support for any t>0 in its lifespan,although the corresponding u0(x)is compactly supported.Then,the global existence and large time behavior for the support of the momentum density are considered.Finally,we study the persistence property of the solution in weighted Sobolev spaces.
基金The research of the paper is supported by National Natural Science Foundation of China(Nos.11931010,11671384,11871047)by the key research project of Academy for Multidisciplinary Studies,Capital Normal University,and by the Capacity Building for Sci-Tech Innovation-Fundamental Scientific Research Funds(No.007/20530290068).
文摘The one-dimensional compressible Navier-Stokes-Vlasov-Fokker-Planck system with density-dependent viscosity and drag force coefficients is investigated in the present paper.The existence,uniqueness,and regularity of global weak solution to the initial value problem for general initial data are established in spatial periodic domain.Moreover,the iong time behavior of the weak solution is analyzed.It is shown that as the time grows,the distri-bution function of the particles converges to the global Maxwellian,and both the fluid velocity and the macroscopic velocity of the particles converge to the same speed.
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.
