In this article, we prove the existence and obtain the expression of its solution formula of global smooth solution for non-homogeneous multi-dimensional（m-D） conservation law with unbounded initial value; our metho...In this article, we prove the existence and obtain the expression of its solution formula of global smooth solution for non-homogeneous multi-dimensional（m-D） conservation law with unbounded initial value; our methods are new and essentially different with the situation of bounded initial value.展开更多
In this paper, we study the Cauchy problem for the following quasi-linear wave equation utt-2kuxxt=β（ux^n）x, where k〉0 and β are real numbers, and n ≥ 2 is an integer. We prove that for any T〉0, the Cauchy prob...In this paper, we study the Cauchy problem for the following quasi-linear wave equation utt-2kuxxt=β（ux^n）x, where k〉0 and β are real numbers, and n ≥ 2 is an integer. We prove that for any T〉0, the Cauchy problem admits a unique global smooth solution u∈C^∞（（0, T]; H^∞（R）） ∩ C（[0, T]; H^2（R）） ∩ C^1（[0, T]; L^2（R）） under suitable assumptions on the initial data.展开更多
This paper is concerned with the bipolar compressible Navier-Stokes-Maxwell system for plasmas. We investigated, by means of the techniques of symmetrizer and elaborate energy method, the Cauchy problem in R^3. Under ...This paper is concerned with the bipolar compressible Navier-Stokes-Maxwell system for plasmas. We investigated, by means of the techniques of symmetrizer and elaborate energy method, the Cauchy problem in R^3. Under the assumption that the initial values are close to a equilibrium solutions, we prove that the smooth solutions of this problem converge to a steady state as the time goes to the infinity. It is shown that the difference of densities of two carriers converge to the equilibrium states with the norm ｜｜·｜｜H^s-1, while the velocities and the electromagnetic fields converge to the equilibrium states with weaker norms than ｜｜·｜｜H^s-1. This phenomenon on the charge transport shows the essential difference between the unipolar Navier-Stokes-Maxwell and the bipolar Navier-Stokes-Maxwell system.展开更多
The existence of a global smooth solution for the initial value problem of generalized Kuramoto-Sivashinsky type equations have been obtained. Similarty siolutions and the structure of the traveling waves solution for...The existence of a global smooth solution for the initial value problem of generalized Kuramoto-Sivashinsky type equations have been obtained. Similarty siolutions and the structure of the traveling waves solution for the generalized KS equations are discussed and analysed by using the qualitative theory of ODE and Lie's infinitesimal transformation respectively.展开更多
The authors study the asymptotic behavior of the smooth solutions to the Cauchy problems for two macroscopic models (hydrodynamic and drift-diffusion models) for semiconductors and the related relaxation limit problem...The authors study the asymptotic behavior of the smooth solutions to the Cauchy problems for two macroscopic models (hydrodynamic and drift-diffusion models) for semiconductors and the related relaxation limit problem. First, it is proved that the solutions to these two systems converge to the unique stationary solution time asymptotically without the smallness assump- tion on doping profile. Then, very sharp estimates on the smooth solutions, independent of the relaxation time, are obtained and used to establish the zero relaxation limit.展开更多
Under the assumptions that the initial density ρ0 is close enough to 1 and ρ0-1∈Hs+1(R2);u0∈Hs(R2)∩H_∈(R2) for s>2 and 0<ε<1;the authors prove the global existence and uniqueness of smooth solutions to...Under the assumptions that the initial density ρ0 is close enough to 1 and ρ0-1∈Hs+1(R2);u0∈Hs(R2)∩H_∈(R2) for s>2 and 0<ε<1;the authors prove the global existence and uniqueness of smooth solutions to the 2-D inhomogeneous Navier-Stokes equations with the viscous coefficient depending on the density of the fluid.Furthermore,the L2 decay rate of the velocity field is obtained.展开更多
In this paper, we will investigate the incompressible Navier-Stokes-Landau-Lifshitz equations, which is a system of the incompressible Navier-Stokes equations coupled with the Landau-Lifshitz-Gilbert equations. We wil...In this paper, we will investigate the incompressible Navier-Stokes-Landau-Lifshitz equations, which is a system of the incompressible Navier-Stokes equations coupled with the Landau-Lifshitz-Gilbert equations. We will prove global existence of the smooth solution to the incompressible Navier-Stokes-Landau-Lifshitz equation with small initial data in T2or R2and R3.展开更多
We obtain the global smooth solution of a nonlinear SchrSdinger equations in atomic Bose-Einstein condensates with two-dimensional spaces. By using the Galerkin method and a priori estimates, we establish the global e...We obtain the global smooth solution of a nonlinear SchrSdinger equations in atomic Bose-Einstein condensates with two-dimensional spaces. By using the Galerkin method and a priori estimates, we establish the global existence and uniqueness of the smooth solution.展开更多
In this paper,we study the global smooth solutions of the Cauchy problem for two important nonstrictly quasilinear hyperbolic systems.i.e.,the isentropic gas dynamics system in Euler coordinates (Ⅰ) and the rotationa...In this paper,we study the global smooth solutions of the Cauchy problem for two important nonstrictly quasilinear hyperbolic systems.i.e.,the isentropic gas dynamics system in Euler coordinates (Ⅰ) and the rotational degeneracy of hyperbolic systems of conservation laws(Ⅱ).sufficient conditions which guarantee the existence of gloats smooth solutions of the Cauchy problems (Ⅰ) and (Ⅱ) are obtained by employing the characteristic method.展开更多
In this paper, we study a class of coupled fractional nonlinear Schr^dinger system with periodic boundary condition. Using Galerkin method, the existence of global smooth solution is obtained. We also prove the unique...In this paper, we study a class of coupled fractional nonlinear Schr^dinger system with periodic boundary condition. Using Galerkin method, the existence of global smooth solution is obtained. We also prove the uniqueness of the solution.展开更多
The bipolar compressible Euler-Maxwell equations as a fluid dynamic model arising from plasma physics to describe the dynamics of the compressible electrons and ions is investigated. This work is concerned with three-...The bipolar compressible Euler-Maxwell equations as a fluid dynamic model arising from plasma physics to describe the dynamics of the compressible electrons and ions is investigated. This work is concerned with three-dimensional Euler-Maxwell equations with smooth periodic solutions. With the help of the symmetry operator techniques and energy method, the global smooth solution with small amplitude is constructed around a constant equilibrium solution with asymptotic stability property.展开更多
In this paper,we consider the global well-posedness of smooth solutions for the Cauchy problem of a sixth order convective Cahn-Hilliard equation with small initial data.We first construct a local smooth solution,then...In this paper,we consider the global well-posedness of smooth solutions for the Cauchy problem of a sixth order convective Cahn-Hilliard equation with small initial data.We first construct a local smooth solution,then by combining some a priori estimates,continuity argument,the local smooth solution is extended step by step to all t>0 provided that the L1 norm of initial data is suitably small and the smooth nonlinear functions f(u)and g(u)satisfy certain local growth conditions at some fixed point■.展开更多
This paper is concerned with an Initial Boundary Value Problem (IBVP) for a strongly coupled semilinear reaction-diffusion system. By using the upper and lower solutions method and Leray-Schauder fixed point theorem a...This paper is concerned with an Initial Boundary Value Problem (IBVP) for a strongly coupled semilinear reaction-diffusion system. By using the upper and lower solutions method and Leray-Schauder fixed point theorem and so on, the authors prove the global existence and uniqueness of a. smooth. solution for this IBVP under some appropriate conditions.展开更多
We consider a non-isentropic Euler-Poisson system with two small parameters arising in the modeling of unmagnetized plasmas and semiconductors.On the basis of the energy estimates and the compactness theorem,the unifo...We consider a non-isentropic Euler-Poisson system with two small parameters arising in the modeling of unmagnetized plasmas and semiconductors.On the basis of the energy estimates and the compactness theorem,the uniform global existence of the solutions and the combined quasi-neutral and zero-electron-mass limit of the system are proved when the initial data are close to the constant equilibrium state.In particular,the limit is rigorously justified as the two parameters tend to zero independently.展开更多
In this paper, the asymptotic behavior of the global smooth solutions to the Cauchy problem for the one-dimensional nonisentropic Euler-Poisson （or full hydrodynamic） model for semiconductors, where the energy equat...In this paper, the asymptotic behavior of the global smooth solutions to the Cauchy problem for the one-dimensional nonisentropic Euler-Poisson （or full hydrodynamic） model for semiconductors, where the energy equation with non-zero thermal conductivity coefficient are contained, is discussed. The global existence of smooth solutions for the Cauchy problem with small perturbed initial data is proved. In particular, that the solutions converge to the corresponding stationary solutions exponentially fast as t → ∞ is showed.展开更多
We couple together existing ideas,existing results,special structure and novel ideas to accomplish the exact limits and improved decay estimates with sharp rates for all order derivatives of the global weak solutions ...We couple together existing ideas,existing results,special structure and novel ideas to accomplish the exact limits and improved decay estimates with sharp rates for all order derivatives of the global weak solutions of the Cauchy problem for an n-dimensional incompressible Navier-Stokes equations.We also use the global smooth solution of the corresponding heat equation to approximate the global weak solutions of the incompressible Navier-Stokes equations.展开更多
The radiative Euler equations is a typical model describing the motion of astrophysical flows.For its mathematical studies,it is now well-understood that the radiation effect can indeed induce some dissipative mechani...The radiative Euler equations is a typical model describing the motion of astrophysical flows.For its mathematical studies,it is now well-understood that the radiation effect can indeed induce some dissipative mechanism,which can guarantee the global regularity of smooth solutions to the radiative Euler equations for small initial data.Thus a problem of interest is to see to what extent does the viscosity and/or thermal conductivity influence the global regularity of smooth solutions to the one-dimensional radiative Euler equations for large initial data.For results in this direction,it is shown in[30]that,for a class of state equations,even if a special class of thermal conductivity is further added to the radiative Euler equations,its smooth solutions will still blow up in finite time for large initial data.The main purpose of this paper focuses on the case when both viscosity and thermal conductivity are considered.We first show that,for the state equations and the heat conductivity considered in[30],if the viscosity is further taken into account,the corresponding radiative Navier-Stokes equations does admit a unique global smooth solution for any large initial data provided that the viscosity is a smooth function of the density satisfying certain growth conditions as the density tends to zero and infinity.Moreover,we also show that similar result still holds for the case when the thermodynamics variables satisfy the state equations for ideal polytropic gases,the heat conductivity takes the form studied in[30],and the viscosity is assumed to satisfy the same conditions imposed in the first result.展开更多
The global well-posedness of another class of initial-boundary value problem on two/three-dimensional incompressible MHD-Boussinesq equations in the bounded domain with the smooth boundary is studied. The existence of...The global well-posedness of another class of initial-boundary value problem on two/three-dimensional incompressible MHD-Boussinesq equations in the bounded domain with the smooth boundary is studied. The existence of a class of global weak solution to the initial boundary value problem for two/three-dimensional incompressible MHD-Boussinesq equation with the given pressure-velocity’s relation boundary condition for the fluid field,one generalized perfectly conducting boundary condition for the magnetic field and one density/temperature-velocity’s relation boundary condition for the density/temapture at the boundary is obtained, and the global existence and uniqueness of the smooth solution to the corresponding problem in two-dimensional case for the smooth initial data is also proven.展开更多
In this paper, we are concerned with the necessary and sufficient condition of the global existence of smooth solutions of the Cauchy problem of the multi-dimensional scalar conservation law with source-term,where the...In this paper, we are concerned with the necessary and sufficient condition of the global existence of smooth solutions of the Cauchy problem of the multi-dimensional scalar conservation law with source-term,where the initial data lies in W1,∞(Rn) ∩ C1(Rn). We obtain the solution formula for smooth solution, and then apply it to establish and prove the necessary and sufficient condition for the global existence of smooth solution. Moreover, if the smooth solution blows up at a finite time, the exact lifespan of the smooth solution can be obtained. In particular, when the source term vanishes, the corresponding theorem for the homogeneous case is obtained too. Finally, we give two examples as its applications, one for the global existence of the smooth solution and the other one for the blowup of the smooth solutions at any given positive time.展开更多
In this article, first, the authors prove that there exists a unique global smooth solution for the Cauthy problem to the hyperbolic conservation laws systems with relaxation; second, in the large time station, they p...In this article, first, the authors prove that there exists a unique global smooth solution for the Cauthy problem to the hyperbolic conservation laws systems with relaxation; second, in the large time station, they prove that the global smooth solutions of the hyperbolic conservation laws systems with relaxation converge to rarefaction waves solution at a determined L^P（p ≥ 2） decay rate.展开更多
基金partly supported by Natural Science Foundation of China(11471332 and 11071246)
文摘In this article, we prove the existence and obtain the expression of its solution formula of global smooth solution for non-homogeneous multi-dimensional（m-D） conservation law with unbounded initial value; our methods are new and essentially different with the situation of bounded initial value.
基金Supported by the National Natural Science Foundation of China(10371073)
文摘In this paper, we study the Cauchy problem for the following quasi-linear wave equation utt-2kuxxt=β（ux^n）x, where k〉0 and β are real numbers, and n ≥ 2 is an integer. We prove that for any T〉0, the Cauchy problem admits a unique global smooth solution u∈C^∞（（0, T]; H^∞（R）） ∩ C（[0, T]; H^2（R）） ∩ C^1（[0, T]; L^2（R）） under suitable assumptions on the initial data.
基金supported by the Collaborative Innovation Center on Beijing Society-building and Social GovernanceNSFC(11371042)+2 种基金BNSF(1132006)the key fund of the Beijing education committee of ChinaChina Postdoctoral Science Foundation funded project
文摘This paper is concerned with the bipolar compressible Navier-Stokes-Maxwell system for plasmas. We investigated, by means of the techniques of symmetrizer and elaborate energy method, the Cauchy problem in R^3. Under the assumption that the initial values are close to a equilibrium solutions, we prove that the smooth solutions of this problem converge to a steady state as the time goes to the infinity. It is shown that the difference of densities of two carriers converge to the equilibrium states with the norm ｜｜·｜｜H^s-1, while the velocities and the electromagnetic fields converge to the equilibrium states with weaker norms than ｜｜·｜｜H^s-1. This phenomenon on the charge transport shows the essential difference between the unipolar Navier-Stokes-Maxwell and the bipolar Navier-Stokes-Maxwell system.
文摘The existence of a global smooth solution for the initial value problem of generalized Kuramoto-Sivashinsky type equations have been obtained. Similarty siolutions and the structure of the traveling waves solution for the generalized KS equations are discussed and analysed by using the qualitative theory of ODE and Lie's infinitesimal transformation respectively.
基金Project supported by the National Natural Science Foundation of China, the Grant of MST of China,the National Natural Science
文摘The authors study the asymptotic behavior of the smooth solutions to the Cauchy problems for two macroscopic models (hydrodynamic and drift-diffusion models) for semiconductors and the related relaxation limit problem. First, it is proved that the solutions to these two systems converge to the unique stationary solution time asymptotically without the smallness assump- tion on doping profile. Then, very sharp estimates on the smooth solutions, independent of the relaxation time, are obtained and used to establish the zero relaxation limit.
基金Project supported by the National Natural Science Foundation of China (Nos.10525101,10421101)the 973 project of the Ministry of Science and Technology of Chinathe innovation grant from Chinese Academy of Sciences
文摘Under the assumptions that the initial density ρ0 is close enough to 1 and ρ0-1∈Hs+1(R2);u0∈Hs(R2)∩H_∈(R2) for s>2 and 0<ε<1;the authors prove the global existence and uniqueness of smooth solutions to the 2-D inhomogeneous Navier-Stokes equations with the viscous coefficient depending on the density of the fluid.Furthermore,the L2 decay rate of the velocity field is obtained.
基金supported by the National Natural Science Foundation of China(No.11801107)Science and Technology Projects in Guangzhou(No.202102010467)+1 种基金the second author is supported by the National Natural Science Foundation of China(Grant No.11971400)Guangdong Basic and Applied Basic Research Foundation(Grant No.2020A1515011019)。
文摘In this paper, we will investigate the incompressible Navier-Stokes-Landau-Lifshitz equations, which is a system of the incompressible Navier-Stokes equations coupled with the Landau-Lifshitz-Gilbert equations. We will prove global existence of the smooth solution to the incompressible Navier-Stokes-Landau-Lifshitz equation with small initial data in T2or R2and R3.
基金Acknowledgements This work was supported in part by the National Natural Science Foundation of China （Grant No. 11571254）.
文摘We obtain the global smooth solution of a nonlinear SchrSdinger equations in atomic Bose-Einstein condensates with two-dimensional spaces. By using the Galerkin method and a priori estimates, we establish the global existence and uniqueness of the smooth solution.
文摘In this paper,we study the global smooth solutions of the Cauchy problem for two important nonstrictly quasilinear hyperbolic systems.i.e.,the isentropic gas dynamics system in Euler coordinates (Ⅰ) and the rotational degeneracy of hyperbolic systems of conservation laws(Ⅱ).sufficient conditions which guarantee the existence of gloats smooth solutions of the Cauchy problems (Ⅰ) and (Ⅱ) are obtained by employing the characteristic method.
文摘In this paper, we study a class of coupled fractional nonlinear Schr^dinger system with periodic boundary condition. Using Galerkin method, the existence of global smooth solution is obtained. We also prove the uniqueness of the solution.
基金Supported by the Foundation Project of Doctor Graduate Student Innovation of Beijing University of Technology(ykj-2012-6724)Supported by the NSFC(10771009)Supported by the BSF(1082001)
文摘The bipolar compressible Euler-Maxwell equations as a fluid dynamic model arising from plasma physics to describe the dynamics of the compressible electrons and ions is investigated. This work is concerned with three-dimensional Euler-Maxwell equations with smooth periodic solutions. With the help of the symmetry operator techniques and energy method, the global smooth solution with small amplitude is constructed around a constant equilibrium solution with asymptotic stability property.
基金Supported by the Fundamental Research Funds for the Central Universities(Grant No.N2005031)。
文摘In this paper,we consider the global well-posedness of smooth solutions for the Cauchy problem of a sixth order convective Cahn-Hilliard equation with small initial data.We first construct a local smooth solution,then by combining some a priori estimates,continuity argument,the local smooth solution is extended step by step to all t>0 provided that the L1 norm of initial data is suitably small and the smooth nonlinear functions f(u)and g(u)satisfy certain local growth conditions at some fixed point■.
文摘This paper is concerned with an Initial Boundary Value Problem (IBVP) for a strongly coupled semilinear reaction-diffusion system. By using the upper and lower solutions method and Leray-Schauder fixed point theorem and so on, the authors prove the global existence and uniqueness of a. smooth. solution for this IBVP under some appropriate conditions.
基金partially supported by the ISFNSFC joint research program(11761141008)NSFC(12071044 and 12131007)the NSF of Jiangsu Province(BK20191296)。
文摘We consider a non-isentropic Euler-Poisson system with two small parameters arising in the modeling of unmagnetized plasmas and semiconductors.On the basis of the energy estimates and the compactness theorem,the uniform global existence of the solutions and the combined quasi-neutral and zero-electron-mass limit of the system are proved when the initial data are close to the constant equilibrium state.In particular,the limit is rigorously justified as the two parameters tend to zero independently.
基金the Youngth Program of Hubei Provincial Department of Education (Q200628002)the Innovation Program of Shanghai Municipal Education Commission (08YZ72)
文摘In this paper, the asymptotic behavior of the global smooth solutions to the Cauchy problem for the one-dimensional nonisentropic Euler-Poisson （or full hydrodynamic） model for semiconductors, where the energy equation with non-zero thermal conductivity coefficient are contained, is discussed. The global existence of smooth solutions for the Cauchy problem with small perturbed initial data is proved. In particular, that the solutions converge to the corresponding stationary solutions exponentially fast as t → ∞ is showed.
文摘We couple together existing ideas,existing results,special structure and novel ideas to accomplish the exact limits and improved decay estimates with sharp rates for all order derivatives of the global weak solutions of the Cauchy problem for an n-dimensional incompressible Navier-Stokes equations.We also use the global smooth solution of the corresponding heat equation to approximate the global weak solutions of the incompressible Navier-Stokes equations.
基金supported by the NSFC(12001495)supported by the NSFC(12221001,12371225)the Science and Technology Department of Hubei Province(2020DFH002)。
文摘The radiative Euler equations is a typical model describing the motion of astrophysical flows.For its mathematical studies,it is now well-understood that the radiation effect can indeed induce some dissipative mechanism,which can guarantee the global regularity of smooth solutions to the radiative Euler equations for small initial data.Thus a problem of interest is to see to what extent does the viscosity and/or thermal conductivity influence the global regularity of smooth solutions to the one-dimensional radiative Euler equations for large initial data.For results in this direction,it is shown in[30]that,for a class of state equations,even if a special class of thermal conductivity is further added to the radiative Euler equations,its smooth solutions will still blow up in finite time for large initial data.The main purpose of this paper focuses on the case when both viscosity and thermal conductivity are considered.We first show that,for the state equations and the heat conductivity considered in[30],if the viscosity is further taken into account,the corresponding radiative Navier-Stokes equations does admit a unique global smooth solution for any large initial data provided that the viscosity is a smooth function of the density satisfying certain growth conditions as the density tends to zero and infinity.Moreover,we also show that similar result still holds for the case when the thermodynamics variables satisfy the state equations for ideal polytropic gases,the heat conductivity takes the form studied in[30],and the viscosity is assumed to satisfy the same conditions imposed in the first result.
基金Supported by National Natural Science Foundation of China(Grant Nos.11831003,12171111)Natural Science Foundation of Beijing in China(Grant No.KZ202110005011)。
文摘The global well-posedness of another class of initial-boundary value problem on two/three-dimensional incompressible MHD-Boussinesq equations in the bounded domain with the smooth boundary is studied. The existence of a class of global weak solution to the initial boundary value problem for two/three-dimensional incompressible MHD-Boussinesq equation with the given pressure-velocity’s relation boundary condition for the fluid field,one generalized perfectly conducting boundary condition for the magnetic field and one density/temperature-velocity’s relation boundary condition for the density/temapture at the boundary is obtained, and the global existence and uniqueness of the smooth solution to the corresponding problem in two-dimensional case for the smooth initial data is also proven.
基金The research of Gaowei Cao was supported in part by the NSFC(Grant 11701551 and Grant 11971024)the China Scholarship Council No.202004910200.The research of Hui Kan was supported in part by the NSFC(Grant 11801551)+1 种基金The research of Wei Xiang was supported in part by the Research Grants Council of the HKSAR,China(Project No.City U 11332916,Project No.City U 11304817 and Project No.City U 11303518)The research of X.Z.Yang was supported in part by the NSFC(Grant 11471332)。
文摘In this paper, we are concerned with the necessary and sufficient condition of the global existence of smooth solutions of the Cauchy problem of the multi-dimensional scalar conservation law with source-term,where the initial data lies in W1,∞(Rn) ∩ C1(Rn). We obtain the solution formula for smooth solution, and then apply it to establish and prove the necessary and sufficient condition for the global existence of smooth solution. Moreover, if the smooth solution blows up at a finite time, the exact lifespan of the smooth solution can be obtained. In particular, when the source term vanishes, the corresponding theorem for the homogeneous case is obtained too. Finally, we give two examples as its applications, one for the global existence of the smooth solution and the other one for the blowup of the smooth solutions at any given positive time.
基金This research is supported by "Foundation of office of overseas Chinese affair under the state council: 03QZR09"
文摘In this article, first, the authors prove that there exists a unique global smooth solution for the Cauthy problem to the hyperbolic conservation laws systems with relaxation; second, in the large time station, they prove that the global smooth solutions of the hyperbolic conservation laws systems with relaxation converge to rarefaction waves solution at a determined L^P（p ≥ 2） decay rate.
