This paper is concerned with the dissipation of solutions of the isentropic Navier-Stokes equations in even and bigger than two multi-dimensions. Pointwise estimates of the time-asymptotic shape of the solutions are o...This paper is concerned with the dissipation of solutions of the isentropic Navier-Stokes equations in even and bigger than two multi-dimensions. Pointwise estimates of the time-asymptotic shape of the solutions are obtained and the generalized Huygan's principle is exhibited. The approch of the paper is based on the detailed analysis of the Green function of Iinearized system. This is used to study the coupling of nonlinear diffesion waves.展开更多
This paper studies the time asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in one dimension.First,the pointwise estimates of solutions are obtained,furthermore,we obtain the op...This paper studies the time asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in one dimension.First,the pointwise estimates of solutions are obtained,furthermore,we obtain the optimal Lp,1≤ p ≤ ＋∞,convergence rate of solutions for small initial data.Then we establish the local existence of solutions,the blow up criterion and the sufficient condition to ensure the nonnegativity of solutions for large initial data.Our approach is based on the detailed analysis of the Green function of the linearized equation and some energy estimates.展开更多
In this article,we study the pointwise estimates of solutions to the nonlinear viscous wave equation in even dimensions(n≥4).We use the Green’s function method.Our approach is on the basis of the detailed analysis o...In this article,we study the pointwise estimates of solutions to the nonlinear viscous wave equation in even dimensions(n≥4).We use the Green’s function method.Our approach is on the basis of the detailed analysis of the Green’s function of the linearized system.We show that the decay rates of the solution for the same problem are different in even dimensions and odd dimensions.It is shown that the solution exhibits a generalized Huygens principle.展开更多
In this paper, we study the linear thermo-visco-elastic system in one-dimensional space variable. The mathematical model is a hyperbolic-parabolic partial differential system. The solutions of the system show some dec...In this paper, we study the linear thermo-visco-elastic system in one-dimensional space variable. The mathematical model is a hyperbolic-parabolic partial differential system. The solutions of the system show some decay property due to the parabolicity. Based on detailed analysis on the Green function of the system, the pointwise estimates of the solutions are obtained, from which the generalized Huygens’ principle is shown.展开更多
For linear combinations of Gamma operators, if 0<a<2r, 1/2-1/2r≤λ≤1. or 0≤λ≤1/2-1/2r(r≥2),0 <a<r+1/1-λ, we obtain an equivalent theorem with ω(?)(f,t) instead of ω(?)(f,t), where ω(?)(f,t) is th...For linear combinations of Gamma operators, if 0<a<2r, 1/2-1/2r≤λ≤1. or 0≤λ≤1/2-1/2r(r≥2),0 <a<r+1/1-λ, we obtain an equivalent theorem with ω(?)(f,t) instead of ω(?)(f,t), where ω(?)(f,t) is the Ditzian-Totik moduli of smoothness.展开更多
Explicit bounds on bounded solutions to a new class of Volterra-type linear and nonlinear discrete inequalities involving infinite sums are established. These inequalities can be viewed as discrete analogues of some V...Explicit bounds on bounded solutions to a new class of Volterra-type linear and nonlinear discrete inequalities involving infinite sums are established. These inequalities can be viewed as discrete analogues of some Volterra-type inequalities having improper integral functionals,which are new to the literature.展开更多
This paper is devoted to the pointwise estimate of solutions for the initial value problem to the three-dimensional compressible magnetohydrodynamic equations, which models the dynamics of compressible quasi-neutrally...This paper is devoted to the pointwise estimate of solutions for the initial value problem to the three-dimensional compressible magnetohydrodynamic equations, which models the dynamics of compressible quasi-neutrally ionized fluids under the influence of electromagnetic fields. Based on the detailed analysis of the Green function of the linearized system, we obtain the pointwise estimates of smooth solutions when the initial data is sufficiently small with the algebraic decay to the constant equilibrium. As the by-product, we also show the corresponding pL-estimates of the smooth solutions.展开更多
The object of this paper is to establish the pointwise estimations of approximation of functions in C^1 and their derivatives by Hermite interpolation polynomials. The given orders have been proved to be exact in gen-...The object of this paper is to establish the pointwise estimations of approximation of functions in C^1 and their derivatives by Hermite interpolation polynomials. The given orders have been proved to be exact in gen- eral.展开更多
In this paper, we will introduce how to apply Green＇s function method to get the pointwise estimates for the solutions of Cauchy problem of nonlinear evolution equations with dissipative structure. First of all, we i...In this paper, we will introduce how to apply Green＇s function method to get the pointwise estimates for the solutions of Cauchy problem of nonlinear evolution equations with dissipative structure. First of all, we introduce the pointwise estimates of the time-asymptotic shape of the solutions of the isentropic Navier-Stokes equations and show to exhibit the generalized Huygen＇s principle. Then, for other nonlinear dissipative evolution equations, we will only introduce the result and give some brief explanations. Our approach is based on the detailed analysis of the Green＇s function of the linearized system and micro-local analysis, such as frequency decomposition and so on.展开更多
We consider the Cauchy problem of Euler equations with damping. Based on the Green function and energy estimates of solutions, we improve the pointwise estimates and obtainL 1 estimate of solutions.
The large time behavior of solutions to the two-dimensional perturbed Hasegawa- Mima equation with large initial data is studied in this paper. Based on the time-frequency decomposition and the method of Green functio...The large time behavior of solutions to the two-dimensional perturbed Hasegawa- Mima equation with large initial data is studied in this paper. Based on the time-frequency decomposition and the method of Green function, we not only obtain the optimal decay rate but also establish the pointwise estimate of global classical solutions.展开更多
Let a function f E C[-1, 1], changes its monotonisity at the finite collection Y ：= {y1,……, ys} of s points Yi ∈ （-1, 1）. For each n 〉 N（Y）, we construct an algebraic polynomial Pn, of degree 〈 n, which is c...Let a function f E C[-1, 1], changes its monotonisity at the finite collection Y ：= {y1,……, ys} of s points Yi ∈ （-1, 1）. For each n 〉 N（Y）, we construct an algebraic polynomial Pn, of degree 〈 n, which is comonotone with f, that is changes its monotonisity at the same points yi as f, and ｜f（x） - Pn（x）｜ ≤ c（s）ω2 （f1 √1-x^2/n）,x∈ [-1,1] where N（Y） is a constant depending only on Y, c（s） is a constant depending only on s and ω2 （f, t） is the second modulus of smoothness of f.展开更多
In this paper,we analyze and test a high-order compact difference scheme numerically for solving a two-dimensional nonlinear Kuramoto-Tsuzuki equation under the Neumann boundary condition.A three-level average techniq...In this paper,we analyze and test a high-order compact difference scheme numerically for solving a two-dimensional nonlinear Kuramoto-Tsuzuki equation under the Neumann boundary condition.A three-level average technique is utilized,thereby leading to a linearized difference scheme.The main work lies in the pointwise error estimate in H^(2)-norm.The optimal fourth-order convergence order is proved in combination of induction,the energy method and the embedded inequality.Moreover,we establish the stability of the difference scheme with respect to the initial value under very mild condition,however,does not require any step ratio restriction.Extensive numerical examples with/without exact solutions under diverse cases are implemented to validate the theoretical results.展开更多
基金Supported in part by National Natural Science Foundationof China (19871065) Hua-Cheng Grant
文摘This paper is concerned with the dissipation of solutions of the isentropic Navier-Stokes equations in even and bigger than two multi-dimensions. Pointwise estimates of the time-asymptotic shape of the solutions are obtained and the generalized Huygan's principle is exhibited. The approch of the paper is based on the detailed analysis of the Green function of Iinearized system. This is used to study the coupling of nonlinear diffesion waves.
文摘This paper studies the time asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in one dimension.First,the pointwise estimates of solutions are obtained,furthermore,we obtain the optimal Lp,1≤ p ≤ ＋∞,convergence rate of solutions for small initial data.Then we establish the local existence of solutions,the blow up criterion and the sufficient condition to ensure the nonnegativity of solutions for large initial data.Our approach is based on the detailed analysis of the Green function of the linearized equation and some energy estimates.
文摘In this article,we study the pointwise estimates of solutions to the nonlinear viscous wave equation in even dimensions(n≥4).We use the Green’s function method.Our approach is on the basis of the detailed analysis of the Green’s function of the linearized system.We show that the decay rates of the solution for the same problem are different in even dimensions and odd dimensions.It is shown that the solution exhibits a generalized Huygens principle.
基金Xingwen Hao's research was supported in part by National Natural Science Foundation of China (10571120 and 10971135)Shanghai Shuguang Project (06SG11)+1 种基金the Program for New Century Excellent Talents of Chinese Ministry of Education (NCET-07-0546) Doctorial Foundation of Weifang University (2011BS11)
文摘In this paper, we study the linear thermo-visco-elastic system in one-dimensional space variable. The mathematical model is a hyperbolic-parabolic partial differential system. The solutions of the system show some decay property due to the parabolicity. Based on detailed analysis on the Green function of the system, the pointwise estimates of the solutions are obtained, from which the generalized Huygens’ principle is shown.
基金Supported by the Hebei Provincial Natural Science Foundation of China(101090). Supported by the Major Subject Foundation of Hebei Normal University.
文摘For linear combinations of Gamma operators, if 0<a<2r, 1/2-1/2r≤λ≤1. or 0≤λ≤1/2-1/2r(r≥2),0 <a<r+1/1-λ, we obtain an equivalent theorem with ω(?)(f,t) instead of ω(?)(f,t), where ω(?)(f,t) is the Ditzian-Totik moduli of smoothness.
文摘Explicit bounds on bounded solutions to a new class of Volterra-type linear and nonlinear discrete inequalities involving infinite sums are established. These inequalities can be viewed as discrete analogues of some Volterra-type inequalities having improper integral functionals,which are new to the literature.
基金Supported by Research Grant of Department of Education of Hubei Province(Q20142803)
文摘This paper is devoted to the pointwise estimate of solutions for the initial value problem to the three-dimensional compressible magnetohydrodynamic equations, which models the dynamics of compressible quasi-neutrally ionized fluids under the influence of electromagnetic fields. Based on the detailed analysis of the Green function of the linearized system, we obtain the pointwise estimates of smooth solutions when the initial data is sufficiently small with the algebraic decay to the constant equilibrium. As the by-product, we also show the corresponding pL-estimates of the smooth solutions.
文摘The object of this paper is to establish the pointwise estimations of approximation of functions in C^1 and their derivatives by Hermite interpolation polynomials. The given orders have been proved to be exact in gen- eral.
基金supported by National Science Foundation of China(11071162)Shanghai Municipal Natural Science Foundation (09ZR1413500)
文摘In this paper, we will introduce how to apply Green＇s function method to get the pointwise estimates for the solutions of Cauchy problem of nonlinear evolution equations with dissipative structure. First of all, we introduce the pointwise estimates of the time-asymptotic shape of the solutions of the isentropic Navier-Stokes equations and show to exhibit the generalized Huygen＇s principle. Then, for other nonlinear dissipative evolution equations, we will only introduce the result and give some brief explanations. Our approach is based on the detailed analysis of the Green＇s function of the linearized system and micro-local analysis, such as frequency decomposition and so on.
基金the National Natural Science Foundation of China(1013105)
文摘We consider the Cauchy problem of Euler equations with damping. Based on the Green function and energy estimates of solutions, we improve the pointwise estimates and obtainL 1 estimate of solutions.
基金supported by the National Natural Science Foundation of China(11231006)
文摘The large time behavior of solutions to the two-dimensional perturbed Hasegawa- Mima equation with large initial data is studied in this paper. Based on the time-frequency decomposition and the method of Green function, we not only obtain the optimal decay rate but also establish the pointwise estimate of global classical solutions.
文摘Let a function f E C[-1, 1], changes its monotonisity at the finite collection Y ：= {y1,……, ys} of s points Yi ∈ （-1, 1）. For each n 〉 N（Y）, we construct an algebraic polynomial Pn, of degree 〈 n, which is comonotone with f, that is changes its monotonisity at the same points yi as f, and ｜f（x） - Pn（x）｜ ≤ c（s）ω2 （f1 √1-x^2/n）,x∈ [-1,1] where N（Y） is a constant depending only on Y, c（s） is a constant depending only on s and ω2 （f, t） is the second modulus of smoothness of f.
基金supported in part by Natural Sciences Foundation of Zhejiang Province(No.LZ23A010007)in part by the National Natural Science Foundation of China(No.12271518)+1 种基金Natural Science Foundation of Jiangsu Province(No.BK20201149)the Fundamental Research Funds of Xuzhou(No.KC21019)
文摘In this paper,we analyze and test a high-order compact difference scheme numerically for solving a two-dimensional nonlinear Kuramoto-Tsuzuki equation under the Neumann boundary condition.A three-level average technique is utilized,thereby leading to a linearized difference scheme.The main work lies in the pointwise error estimate in H^(2)-norm.The optimal fourth-order convergence order is proved in combination of induction,the energy method and the embedded inequality.Moreover,we establish the stability of the difference scheme with respect to the initial value under very mild condition,however,does not require any step ratio restriction.Extensive numerical examples with/without exact solutions under diverse cases are implemented to validate the theoretical results.
