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
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
An essential feature of the subdiffusion equations with theα-order time fractional derivative is the weak singularity at the initial time.The weak regularity of the solution is usually characterized by a regularity p...An essential feature of the subdiffusion equations with theα-order time fractional derivative is the weak singularity at the initial time.The weak regularity of the solution is usually characterized by a regularity parameterσ∈(0,1)∪(1,2).Under this general regularity assumption,we present a rigorous analysis for the truncation errors and develop a new tool to obtain the stability results,i.e.,a refined discrete fractional-type Grönwall inequality(DFGI).After that,we obtain the pointwise-in-time error estimate of the widely used L1 scheme for nonlinear subdiffusion equations.The present results fill the gap on some interesting convergence results of L1 scheme onσ∈(0,α)∪(α,1)∪(1,2].Numerical experiments are provided to demonstrate the effectiveness of our theoretical analysis.展开更多
This article is concerned with the pointwise error estimates for vanishing vis- cosity approximations to scalar convex conservation laws with boundary.By the weighted error function and a bootstrap extrapolation techn...This article is concerned with the pointwise error estimates for vanishing vis- cosity approximations to scalar convex conservation laws with boundary.By the weighted error function and a bootstrap extrapolation technique introduced by Tadmor-Tang,an optimal pointwise convergence rate is derived for the vanishing viscosity approximations to the initial-boundary value problem for scalar convex conservation laws,whose weak entropy solution is piecewise C 2 -smooth with interaction of elementary waves and the ...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.
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
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
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
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
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.
基金supported by the National Natural Science Foundation of China under grants 11771162,11771035,12171376 and 2020-JCJQ-ZD-029.
文摘An essential feature of the subdiffusion equations with theα-order time fractional derivative is the weak singularity at the initial time.The weak regularity of the solution is usually characterized by a regularity parameterσ∈(0,1)∪(1,2).Under this general regularity assumption,we present a rigorous analysis for the truncation errors and develop a new tool to obtain the stability results,i.e.,a refined discrete fractional-type Grönwall inequality(DFGI).After that,we obtain the pointwise-in-time error estimate of the widely used L1 scheme for nonlinear subdiffusion equations.The present results fill the gap on some interesting convergence results of L1 scheme onσ∈(0,α)∪(α,1)∪(1,2].Numerical experiments are provided to demonstrate the effectiveness of our theoretical analysis.
基金supported by the NSF China#10571075NSF-Guangdong China#04010473+1 种基金The research of the second author was supported by Jinan University Foundation#51204033the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State education Ministry#2005-383
文摘This article is concerned with the pointwise error estimates for vanishing vis- cosity approximations to scalar convex conservation laws with boundary.By the weighted error function and a bootstrap extrapolation technique introduced by Tadmor-Tang,an optimal pointwise convergence rate is derived for the vanishing viscosity approximations to the initial-boundary value problem for scalar convex conservation laws,whose weak entropy solution is piecewise C 2 -smooth with interaction of elementary waves and the ...
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 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.
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.
