Presents the iterative method of solving Cauchy problem with reproducing kernel for nonlinear hyperbolic equations, and the application of the computational technique of reproducing kernel space to simplify, the itera...Presents the iterative method of solving Cauchy problem with reproducing kernel for nonlinear hyperbolic equations, and the application of the computational technique of reproducing kernel space to simplify, the iterative computation and increase the convergence rate and points out that this method is still effective. Even if the initial condition is discrete.展开更多
In this paper we study the Cauchy problem for a class of semi-linear parabolic type equations withweak data in the homogeneous spaces. We give a method which can be used to construct local mild solutionsof the abstrac...In this paper we study the Cauchy problem for a class of semi-linear parabolic type equations withweak data in the homogeneous spaces. We give a method which can be used to construct local mild solutionsof the abstract Cauchy problem in Cσ,s,p and Lq([O, T);Hs,p) by introducing the concept of both admissiblequintuplet and compatible space and establishing time-space estimates for solutions to the linear parabolic typeequations. For the small data, we prove that these results can be extended globally in time. We also study theregularity of the solution to the abstract Cauchy problem for nonlinear parabolic type equations in Cσ,s,p. Asan application, we obtain the same result for Navier-Stokes equations with weak initial data in homogeneousSobolev spaces.展开更多
In this paper, we consider the local and global solution for the nonlinear Schrodinger equation with data in the homogeneous and nonhomogeneous Besov space and the scattering result for small data. The techniques to b...In this paper, we consider the local and global solution for the nonlinear Schrodinger equation with data in the homogeneous and nonhomogeneous Besov space and the scattering result for small data. The techniques to be used are adapted from the Strichartz type estimate, Kato＇s smoothing effect and the maximal function （in time） estimate for the free SchrSdinger operator.展开更多
In this paper the Cauchy problem for a class of nonhomogenous Navier-Stokes equationsin the infinite cylinder is considered. We construct a unique local solution infor a class of nonhomogeneous Navier-Stokes equation...In this paper the Cauchy problem for a class of nonhomogenous Navier-Stokes equationsin the infinite cylinder is considered. We construct a unique local solution infor a class of nonhomogeneous Navier-Stokes equations provided that initialdata are in, where is an exponent determined by the structure of nonlinear termsand p,q are such that . Meanwhile under suitable conditions we also obtain thatprovided that initial data are sufficiently small.展开更多
In this work, we study the smoothing effect of Cauchy problem in Sobolev space for the spatially homogeneous Landau equation in the Maxwellian case. We obtain a precise estimate with respect to time variable, which im...In this work, we study the smoothing effect of Cauchy problem in Sobolev space for the spatially homogeneous Landau equation in the Maxwellian case. We obtain a precise estimate with respect to time variable, which implies the ultra-analytic effect of weak solutions.展开更多
In this paper, the necessary conditions of the existence of C￣2 solution. in someinitial problems of Navier-Stokes equations are given. and examples of instability ofinitial value (at t=0) problems are also given. Th...In this paper, the necessary conditions of the existence of C￣2 solution. in someinitial problems of Navier-Stokes equations are given. and examples of instability ofinitial value (at t=0) problems are also given. The initial value problem ofNavier-Stokes equation is one of the most fundamental problem for this equationvarious authors studies this problem and contributed a number of results .J.Leray .aFrench professor, proved the existence of Navier-Stokes equation under certain definedinitial and boundary value conditions .In this paper,with certain rigorously definedkey.concepts,based upon the basic theory of J.Hadmard partial differentialequanous￣[1], gives a fundamental theory of instability of Navier-Stokes equations.Finally,many examples are given,proofs referring to Ref.[4] .展开更多
In this paper, we study the Cauchy problem for the modified Camassa-Holm equation mt ＋ umx ＋ 2ux m = 0, m =（1- δx^2）^2u,u（x, 0） = u0（x） ∈ H^s（R）, x ∈ R, t 〉 0,and show that the solution map is not unifor...In this paper, we study the Cauchy problem for the modified Camassa-Holm equation mt ＋ umx ＋ 2ux m = 0, m =（1- δx^2）^2u,u（x, 0） = u0（x） ∈ H^s（R）, x ∈ R, t 〉 0,and show that the solution map is not uniformly continuous in Sobolev spaces H^s（R） for s 〉 7/2. Compared with the periodic problem, the non-periodic problem is more difficult,e.g., it depends on the conservation law. Our proof is based on the estimates for the actual solutions and the approximate solutions, which consist of a low frequency and a high frequency part.展开更多
In this paper, a Darbao type random fixed point theorem for a system of weak continuous random operators with random domain is first proved. When, by using the theorem, some existence criteria of random solutions for ...In this paper, a Darbao type random fixed point theorem for a system of weak continuous random operators with random domain is first proved. When, by using the theorem, some existence criteria of random solutions for a systems of nonlinear random Volterra integral equations relative to the weak topology in Banach spaces are given. As applications, some existence theorems of weak random solutions for the random Cauchy problem of a system of nonlinear random differential equations are obtained, as well as the existence of extremal random solutions and random comparison results for these systems of random equations relative to weak topology in Banach spaces. The corresponding results of Szep, Mitchell-Smith, Cramer-Lakshmikantham, Lakshmikantham-Leela and Ding are improved and generalized by these theorems.展开更多
文摘Presents the iterative method of solving Cauchy problem with reproducing kernel for nonlinear hyperbolic equations, and the application of the computational technique of reproducing kernel space to simplify, the iterative computation and increase the convergence rate and points out that this method is still effective. Even if the initial condition is discrete.
基金This work was supported by the National Natural Science Foundation of China(Grant No.19971001)the Special Funds for Major State Basic Research Projects of China.
文摘In this paper we study the Cauchy problem for a class of semi-linear parabolic type equations withweak data in the homogeneous spaces. We give a method which can be used to construct local mild solutionsof the abstract Cauchy problem in Cσ,s,p and Lq([O, T);Hs,p) by introducing the concept of both admissiblequintuplet and compatible space and establishing time-space estimates for solutions to the linear parabolic typeequations. For the small data, we prove that these results can be extended globally in time. We also study theregularity of the solution to the abstract Cauchy problem for nonlinear parabolic type equations in Cσ,s,p. Asan application, we obtain the same result for Navier-Stokes equations with weak initial data in homogeneousSobolev spaces.
文摘In this paper, we consider the local and global solution for the nonlinear Schrodinger equation with data in the homogeneous and nonhomogeneous Besov space and the scattering result for small data. The techniques to be used are adapted from the Strichartz type estimate, Kato＇s smoothing effect and the maximal function （in time） estimate for the free SchrSdinger operator.
文摘In this paper the Cauchy problem for a class of nonhomogenous Navier-Stokes equationsin the infinite cylinder is considered. We construct a unique local solution infor a class of nonhomogeneous Navier-Stokes equations provided that initialdata are in, where is an exponent determined by the structure of nonlinear termsand p,q are such that . Meanwhile under suitable conditions we also obtain thatprovided that initial data are sufficiently small.
文摘In this work, we study the smoothing effect of Cauchy problem in Sobolev space for the spatially homogeneous Landau equation in the Maxwellian case. We obtain a precise estimate with respect to time variable, which implies the ultra-analytic effect of weak solutions.
文摘In this paper, the necessary conditions of the existence of C￣2 solution. in someinitial problems of Navier-Stokes equations are given. and examples of instability ofinitial value (at t=0) problems are also given. The initial value problem ofNavier-Stokes equation is one of the most fundamental problem for this equationvarious authors studies this problem and contributed a number of results .J.Leray .aFrench professor, proved the existence of Navier-Stokes equation under certain definedinitial and boundary value conditions .In this paper,with certain rigorously definedkey.concepts,based upon the basic theory of J.Hadmard partial differentialequanous￣[1], gives a fundamental theory of instability of Navier-Stokes equations.Finally,many examples are given,proofs referring to Ref.[4] .
基金supported by the National Natural Science Foundation of China(11226159)
文摘In this paper, we study the Cauchy problem for the modified Camassa-Holm equation mt ＋ umx ＋ 2ux m = 0, m =（1- δx^2）^2u,u（x, 0） = u0（x） ∈ H^s（R）, x ∈ R, t 〉 0,and show that the solution map is not uniformly continuous in Sobolev spaces H^s（R） for s 〉 7/2. Compared with the periodic problem, the non-periodic problem is more difficult,e.g., it depends on the conservation law. Our proof is based on the estimates for the actual solutions and the approximate solutions, which consist of a low frequency and a high frequency part.
文摘In this paper, a Darbao type random fixed point theorem for a system of weak continuous random operators with random domain is first proved. When, by using the theorem, some existence criteria of random solutions for a systems of nonlinear random Volterra integral equations relative to the weak topology in Banach spaces are given. As applications, some existence theorems of weak random solutions for the random Cauchy problem of a system of nonlinear random differential equations are obtained, as well as the existence of extremal random solutions and random comparison results for these systems of random equations relative to weak topology in Banach spaces. The corresponding results of Szep, Mitchell-Smith, Cramer-Lakshmikantham, Lakshmikantham-Leela and Ding are improved and generalized by these theorems.
