【作者】 王建峰；

【导师】 何炳生；

【作者基本信息】 南京大学 ， 应用数学， 2013， 博士

【摘要】 多项式微分方程组是非线性微分方程组的重要情形。本篇学位论文首先讨论了一阶多项式常微分方程组光滑解的存在性。我们利用与一元函数有关的常系数线性方程组的理论,将一阶多项式常微分方程组转化为二次多项式方程组与一阶常系数线性常微分方程组的合成。接着我们利用傅里叶变换将一阶常系数线性常微分方程组转化成线性方程组,并且求出其通解,然后根据该通解,我们将二次方程组转化为与非线性Fredholm积分方程组等价的不动点问题。我们利用压缩映射不动点原理和绍德尔不动点定理以及勒雷-邵德尔拓扑度理论,可以得到一阶多项式常微分方程组的光滑解的存在性有如下三种可能：(1)在有界闭区间[a,b]上,(?)a1>0,当b-a<a1时,光滑解都存在并且唯一。(2)(?)a2>a1>0,当b-a<a2时,光滑解在有界闭区间[a,b]上存在。(3)在有界闭区间[a,b]上,光滑解存在。并且,我们可以写出光滑解的与非线性Fredholm积分方程组有关的显式表达式。接着,我们将结论推广到一阶的多项式偏微分方程组。我们利用与多元函数有关的常系数线性方程组的理论,将一阶多项式偏微分方程组转化为二次多项式方程组与一阶常系数线性偏微分方程组的合成。接着我们利用傅里叶变换将一阶常系数线性偏微分方程组转化成线性方程组,并且求出其通解,然后根据该通解,我们将二次方程组转化为与非线性Fredholm积分方程组等价的不动点问题。我们利用压缩映射不动点原理和绍德尔不动点定理以及勒雷-邵德尔拓扑度理论,可以得到一阶多项式偏微分方程组的光滑解的存在性有如下三种可能：(1)在有界闭区域Ω上,(?)a1>0,当m(Ω)<a1时,光滑解都存在并且唯一(2)(?)a2>a1>0,当m(Ω)<a2时,光滑解在有界闭区域Ω上存在。(3)在有界闭区域Ω上,光滑解存在。同样,我们可以写出光滑解的与非线性Fredholm积分方程组有关的显式表达式。最后,我们讨论了纳维-斯托克斯方程组。我们利用与四元函数有关的常系数线性方程组的理论,将纳维-斯托克斯方程组转化为二次多项式方程组与一阶常系数线性偏微分方程组的合成。接着我们利用傅里叶变换将一阶常系数线性偏微分方程组转化成线性方程组,并且求出其通解,然后根据该通解,我们将二次方程组转化为与非线性Fredholm积分方程组等价的不动点问题。我们利用压缩映射不动点原理和绍德尔不动点定理以及勒雷-邵德尔拓扑度理论,可以得到纳维-斯托克斯方程组的光滑解的存在性有如下三种可能：(1)在Ω×[0,T]上,(?)a1>0,当m(Ω)<a1时,光滑解都存在并且唯一。(2)(?)a2>a1>0,当m(Q)<u2时,光滑解在Ω×[0,T]上存在。(3)在Ω×[0,T]上,光滑解存在。同样,我们可以写出光滑解的与非线性Fredholm积分方程组有关的显式表达式。更多还原

【Abstract】 The polynomial differential equations are the important scenarios of the nonlinear differential equations. This dissertation discussed how to solve the first order polyno-mial ordinary differential equations. At first, we put forward the theory of the linear equations about the unknown one variable functions with constant coefficients. Sec-ondly, we use this theory to convert the first order polynomial ordinary differential equations into the simultaneous of the first order linear ordinary differential equation-s with constant coefficients and the quadratic equations. Thirdly, we use the Fourier transformation to convert the first order linear ordinary differential equations with con-stant coefficients into the linear equations, and we work out the explicit general solution of it. With this general solution, we convert the quadratic equations into the question of the fixed-point of a continuous mapping. And it is equivalent to the nonlinear Fred-holm integral equations. We use the theories about the contraction mapping principle, the Schauder fixed-point theorem and the Leray-Schauder degree to get the existence and the uniqueness of the fixed-point. Finally, we find that there exist three scenarios about the smoothing-solution of the first order polynomial ordinary differential equa-tions:(l)In the region [a, b], where [a, b] is a bounded and closed set,(?)a1>0, if b-a<a1, then the smoothing-solution exists and unique.(2)In the region [a, b], where [a, b] is a bounded and closed set,(?)a2> a1>0, if b-a<a2, then the smoothing-solution exists.(3)In the region [a,b], where [a, b] is a bounded and closed set, the smoothing-solution exists. Moreover we can write out the explicit expressions of the smoothing-solutions which are related with the nonlinear Fredholm integral equations.Consequently we discussed how to solve the first order polynomial partial differential equations. At first, we put forward the theory of the linear equations about the un-known multivariate functions with constant coefficients. Secondly, we use this theory to convert the first order polynomial partial differential equations into the simultane-ous of the first order linear partial differential equations with constant coefficients and the quadratic equations. Thirdly, we use the Fourier transformation to convert the first order linear partial differential equations with constant coefficients into the linear equa-tions, and we work out the explicit general solution of it. With this general solution, we convert the quadratic equations into the question of the fixed-point of a continuous mapping. And it is equivalent to the nonlinear Fredholm integral equations. We use the theories about the contraction mapping principle, the Schauder fixed-point theorem and the Leray-Schauder degree to get the existence and the uniqueness of the fixed-point. Finally, we find that there exist three scenarios about the smoothing-solution of the first order polynomial partial differential equations:(1)In the region Ω, where Ω is a bounded and closed set in Rn,3a1>0, if m(Ω)<a1, then the solution exists and unique, where m(Ω) is the Lebesgue measure of Ω.(2)In the region Ω, where Ω is a bounded and closed set in Rn,(?)a2> a1>0, if m(Ω)<a2, then the solution exists, where m(Q) is the Lebesgue measure of Ω.(3)In the region Q, where Ω. is a bounded and closed set in Rn, the solution exists. Moreover we can write out the explicit expressions of the smoothing-solutions which are related with the nonlinear Fredholm integral equations.Finally we discussed how to solve the Navier-Stokes equations. At first, we put forward the theory of the linear equations which is about the unknown multivariate functions with constant coefficients. Secondly, we use this theory to convert the Navier-Stokes equations into the simultaneous of the first order linear partial differential equation-s with constant coefficients and the quadratic equations. Thirdly, we use the Fourier transformation to convert the first order linear partial differential equations with con-stant coefficients into the linear equations, and we work out the explicit solution of it. With this general solution, we convert the quadratic equations into the question of the fixed-point of a continuous mapping. And it is equivalent to the nonlinear Fredholm integral equations. We use the theories about the contraction mapping principle, the Schauder fixed-point theorem and the Leray-Schauder degree to get the existence and the uniqueness of the fixed-point. Finally, we find that there exist three scenarios about the smoothing-solution of the Navier-Stokes equations:(1)In the region Ω×[0, T], where T>0, Ω is a bounded and closed set in R3,3a1>0, if m(Ω)<a1, then the smoothing-solution exists and unique, where m(Ω) is the Lebesgue measure of Ω.(2)In the region Ω×[0, T], where T>0, Ω is a bounded and closed set in R3,3a2> a1>0, if m(Ω)<a2, then the smoothing-solution exists, where m(Ω) is the Lebesgue measure of Ω.(3)In the region Ω x [0,T], where T>0, Ω is a bounded and closed set in R3, the smoothing-solution exists.Moreover we can write out the explicit expressions of the smoothing-solutions which are related with the nonlinear Fredholm integral equations.更多还原