高阶非线性抛物方程解的性质及其数值解法
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摘要
非线性高阶抛物型方程作为数学模型描述了很多物理学、化学、信息科学、生命科学、地理科学、环境科学以及空间科学等领域中出现的现象,是非线性科学的重要组成部分.
本文研究了三种具有广泛物理背景的高阶非线性抛物型方程解的性质及其数值解法.
首先,我们研究了扩散系数依赖于未知函数本身的薄膜外延增长方程的初边值问题利用Leyar-Schauder不动点定理,Campanato空间的性质和先验估计,我们得到了初边值问题(1)整体解的存在唯一性以及古典解的存在性.
然后,我们研究了描述薄膜外延增长的方程和晶体表面生长的方程的解的长时间行为问题.借用[1,2]中关于整体吸引子的存在性定理,我们得到了在分数维空间Hk(0≤k<5)中方程(2)的整体吸引子的存在性以及在分数维空间Hk(0≤k<+∞)中方程(3)整体吸引子的存在性.需要强调的是方程(2)中p的取值不但可以是整数,还可以是分数.当我们对方程(2)进行先验估计的时候,如果对非线性项div(|(?)u|p-2(?)u)微分多次,则会出现负指数的情形(这种情形的先验估计很难得到).因此,我们只能得到在Hk(0≤k<5)空间中方程(2)的整体吸引子存在性.而方程(3)就不存在这种情况,我们可以利用迭代法得到其在更广泛的分数维空间Hk(0≤k<+∞)中整体吸引子的存在性.
另外,我们考虑了晶体表面生长方程和薄膜外延增长方程的数值解法.文献[3]曾经对Cahn-Hilliard方程进行过有限元分析.借助于一个双调和问题的有限元投影逼近,作者得到了方程(4)的最优阶L2模误差估计.本文中,我们考虑方程(2)和方程(3)的一维情形的有限元方法.我们也借助于一个双调和问题的有限元投影逼近,得到了方程的最优阶L2模误差估计,并进行了数值分析.需要特别指出的是,在[3]中,方程的非线性项是(?)2/(?)x2φ(u),在建立逼近格式的时候,可以通过两次分部积分(在适当的边界条件下)和先验估计结果来进行分析,即其中Sh(k)为剖分上分段k≥3次多项式构成的有限元空间.而后利用双调和方程标准的有限元分析方法[4]得到想要的结果.而我们所考虑的两个方程的非线性项与[3]中不同.在建立逼近格式的时候只能分部积分一次(在适当的边界条件下),利用先验估计结果可得由于有一阶导数项的存在,此时利用双调和方程标准的有限元分析方法[4],空间上的误差精度会下降一阶.
修正Swift-Hohenberg方程也是一类带有一阶导数项的高阶抛物型方程.我们利用Fourier谱方法对该方程的数值解进行分析,建立了半离散格式和全离散格式,给出了误差估计.在对修正Swift-Hohenberg方程进行Fourier谱分析的时候也会遇到类似上面的问题.但是,方程(5)的一阶导数项是二次的,运用合适的分部积分和先验估计,可以避免空间上误差精度的下降.
As mathematics models, higher-order nonlinear parabolic type equations describe many phenomenons which exist in Physics, Chemistry, Informatics, Bioscience, Geo-science, Environmental science, Space science and so on. Higher-order nonlinear parabol-ic type equations is an important part of Nonlinear science.
In this paper, we consider the properties of solutions and numerical solutions for three types of higher order nonlinear parabolic equation with extensive physical back-ground.
Firstly, we consider the higher order nonlinear parabolic equation describing thin-film epitaxial growth whose diffusion coefficient dependents on the unknown function Using Leyar-Schauder fixed point theorem, the qualities of Campanato space and a prior estimates, we get the existence and uniqueness of global solutions and the existence of classical solutions for IBVP (1).
Secondly, we study the long time behavior of solutions for the higher order non-linear parabolic equation describing thin-film epitaxial growth and the higher order nonlinear parabolic equation describing process growing of a crys-tal surface Using the theorem of existence of global attractor in [1,2], we obtain the existence of global attractor for equation (2) in fractal dimension space Hk(0≤k<5) and the existence of global attractor for equation (3) in fractal dimension space Hk(0≤k<+∞). Noticing that the value of p is not only integer, but also fraction in (2), when we consider the a prior estimates for equation (2), if differential many times for term div(|(?)u|p-2(?)u), it maybe exists negative exponential (the a priori estimates of this kind of situation are hard to obtain). So, we have to obtain the existence of global attractor for (2) in space Hk(0≤k<5). On the other hand, there is no this kind of situation for equation (3), we can use iteration method get the existence of global attractor for equation (3) in space Hk(0≤k<+∞).
Thirdly, we consider the numerical solutions for the higher order nonlinear parabol-ic equations describing thin-film epitaxial growth and process growing of a crystal sur-face. Using finite element method,[3] studied the Cahn-Hilliard equation Basing on the finite element projection approximation of a biharmonic problem, he ob-tain the optimal order L2-norm error estimate for equation4. In this paper, we consider the finite element method for equation (2) and equation (3). We also use the finite ele-ment projection approximation of a biharmonic problem, obtain the optimal order L2-norm error estimate, get the numerical analysis. Moreover, in [3], the nonlinear term of equation is (?)2/(?)x2φ(u). when establishing the approximation form, we can integration by parts two times (under sufficiently boundary value conditions) and the a prior estimates, that is where Sh(k) is a finite element space which constituted by the k≥3piecewise polyno-mial. Then, using the finite element analysis method for biharmonic equation [4], we can obtain the result. But the nonlinear terms of the two equations considered by us are different from [3]. When we establish the approximation form, we can only integration by parts one time (under sufficiently boundary value conditions), using the result of a prior estimate, we get Because there exist the derivative term, by the finite element analysis method for bihar-monic equation [4], the error precision of space will decline in one order.
The modified Swift-Hohenberg equation is also a higher order nonlinear parabolic equation with a derivative term. Using Fourier spectral method, we study the numerical solutions for equation (5). We establish the semi-discrete form and full-discrete form, get the error estimates. We can also meet the same problem when we analyse the equation. But, the derivative term is a quadratic term, using sufficient integration by parts and the a prior estimates, we can avoid the decline of error precision of space.
本文研究了三种具有广泛物理背景的高阶非线性抛物型方程解的性质及其数值解法.
首先,我们研究了扩散系数依赖于未知函数本身的薄膜外延增长方程的初边值问题利用Leyar-Schauder不动点定理,Campanato空间的性质和先验估计,我们得到了初边值问题(1)整体解的存在唯一性以及古典解的存在性.
然后,我们研究了描述薄膜外延增长的方程和晶体表面生长的方程的解的长时间行为问题.借用[1,2]中关于整体吸引子的存在性定理,我们得到了在分数维空间Hk(0≤k<5)中方程(2)的整体吸引子的存在性以及在分数维空间Hk(0≤k<+∞)中方程(3)整体吸引子的存在性.需要强调的是方程(2)中p的取值不但可以是整数,还可以是分数.当我们对方程(2)进行先验估计的时候,如果对非线性项div(|(?)u|p-2(?)u)微分多次,则会出现负指数的情形(这种情形的先验估计很难得到).因此,我们只能得到在Hk(0≤k<5)空间中方程(2)的整体吸引子存在性.而方程(3)就不存在这种情况,我们可以利用迭代法得到其在更广泛的分数维空间Hk(0≤k<+∞)中整体吸引子的存在性.
另外,我们考虑了晶体表面生长方程和薄膜外延增长方程的数值解法.文献[3]曾经对Cahn-Hilliard方程进行过有限元分析.借助于一个双调和问题的有限元投影逼近,作者得到了方程(4)的最优阶L2模误差估计.本文中,我们考虑方程(2)和方程(3)的一维情形的有限元方法.我们也借助于一个双调和问题的有限元投影逼近,得到了方程的最优阶L2模误差估计,并进行了数值分析.需要特别指出的是,在[3]中,方程的非线性项是(?)2/(?)x2φ(u),在建立逼近格式的时候,可以通过两次分部积分(在适当的边界条件下)和先验估计结果来进行分析,即其中Sh(k)为剖分上分段k≥3次多项式构成的有限元空间.而后利用双调和方程标准的有限元分析方法[4]得到想要的结果.而我们所考虑的两个方程的非线性项与[3]中不同.在建立逼近格式的时候只能分部积分一次(在适当的边界条件下),利用先验估计结果可得由于有一阶导数项的存在,此时利用双调和方程标准的有限元分析方法[4],空间上的误差精度会下降一阶.
修正Swift-Hohenberg方程也是一类带有一阶导数项的高阶抛物型方程.我们利用Fourier谱方法对该方程的数值解进行分析,建立了半离散格式和全离散格式,给出了误差估计.在对修正Swift-Hohenberg方程进行Fourier谱分析的时候也会遇到类似上面的问题.但是,方程(5)的一阶导数项是二次的,运用合适的分部积分和先验估计,可以避免空间上误差精度的下降.
As mathematics models, higher-order nonlinear parabolic type equations describe many phenomenons which exist in Physics, Chemistry, Informatics, Bioscience, Geo-science, Environmental science, Space science and so on. Higher-order nonlinear parabol-ic type equations is an important part of Nonlinear science.
In this paper, we consider the properties of solutions and numerical solutions for three types of higher order nonlinear parabolic equation with extensive physical back-ground.
Firstly, we consider the higher order nonlinear parabolic equation describing thin-film epitaxial growth whose diffusion coefficient dependents on the unknown function Using Leyar-Schauder fixed point theorem, the qualities of Campanato space and a prior estimates, we get the existence and uniqueness of global solutions and the existence of classical solutions for IBVP (1).
Secondly, we study the long time behavior of solutions for the higher order non-linear parabolic equation describing thin-film epitaxial growth and the higher order nonlinear parabolic equation describing process growing of a crys-tal surface Using the theorem of existence of global attractor in [1,2], we obtain the existence of global attractor for equation (2) in fractal dimension space Hk(0≤k<5) and the existence of global attractor for equation (3) in fractal dimension space Hk(0≤k<+∞). Noticing that the value of p is not only integer, but also fraction in (2), when we consider the a prior estimates for equation (2), if differential many times for term div(|(?)u|p-2(?)u), it maybe exists negative exponential (the a priori estimates of this kind of situation are hard to obtain). So, we have to obtain the existence of global attractor for (2) in space Hk(0≤k<5). On the other hand, there is no this kind of situation for equation (3), we can use iteration method get the existence of global attractor for equation (3) in space Hk(0≤k<+∞).
Thirdly, we consider the numerical solutions for the higher order nonlinear parabol-ic equations describing thin-film epitaxial growth and process growing of a crystal sur-face. Using finite element method,[3] studied the Cahn-Hilliard equation Basing on the finite element projection approximation of a biharmonic problem, he ob-tain the optimal order L2-norm error estimate for equation4. In this paper, we consider the finite element method for equation (2) and equation (3). We also use the finite ele-ment projection approximation of a biharmonic problem, obtain the optimal order L2-norm error estimate, get the numerical analysis. Moreover, in [3], the nonlinear term of equation is (?)2/(?)x2φ(u). when establishing the approximation form, we can integration by parts two times (under sufficiently boundary value conditions) and the a prior estimates, that is where Sh(k) is a finite element space which constituted by the k≥3piecewise polyno-mial. Then, using the finite element analysis method for biharmonic equation [4], we can obtain the result. But the nonlinear terms of the two equations considered by us are different from [3]. When we establish the approximation form, we can only integration by parts one time (under sufficiently boundary value conditions), using the result of a prior estimate, we get Because there exist the derivative term, by the finite element analysis method for bihar-monic equation [4], the error precision of space will decline in one order.
The modified Swift-Hohenberg equation is also a higher order nonlinear parabolic equation with a derivative term. Using Fourier spectral method, we study the numerical solutions for equation (5). We establish the semi-discrete form and full-discrete form, get the error estimates. We can also meet the same problem when we analyse the equation. But, the derivative term is a quadratic term, using sufficient integration by parts and the a prior estimates, we can avoid the decline of error precision of space.
引文
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