几类随机和确定性非线性偏微分方程的定性研究
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摘要
我们知道许多数学模型都是在理想状态下建立的,然而现实环境通常存在很多不确定性因素,因此有必要在随机环境中考虑已有的模型.本文首先考虑一些重要的浅水波方程,包括：Camassa-Holm方程,Dullin-Gottwald-Holm方程组,b-famaily方程组,Degasperis-Procesi方程和Ostrovsky方程.接下来,分别考虑带随机化初值的Hartree方程和带随机势的Hartree方程.最后,我们考虑分数次布朗运动驱动的随机发展方程.通过对解的适定性,大偏差和随机吸引子的研究,讨论了随机效应对方程解的影响.具体结构如下：
第1章给出本文研究的物理背景,研究现状和预备知识.在第一部分,我们回顾了Camassa-Holm方程,Dullin-Gottwald-Holm方程组,b-famaily方程组,Ostrovsky方程,Degasperis-Procesi方程等重要的浅水波方程,Hartree方程,和一般的发展方程的研究进展,指出本文的研究内容和意义所在.第二部分给出了文中将要用到的一些预备知识,包括随机Tto积分,对分数次布朗运动的轨道积分,随机动力系统和一些不等式.
第2章,我们利用正则化方法得到带加法噪声的随机Camassa-Holm方程的适定性.首先,由压缩映射原理得到正则化方程,即随机高阶Camassa-Holm方程的适定性,其中的关键是得到：Bourgain空间Xs,b,b<1/2中的双线性估计.其次,我们得到正则化方程解的一致估计,从而可得它为一柯西列,由此得它的极限为随机Camassa-Holm方程的解.最后,我们指出当初值满足一定条件,随机Camassa-Holm方程的解将在有限时间爆破.
第3章,我们利用第2章中的正则化方法得到Dullin-Gottwald-Holm方程组的适定性且在相应的空间中建立波的爆破准则和整体适定性.最后,我们得到Dullin-Gottwald-Holm方程组的孤立波解.
第4章,我们考虑随机效应对b-family方程组的影响,即带乘法噪声的随机b-family方程组解的大偏差原理.这里,我们将再次用到正则化技术得到带乘法噪声的随机b-family方程组解的适定性.由弱收敛方法和随机控制理论,我们得到随机b-family方程组的解满足大偏差原理.
第5章,不同于前面的正则化方法,我们利用迭代技术得到随机Degasperis-Procesi方程的局部适定性.最后,通过建立相应解精确的爆破准则,我们得到整体解的存在性.
第6章,我们探讨带阻尼的随机Ostrovsky方程的长时间行为.首先,我们得到解的整体存在性,且形成随机动力系统.其次,由能量不等式,我们得到解的一致有界性.最后,因为考虑的区域为全空间,我们需要建立解的渐近紧性.已有文献由截尾估计得到渐近紧,这里我们将通过解的分解得到渐近紧,即将解分解为衰减部分和正则部分.
第7章分为两部分.在第一部分,我们利用Burq和Tzvetkov在文献[11]中提出的随机化初值方法建立Hartree方程在超临界空间中解的适定性.文献[139]指出确定性Hartree方程在超临界空间中是不适定的,我们的结果在某种意义下提升了以上结果.第二部分,我们考虑带随机势的Hartree方程.由Strichartz估计和压缩映射原理得到随机Hartree方程在空间Lρ(Ω;C([0,τ];H8(Rn))∩Lq([0,τ];Ws,r(Rn))),其中中的局部适定性.由动量和能量方程,我们得到随机Hartree方程在H1(Rn)中的整体存在唯一性.由一般的变分等式,我们讨论了爆破解.
第8章,我们考虑一类带分数次布朗运动的随机发展方程的随机吸引子的存在性.首先,我们在Holder连续函数空间中利用不动点定理证明解的存在唯一性且得出这些解产生一个随机动力系统.接下来,我们考虑当样本轨道满足特定性质时随机吸引子的存在性.这里,为了得到解的先验估计,我们需要建立离散形式的Grownall引理,从而得到离散形式的非自治动力系统解的吸收集和后拉吸引子的存在性.由此,我们将离散形式后拉吸引子的存在性推广到原始的连续非自治动力系统.最后,我们证明假设带Hurst参数H>1/2的分数次布朗运动的协方差小,则带分数次布朗运动的随机偏微分方程的随机吸引子存在.
As we know, many mathematical models were constructed under the ideal con-ditions. However, due to uncertainty in the modelling and external environment, this modelling could be subject to random fluctuations. Firstly, we consider some im-portant shallow water equations, including:the Camassa-Holm equation, the two-component Dullin-Gottwald-Holm system, the two-component b-famaily system, the Degasperis-Procesi equation and the Ostrovsky equation. Next, we consider the Hartree equation with the randomizated initial value and stochastic potential, respec-tively. At last, wo consier the general stochastic evolution equations with fractional Brownian motion. We investigate some characteristic of solutions, such as the well-posedness, large deviation principle and random attractors. The organization of this thesis is as follows:
In Chapter1, some physical background and the preliminaries are given. In Section1, we recall the research progress of the Camassa-Holm equation, the two-component Dullin-Gottwald-Holm system, the two-component b-famaily system, the Degasperis-Procesi equation, the Ostrovsky equation, the Hartree equation and the general evolution equations, and point out the research contents and significance of the thesis. In Section2, we introduce some preliminary which will be used in the the-sis, including stochastic Ito integral, the stochastic integration for fractional Brownian motions defined pathwise and some inequalities.
In Chapter2, the well-posedness of the stochastic Camassa-Holm equation with additive noise is proved by regularization method. Firstly, the well-posedness for the regularization equation, i.e., stochastic high-order Camassa-Holm equation is ob-tained by contraction mapping principle, where the key point is the the bilinear es-timates in Bourgain spaces Xs,b,b, b<1/2. Then, the consistency estimates for the solutions of the regularization equation are obtained, from which we can check that the solutions is a Cauchy sequence and the limit is the unique solution of the stochas-tic Camassa-Holm equation. At last, we point out that when initial value satisfies some conditions, the solution for the stochastic Camassa-Holm equation will blow-up at finite time.
In Chapter3, we consider the well-posedness of the Dullin-Gottwald-Holm sys- tems by the regularization method in Chapter2, and construct the wave-breaking cri-teria and global well-posedness in the corresponding space. At last, we obtain the solitary-wave solutions of the Dullin-Gottwald-Holm systems.
In Chapter4, we consider the influence of stochastic effects over two-component b-family system, i.e., the large deviation principle for the solutions of the stochastic two-component b-family system with multiplicative noise. We will also use the regu-larization method to get the well-posedness of the stochastic two-component b-family system. By weak convergence approach along with stochastic control, we can ob-tained the large deviation principle for the solutions of the stochastic two-component b-family system.
In Chapter5, instead of the regularization method used in the previous chapters, we obtained the local well-posedness of the stochastic Degasperis-Procesi equation by the iterative technique. Then, we proved the global existence of the solutions by constructing the precise blow-up scenario.
Chapter6is devoted to the long time behavior of the stochastic damped forced Ostrovsky equation. Firstly, the global existence of solution is obtained, and cre-ates a random dynamical system. Then, by the energy inequality, we can obtain the consistency estimates. Since the domain is R, we need to construct the asymptotic compactness of the solution. And the asymptotic compactness is usually proved by a tail-estimates. Here, the asymptotic compactness is checked by splitting the solutions into a decaying part plus a regular part.
Chapter7is consists of two parts. In Section1, by the randomization of the initial value raised by Burq and Tzvetkov in [11], we can construct the existence of solutions of the Hartree equation in the supercritical spaces, where the authors in [139] obtained the ill-posedness. So, our result improves those in [11] in some sence. In Section1, we consider the Hartree equations with stochastic potential. By the Strichartz esti-mate and the contraction mapping principle, the local well-posedness of the stochastic Hartree equation in the space Lρ(Ω;C([0,τ]; Hs(Rn))∩L9([0,τ]; Ws'r(Rn))), with is obtained. By studying the evolution of mo-mentum and of the energy of the solution, the global existence and uniqueness for solution of the stochastic Hartree equation in H1(Rn) is proved. The blow-up solution is also discussed by generalizing the variance identity.
Chapter8shows the existence of random attractors for a class of stochastic evo- lution equations with fractional Brownian motion. Firstly, we are able to prove the existence of a unique solution in Holder continuous functions space by the contraction mapping principle. Standard arguments then allow us to conclude that this solution is random or more precisely creates a random dynamical system. Next, we consider the existence of the random attractors, while the paths satisfy some specified conditions. Here, to obtain a priori estimates, we have to formulate a special Gronwall lemma for discrete time, then we can check the existence of a pullback attractor for a discrete non-autonomous dynamical system. It is not difficult then to extend the existence of a pullback attractor to the original continuous non-autonomous dynamical system. Fi-nally, we prove that provided that the covariance of the fractional Brownian motion with Hurst parameter H>1/2is small, then there exists a random attractor for the stochastic evolution equations with fractional Brownian motion.
第1章给出本文研究的物理背景,研究现状和预备知识.在第一部分,我们回顾了Camassa-Holm方程,Dullin-Gottwald-Holm方程组,b-famaily方程组,Ostrovsky方程,Degasperis-Procesi方程等重要的浅水波方程,Hartree方程,和一般的发展方程的研究进展,指出本文的研究内容和意义所在.第二部分给出了文中将要用到的一些预备知识,包括随机Tto积分,对分数次布朗运动的轨道积分,随机动力系统和一些不等式.
第2章,我们利用正则化方法得到带加法噪声的随机Camassa-Holm方程的适定性.首先,由压缩映射原理得到正则化方程,即随机高阶Camassa-Holm方程的适定性,其中的关键是得到：Bourgain空间Xs,b,b<1/2中的双线性估计.其次,我们得到正则化方程解的一致估计,从而可得它为一柯西列,由此得它的极限为随机Camassa-Holm方程的解.最后,我们指出当初值满足一定条件,随机Camassa-Holm方程的解将在有限时间爆破.
第3章,我们利用第2章中的正则化方法得到Dullin-Gottwald-Holm方程组的适定性且在相应的空间中建立波的爆破准则和整体适定性.最后,我们得到Dullin-Gottwald-Holm方程组的孤立波解.
第4章,我们考虑随机效应对b-family方程组的影响,即带乘法噪声的随机b-family方程组解的大偏差原理.这里,我们将再次用到正则化技术得到带乘法噪声的随机b-family方程组解的适定性.由弱收敛方法和随机控制理论,我们得到随机b-family方程组的解满足大偏差原理.
第5章,不同于前面的正则化方法,我们利用迭代技术得到随机Degasperis-Procesi方程的局部适定性.最后,通过建立相应解精确的爆破准则,我们得到整体解的存在性.
第6章,我们探讨带阻尼的随机Ostrovsky方程的长时间行为.首先,我们得到解的整体存在性,且形成随机动力系统.其次,由能量不等式,我们得到解的一致有界性.最后,因为考虑的区域为全空间,我们需要建立解的渐近紧性.已有文献由截尾估计得到渐近紧,这里我们将通过解的分解得到渐近紧,即将解分解为衰减部分和正则部分.
第7章分为两部分.在第一部分,我们利用Burq和Tzvetkov在文献[11]中提出的随机化初值方法建立Hartree方程在超临界空间中解的适定性.文献[139]指出确定性Hartree方程在超临界空间中是不适定的,我们的结果在某种意义下提升了以上结果.第二部分,我们考虑带随机势的Hartree方程.由Strichartz估计和压缩映射原理得到随机Hartree方程在空间Lρ(Ω;C([0,τ];H8(Rn))∩Lq([0,τ];Ws,r(Rn))),其中中的局部适定性.由动量和能量方程,我们得到随机Hartree方程在H1(Rn)中的整体存在唯一性.由一般的变分等式,我们讨论了爆破解.
第8章,我们考虑一类带分数次布朗运动的随机发展方程的随机吸引子的存在性.首先,我们在Holder连续函数空间中利用不动点定理证明解的存在唯一性且得出这些解产生一个随机动力系统.接下来,我们考虑当样本轨道满足特定性质时随机吸引子的存在性.这里,为了得到解的先验估计,我们需要建立离散形式的Grownall引理,从而得到离散形式的非自治动力系统解的吸收集和后拉吸引子的存在性.由此,我们将离散形式后拉吸引子的存在性推广到原始的连续非自治动力系统.最后,我们证明假设带Hurst参数H>1/2的分数次布朗运动的协方差小,则带分数次布朗运动的随机偏微分方程的随机吸引子存在.
As we know, many mathematical models were constructed under the ideal con-ditions. However, due to uncertainty in the modelling and external environment, this modelling could be subject to random fluctuations. Firstly, we consider some im-portant shallow water equations, including:the Camassa-Holm equation, the two-component Dullin-Gottwald-Holm system, the two-component b-famaily system, the Degasperis-Procesi equation and the Ostrovsky equation. Next, we consider the Hartree equation with the randomizated initial value and stochastic potential, respec-tively. At last, wo consier the general stochastic evolution equations with fractional Brownian motion. We investigate some characteristic of solutions, such as the well-posedness, large deviation principle and random attractors. The organization of this thesis is as follows:
In Chapter1, some physical background and the preliminaries are given. In Section1, we recall the research progress of the Camassa-Holm equation, the two-component Dullin-Gottwald-Holm system, the two-component b-famaily system, the Degasperis-Procesi equation, the Ostrovsky equation, the Hartree equation and the general evolution equations, and point out the research contents and significance of the thesis. In Section2, we introduce some preliminary which will be used in the the-sis, including stochastic Ito integral, the stochastic integration for fractional Brownian motions defined pathwise and some inequalities.
In Chapter2, the well-posedness of the stochastic Camassa-Holm equation with additive noise is proved by regularization method. Firstly, the well-posedness for the regularization equation, i.e., stochastic high-order Camassa-Holm equation is ob-tained by contraction mapping principle, where the key point is the the bilinear es-timates in Bourgain spaces Xs,b,b, b<1/2. Then, the consistency estimates for the solutions of the regularization equation are obtained, from which we can check that the solutions is a Cauchy sequence and the limit is the unique solution of the stochas-tic Camassa-Holm equation. At last, we point out that when initial value satisfies some conditions, the solution for the stochastic Camassa-Holm equation will blow-up at finite time.
In Chapter3, we consider the well-posedness of the Dullin-Gottwald-Holm sys- tems by the regularization method in Chapter2, and construct the wave-breaking cri-teria and global well-posedness in the corresponding space. At last, we obtain the solitary-wave solutions of the Dullin-Gottwald-Holm systems.
In Chapter4, we consider the influence of stochastic effects over two-component b-family system, i.e., the large deviation principle for the solutions of the stochastic two-component b-family system with multiplicative noise. We will also use the regu-larization method to get the well-posedness of the stochastic two-component b-family system. By weak convergence approach along with stochastic control, we can ob-tained the large deviation principle for the solutions of the stochastic two-component b-family system.
In Chapter5, instead of the regularization method used in the previous chapters, we obtained the local well-posedness of the stochastic Degasperis-Procesi equation by the iterative technique. Then, we proved the global existence of the solutions by constructing the precise blow-up scenario.
Chapter6is devoted to the long time behavior of the stochastic damped forced Ostrovsky equation. Firstly, the global existence of solution is obtained, and cre-ates a random dynamical system. Then, by the energy inequality, we can obtain the consistency estimates. Since the domain is R, we need to construct the asymptotic compactness of the solution. And the asymptotic compactness is usually proved by a tail-estimates. Here, the asymptotic compactness is checked by splitting the solutions into a decaying part plus a regular part.
Chapter7is consists of two parts. In Section1, by the randomization of the initial value raised by Burq and Tzvetkov in [11], we can construct the existence of solutions of the Hartree equation in the supercritical spaces, where the authors in [139] obtained the ill-posedness. So, our result improves those in [11] in some sence. In Section1, we consider the Hartree equations with stochastic potential. By the Strichartz esti-mate and the contraction mapping principle, the local well-posedness of the stochastic Hartree equation in the space Lρ(Ω;C([0,τ]; Hs(Rn))∩L9([0,τ]; Ws'r(Rn))), with is obtained. By studying the evolution of mo-mentum and of the energy of the solution, the global existence and uniqueness for solution of the stochastic Hartree equation in H1(Rn) is proved. The blow-up solution is also discussed by generalizing the variance identity.
Chapter8shows the existence of random attractors for a class of stochastic evo- lution equations with fractional Brownian motion. Firstly, we are able to prove the existence of a unique solution in Holder continuous functions space by the contraction mapping principle. Standard arguments then allow us to conclude that this solution is random or more precisely creates a random dynamical system. Next, we consider the existence of the random attractors, while the paths satisfy some specified conditions. Here, to obtain a priori estimates, we have to formulate a special Gronwall lemma for discrete time, then we can check the existence of a pullback attractor for a discrete non-autonomous dynamical system. It is not difficult then to extend the existence of a pullback attractor to the original continuous non-autonomous dynamical system. Fi-nally, we prove that provided that the covariance of the fractional Brownian motion with Hurst parameter H>1/2is small, then there exists a random attractor for the stochastic evolution equations with fractional Brownian motion.
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