Green函数在带扩散机制的非线性方程中的应用

发布时间：2018-04-12 21:31

本文选题：Green函数 + 扩散方程　；参考：《上海交通大学》2015年博士论文

【摘要】：本文旨在研究Green函数方法及其在带扩散机制的非线性方程中的应用.我们主要考虑了两类带扩散机制的非线性方程.第一类方程是趋化模型,含有线性扩散项和非线性交叉扩散项.它们之间的竞争机制是此类方程的特点之一,也是研究中我们要面对的主要困难.第二类方程是单个粘性守恒律方程.该方程在激波解附近的线性化方程除了含有非常系数项以外,还带有热扩散机制.我们将分别考虑这两类方程的初边值问题和大扰动Cauchy问题.具体内容概括如下：第一章是绪论.这里,我们将着重介绍Green函数方法、趋化模型和粘性守恒律方程的相关背景,并给出一些重要的结果.在第二章,我们将研究一类吸引-排斥趋化模型Cauchy问题解的大时间行为.本章共分为两大部分.第一部分中,我们考虑该模型Cauchy问题小解的逐点估计.通过Green函数的方法,以及对非局部算子的精细估计,我们最终得到了小解的逐点估计以及W即衰减估计.结果表明,解的大时间行为与经典热方程的相似.接着在第二部分中,我们继续考虑了该模型Cauchy间题大初值解的衰减估计和爆破现象.大初值情形与小初值有很大不同,原因在于大初值情况下,非线性项所产生的聚集效应有可能压过排斥效应以及扩散效应,从而导致爆破的发生.具体情况要由方程中的参数之间的关系决定.最终,我们证明了当排斥效应压过聚集效应时,Cauchy问题总是存在一致有界的整体光滑解,并得到了解的衰减估计.这里我们所用方法主要是能量估计,Moser-Alikakos迭代技巧和基于Green函数的半阶导方法.特别需要指出的是,利用半阶导方法,我们可以在初值弱正则的情况下,逐步地提高解的正则性.最后,当聚集效应占据主导地位且方程满足一定条件时,我们利用动量方法得到了大解的爆破结果.第三章,我们考虑Keller-Segel模型在半空问X。lt上的初边值问题.我们提出了一个守恒边界条件,以保证质量仍旧满足守恒性质.在此条件下,我们分别研究了解的全局存在性,正则性和大时间行为.我们首先应用Fourier变换和Laplace变换技巧以及复分析方法,构造了线性初边值问题的Green函数.然后通过Green函数的具体估计和：Duhamel齐次化原理,我们证明了当初始值充分小时,该初边值问题总是存在唯一的全局经典解.更进一步地,我们得到了全局解的衰减估计.我们指出边界的移动方向对解的大时间行为会产生重要的影响.具体来说,当l0时,解的L∞模衰减率为(1+t)-n/2(n为空间维数),与热方程的一致.而当l0时,却有不同的结果.不仅如此,此时解的渐近行为还与空间维数n有关.n≥2时,我们证得L∞模衰减率是(1+t)-(n-1)/2,但是n=1时,解并不会趋于零状态(只要初始质量不为零).相反地,我们证明解会以指数级衰减到唯一一个守恒稳态解.这是一个十分有意思的结果,该结果从某个层面上说明了边界会对解的大时间行为产生本质性的影响.第四章,我们考虑两维的单个粘性守恒律方程激波附近大扰动解的稳定性问题.由于方程特殊的结构,以及小性假设的缺失,使得我们无法单纯依罪Green函数方法或者L2能量估计得到解的全局存在性和衰减估计.幸运的是,我们找到了方程的极大值原理和压缩原理.然后再结合能量估计,可以证明只要初值属于L1∩ H4(R2),大扰动解总是全局存在的.之后,我们进一步考虑了大扰动解的大时间行为.在弱激波条件下,我们对Burgers类型和更一般的类型进行了研究,得到了大扰动解的L2衰减估计t1/4和L∞衰减估计t-1/2证明的思想主要是利用半群-能量相结合的方法得到解的某种小性估计,然后再利用能量不等式得到衰减估计.
[Abstract]:This paper aims to study the application of Green function method for solving nonlinear equation with diffusion mechanism in. We mainly consider two kinds of nonlinear equations with diffusion mechanism. The first equation is a chemotaxis model with linear and nonlinear diffusion and cross diffusion. The competition mechanism between them is one of the characteristics of this kind of equation, the main difficulty is in the study we have to face. Second kinds of equations are scalar viscous conservation law equation. This equation in linear equations near the shock solution in addition to contain very coefficient than those with thermal diffusion mechanism. The problem and the initial boundary perturbed Cauchy problem we will consider the two types of equations. The detailed contents are as follows the first chapter is introduction. Here, we will introduce the method of Green function, background chemotaxis model and viscous conservation equations, and gives some important results in second. Chapter, we will study a kind of attraction repulsion chemotaxis model solutions of the Cauchy problem for large time behavior. This chapter is divided into two parts. The first part, we consider the pointwise solution of Cauchy problem. The model estimated by the method of Green function, and the fine estimation of non local operators, we finally got it solution of the pointwise estimates of W and decay estimates. The results show that the large time behavior of the heat equation and the classical similarity. Then in the second part, we continue to consider the problem of model Cauchy large initial value decay estimates and blasting phenomenon. The case of the large initial value and initial value are very different, the reason lies in the large the initial conditions, the nonlinearity generated by the aggregation effect may pressure rejection effect and diffusion effect, which leads to the occurrence of blasting. The specific circumstances should be decided by the relationship between the parameters in the equations. Finally, we prove that When the pressure over the repulsion effect aggregation effect, Cauchy there is always consistent with the overall and smooth solution circles, get understanding. Here we estimate attenuation method is mainly used in energy estimation, Moser-Alikakos iterative technique and semi derivative method based on Green function. In particular, the use of semi derivative method. We can in the initial weak regularity conditions, and gradually improve the regularity of the solution. Finally, when the aggregation effect is dominant and the equation satisfies some conditions, we obtain the solution of blasting results using momentum method. In the third chapter, we consider the Keller-Segel model in half space on the X.lt initial boundary value problem. We propose a conservation of boundary conditions, in order to ensure the quality still satisfy the conservation properties. Under this condition, we study the global existence, regularity and large time behavior. We first use Fo Urier transform and Laplace transform technique and the method of complex analysis, constructed Green function boundary value problem of linear first. Then through the detailed estimation of Green function and Duhamel homogeneitisation principle, we prove that when the initial value is small enough, there is always a problem only global classical solution of the initial boundary value. Further, we get the global solution of the decay estimates. We pointed out that the mobile direction of the boundary will have a significant impact on the large time behavior of solutions. Specifically, when l0, L - norm of the solution decay rate of (1+t) -n/2 (n dimension), and the heat equation. And when l0. There are different results. Moreover, the asymptotic behavior of solutions and the space dimension n.N larger than 2, we get L for mold decay rate is (1+t) - (n-1) /2, but not n=1, solution and zero state (as long as the initial mass is not zero) instead. We prove that the solutions will be. The exponential decay to only a conserved steady-state solution. This is a very interesting result, the result indicates that the influence of the essence of the large time behavior of solutions of the boundary will be from a level. In the fourth chapter, we consider two dimensional scalar viscous conservation laws equations near the shock wave solutions of large disturbance stability. Due to the special structure of the equation, and the lack of small hypothesis, that we could not simply according to the crime of the method of Green function or L2 energy estimates obtained the global existence and decay estimates. Fortunately, we find the equation of the maximum principle and the principle of compression. Then energy estimate method can prove that if the initial data L1 H4 (R2) belongs to a large disturbance, the solution is global existence. After that, we further consider the large time behavior of solutions of large disturbance. In the condition of weak shocks, we Burgers type and general type. In the study, we get the L2 attenuation estimate of the large disturbance solution t1/4 and L L decay estimate t-1/2. The proof is that we use the method of semigroup energy to get a small estimate of the solution, and then use the energy inequality to get the attenuation estimate.

【学位授予单位】：上海交通大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O175

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