本文選題：Green函數(shù) + 擴散方程��； 參考：《上海交通大學》2015年博士論文

【摘要】：本文旨在研究Green函數(shù)方法及其在帶擴散機制的非線性方程中的應用.我們主要考慮了兩類帶擴散機制的非線性方程.第一類方程是趨化模型,含有線性擴散項和非線性交叉擴散項.它們之間的競爭機制是此類方程的特點之一,也是研究中我們要面對的主要困難.第二類方程是單個粘性守恒律方程.該方程在激波解附近的線性化方程除了含有非常系數(shù)項以外,還帶有熱擴散機制.我們將分別考慮這兩類方程的初邊值問題和大擾動Cauchy問題.具體內(nèi)容概括如下：第一章是緒論.這里,我們將著重介紹Green函數(shù)方法、趨化模型和粘性守恒律方程的相關(guān)背景,并給出一些重要的結(jié)果.在第二章,我們將研究一類吸引-排斥趨化模型Cauchy問題解的大時間行為.本章共分為兩大部分.第一部分中,我們考慮該模型Cauchy問題小解的逐點估計.通過Green函數(shù)的方法,以及對非局部算子的精細估計,我們最終得到了小解的逐點估計以及W即衰減估計.結(jié)果表明,解的大時間行為與經(jīng)典熱方程的相似.接著在第二部分中,我們繼續(xù)考慮了該模型Cauchy間題大初值解的衰減估計和爆破現(xiàn)象.大初值情形與小初值有很大不同,原因在于大初值情況下,非線性項所產(chǎn)生的聚集效應有可能壓過排斥效應以及擴散效應,從而導致爆破的發(fā)生.具體情況要由方程中的參數(shù)之間的關(guān)系決定.最終,我們證明了當排斥效應壓過聚集效應時,Cauchy問題總是存在一致有界的整體光滑解,并得到了解的衰減估計.這里我們所用方法主要是能量估計,Moser-Alikakos迭代技巧和基于Green函數(shù)的半階導方法.特別需要指出的是,利用半階導方法,我們可以在初值弱正則的情況下,逐步地提高解的正則性.最后,當聚集效應占據(jù)主導地位且方程滿足一定條件時,我們利用動量方法得到了大解的爆破結(jié)果.第三章,我們考慮Keller-Segel模型在半空問X。lt上的初邊值問題.我們提出了一個守恒邊界條件,以保證質(zhì)量仍舊滿足守恒性質(zhì).在此條件下,我們分別研究了解的全局存在性,正則性和大時間行為.我們首先應用Fourier變換和Laplace變換技巧以及復分析方法,構(gòu)造了線性初邊值問題的Green函數(shù).然后通過Green函數(shù)的具體估計和：Duhamel齊次化原理,我們證明了當初始值充分小時,該初邊值問題總是存在唯一的全局經(jīng)典解.更進一步地,我們得到了全局解的衰減估計.我們指出邊界的移動方向?qū)獾拇髸r間行為會產(chǎn)生重要的影響.具體來說,當l0時,解的L∞模衰減率為(1+t)-n/2(n為空間維數(shù)),與熱方程的一致.而當l0時,卻有不同的結(jié)果.不僅如此,此時解的漸近行為還與空間維數(shù)n有關(guān).n≥2時,我們證得L∞模衰減率是(1+t)-(n-1)/2,但是n=1時,解并不會趨于零狀態(tài)(只要初始質(zhì)量不為零).相反地,我們證明解會以指數(shù)級衰減到唯一一個守恒穩(wěn)態(tài)解.這是一個十分有意思的結(jié)果,該結(jié)果從某個層面上說明了邊界會對解的大時間行為產(chǎn)生本質(zhì)性的影響.第四章,我們考慮兩維的單個粘性守恒律方程激波附近大擾動解的穩(wěn)定性問題.由于方程特殊的結(jié)構(gòu),以及小性假設(shè)的缺失,使得我們無法單純依罪Green函數(shù)方法或者L2能量估計得到解的全局存在性和衰減估計.幸運的是,我們找到了方程的極大值原理和壓縮原理.然后再結(jié)合能量估計,可以證明只要初值屬于L1∩ H4(R2),大擾動解總是全局存在的.之后,我們進一步考慮了大擾動解的大時間行為.在弱激波條件下,我們對Burgers類型和更一般的類型進行了研究,得到了大擾動解的L2衰減估計t1/4和L∞衰減估計t-1/2證明的思想主要是利用半群-能量相結(jié)合的方法得到解的某種小性估計,然后再利用能量不等式得到衰減估計.
[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.

【學位授予單位】：上海交通大學
【學位級別】：博士
【學位授予年份】：2015
【分類號】：O175

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