四阶抛物方程几类问题研究

发布时间：2018-03-13 13:58

  本文选题：四阶偏微分方程　切入点：退化方程　出处：《大连交通大学》2015年硕士论文　论文类型：学位论文

【摘要】：近几年,随着科技不断发展和创新,四阶抛物方程在许多学科领域中的研究越来越深入,应用越来越普遍,因而受到很多学者的关注。比如来源于固体表面微滴扩散的薄膜方程,用于研究相变的Cahn-Hilliard方程以及模拟半导体电荷运载的量子流体动力学方程等。本文首先研究一类Dirichlet边界条件下的四阶退化椭圆方程组其中解u的边值为1,m0,ε,δ均为大于0的常数。这是一个带有非线性二阶扩散项的薄膜方程的定态形式,为了研究其解的存在性,方法上,需要构造不动点算子,其可行性利用Lax-Milgram定理验证。再以紧嵌入定理为基础,通过Leray-Schauder不动点定理给出弱解存在性。最后,通过选取合适的检验函数及选取特殊不等式,获得弱解唯一性。其次,研究一类与上述模型相关的四阶退化抛物方程其中四阶项的指数可大于1,解u的边值为l,n,ε,δ,l均为正常数,m是非负常数。探究解的存在性用到半离散方法。并且当初始泛函趋近与一个正稳态解时,可获得解的唯一性。最后,在半离散问题中采用迭代方法,就能得到当时间趋于无穷大时,解以指数形式收敛于一个正的稳态解。最后,研究非线性扩散作用下四阶退化抛物方程这里p1,m≥0。这类方程在相变理论及薄膜润滑理论中出现。研究方法上采用对时间的半离散化,根据椭圆型方程解的存在性,构造逼近解,再对逼近解作半离散迭代估计、能量估计以及紧性讨论,获得相应的抛物方程解的存在性及唯一性。
[Abstract]:In recent years, with the development and innovation of science and technology, the research of fourth-order parabolic equation has become more and more in-depth in many disciplines, and its application has become more and more common. For example, the thin film equation derived from the diffusion of microdroplets on solid surface, The Cahn-Hilliard equation for phase transition and the quantum hydrodynamic equation for simulating semiconductor charge transport are studied in this paper. In this paper, we first study a class of fourth-order degenerate elliptic equations under Dirichlet boundary condition, where the boundary value of solution u is 1m0, 蔚, 未 is greater than that of the equation. This is a steady state form of a thin film equation with a nonlinear second-order diffusion term, In order to study the existence of the solution, it is necessary to construct the fixed point operator, and its feasibility is verified by Lax-Milgram theorem. Based on the compact embedding theorem, the existence of weak solution is obtained by Leray-Schauder fixed point theorem. The uniqueness of weak solution is obtained by selecting appropriate test function and special inequality. Secondly, In this paper, we study a class of degenerate parabolic equations of fourth order related to the above model, in which the exponent of the fourth-order term can be greater than 1, and the boundary values of solution u are all normal numbers (n, 蔚, 未 L). The existence of solutions is studied by semi-discrete method. When the initial functional approaches to a positive steady-state solution, The uniqueness of the solution can be obtained. Finally, when the time tends to infinity, the solution converges exponentially to a positive steady-state solution when the iterative method is used in the semi-discrete problem. The fourth-order degenerate parabolic equation under nonlinear diffusion is studied, where p1M 鈮，

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