本文選題：四階偏微分方程　切入點：退化方程　出處：《大連交通大學》2015年碩士論文　論文類型：學位論文

【摘要】：近幾年,隨著科技不斷發(fā)展和創(chuàng)新,四階拋物方程在許多學科領(lǐng)域中的研究越來越深入,應(yīng)用越來越普遍,因而受到很多學者的關(guān)注。比如來源于固體表面微滴擴散的薄膜方程,用于研究相變的Cahn-Hilliard方程以及模擬半導體電荷運載的量子流體動力學方程等。本文首先研究一類Dirichlet邊界條件下的四階退化橢圓方程組其中解u的邊值為1,m0,ε,δ均為大于0的常數(shù)。這是一個帶有非線性二階擴散項的薄膜方程的定態(tài)形式,為了研究其解的存在性,方法上,需要構(gòu)造不動點算子,其可行性利用Lax-Milgram定理驗證。再以緊嵌入定理為基礎(chǔ),通過Leray-Schauder不動點定理給出弱解存在性。最后,通過選取合適的檢驗函數(shù)及選取特殊不等式,獲得弱解唯一性。其次,研究一類與上述模型相關(guān)的四階退化拋物方程其中四階項的指數(shù)可大于1,解u的邊值為l,n,ε,δ,l均為正常數(shù),m是非負常數(shù)。探究解的存在性用到半離散方法。并且當初始泛函趨近與一個正穩(wěn)態(tài)解時,可獲得解的唯一性。最后,在半離散問題中采用迭代方法,就能得到當時間趨于無窮大時,解以指數(shù)形式收斂于一個正的穩(wěn)態(tài)解。最后,研究非線性擴散作用下四階退化拋物方程這里p1,m≥0。這類方程在相變理論及薄膜潤滑理論中出現(xiàn)。研究方法上采用對時間的半離散化,根據(jù)橢圓型方程解的存在性,構(gòu)造逼近解,再對逼近解作半離散迭代估計、能量估計以及緊性討論,獲得相應(yīng)的拋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，

本文編號：1606677