本文選題：無窮維動力系統(tǒng) + 長時間動力行為��； 參考：《太原理工大學》2016年博士論文

【摘要】：梁、板、殼結構是工程領域中基本且至關重要的承重構件,在長期復雜服役環(huán)境下,其穩(wěn)定性直接影響整個構件的使用壽命。因此,研究這些構件的長時間動力行為及其動力穩(wěn)定性具有重要的理論價值和實際意義。大多數(shù)工程中的彈性構件實際上接近力學系統(tǒng)的連續(xù)體,在討論有勢力場作用下的力學系統(tǒng)穩(wěn)定性時,連續(xù)系統(tǒng)的穩(wěn)定性涉及到數(shù)學上的非線性偏微分方程、無限自由度動力系統(tǒng)的定性研究以及無限維空間的幾何理論等,因此,基于動力學觀點對非線性彈性系統(tǒng)穩(wěn)定性開展系統(tǒng)研究成為關注的熱點和焦點。近年來,關于彈性梁、板方程(組)解的存在性、唯一性、漸近性等動力行為的研究取得了許多可喜的成果。然而,對于解的長時間動力行為研究的結果相對較少。由于吸引子是描述時間趨于無窮大時系統(tǒng)的長時間動力行為的重要指標,而分形維數(shù)是刻劃吸引子的幾何特征量,所以吸引子的存在性及其維數(shù)估計成為無窮維動力系統(tǒng)研究的重要課題,也是近年來比較活躍的前沿問題。在本文中,我們針對幾類具有強阻尼、結構阻尼或外阻尼的固體結構系統(tǒng)作了以下工作。首先,研究了一類滿足Dirichlet邊界條件的具有強阻尼和外阻尼Kirchhoff型非自治彈性梁系統(tǒng)解的長時間動力行為。利用算子半群理論證明了系統(tǒng)連續(xù)解的存在唯一性;把自治系統(tǒng)的半群理論推廣到非自治系統(tǒng)的過程理論,通過引入等價范數(shù),在一定條件下,利用能量一致先驗估計得到系統(tǒng)所生成的過程的有界吸收集;通過過程分解技術,構造恰當?shù)哪芰糠汉?將過程分解成兩部分,使得一部分滿足緊致性,而另一部分滿足壓縮性質,成功地證明了所對應過程的緊的核截面的存在性,從而得到系統(tǒng)所生成的過程的一致吸引子的存在性。其次,研究了一類具有強阻尼和結構阻尼Kirchhoff型熱彈梁耦合系統(tǒng)解的長時間動力行為。在系數(shù)的一定范圍內,利用算子半群理論證明了系統(tǒng)存在唯一的mild解;以半群理論為依據(jù),構造合適的泛函,獲得等價的泛函系統(tǒng),利用能量一致先驗估計得到半群的有界吸收集,進而證明了系統(tǒng)所生成的解半群的整體吸引子的存在性。最后,研究了一類具有強阻尼的熱彈板耦合系統(tǒng)解的長時間動力行為。利用算子半群理論證明了系統(tǒng)存在唯一的連續(xù)解;通過引入等價范數(shù),能量方法和一系列精細的先驗估計得到半群的有界吸收集,進而證明了系統(tǒng)所生成的解半群存在整體吸引子;通過變分方法與能量一致先驗估計得到吸引子的Hausdorff維數(shù)估計。
[Abstract]:Beam, plate and shell structure are basic and important load-bearing components in engineering field. Under the long-term complex service environment, their stability directly affects the service life of the whole member. Therefore, it is of great theoretical and practical significance to study the long-term dynamic behavior and its dynamic stability of these components. Most elastic members in engineering are actually close to the continuum of the mechanical system. When discussing the stability of the mechanical system under the action of the force field, the stability of the continuous system is related to the mathematical nonlinear partial differential equation. The qualitative study of infinite degree of freedom dynamical system and the geometric theory of infinite dimensional space, etc., therefore, the systematic research on the stability of nonlinear elastic system based on the viewpoint of dynamics has become a hot spot and focus. In recent years, many gratifying results have been obtained in the study of the existence, uniqueness and asymptotic behavior of the solutions of elastic beam and plate equations. However, there are relatively few results on the long-term dynamic behavior of solutions. Because the attractor is an important index to describe the long-time dynamic behavior of the system when the time tends to infinity, the fractal dimension is the geometric characteristic quantity of the attractor. Therefore, the existence of attractor and its dimension estimation have become an important subject in the study of infinite dimensional dynamical systems, and are also active frontier problems in recent years. In this paper, we have done the following work for several solid structural systems with strong damping, structural damping or external damping. Firstly, the long-time dynamic behavior of a class of Kirchhoff type nonautonomous elastic beam systems with strong damping and external damping satisfying the Dirichlet boundary condition is studied. The existence and uniqueness of continuous solution are proved by using operator semigroup theory, the semigroup theory of autonomous system is extended to the process theory of nonautonomous system, and the equivalent norm is introduced, under certain conditions, The bounded absorption set of the process generated by the system is obtained by using the energy consistent prior estimation, and the proper energy functional is constructed by the process decomposition technique, and the process is decomposed into two parts, so that one part satisfies the compactness. The other part satisfies the squeezing property, and proves the existence of the compact kernel cross section of the corresponding process successfully, and thus obtains the existence of the uniform attractor of the process generated by the system. Secondly, the long-time dynamic behavior of a Kirchhoff type thermoelastic beam coupling system with strong damping and structural damping is studied. In a certain range of coefficients, the existence of a unique mild solution is proved by using the operator semigroup theory, and an equivalent functional system is obtained by constructing a proper functional based on the semigroup theory. The bounded absorption set of Semigroups is obtained by energy uniform prior estimation, and the existence of global attractors of solution Semigroups generated by the system is proved. Finally, the long time dynamic behavior of a coupled thermoelastic plate system with strong damping is studied. The existence of unique continuous solution is proved by using operator semigroup theory, and the bounded absorption set of semigroup is obtained by introducing equivalent norm, energy method and a series of fine prior estimates. Furthermore, it is proved that there exists a global attractor in the solution semigroup generated by the system, and the Hausdorff dimension estimation of the attractor is obtained by using the variational method and the energy consistent prior estimation.
【學位授予單位】：太原理工大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O175.29

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