本文選題：Boussinesq方程組 + 非齊次邊界�。� 參考：《東華大學(xué)》2017年博士論文

【摘要】：隨著科學(xué)的發(fā)展,出現(xiàn)越來越多的流體動力學(xué)方程(組),在實(shí)際應(yīng)用中,包含時(shí)間變量t的方程(組)被稱為非線性發(fā)展方程(組).Boussinesq方程組有著極強(qiáng)的物理背景和數(shù)學(xué)意義,能夠描述大氣科學(xué)與海洋運(yùn)動中旋轉(zhuǎn)和分層兩個(gè)最顯著的特征,一經(jīng)提出就引起廣泛關(guān)注.近年來,很多學(xué)者為擴(kuò)大Boussinesq方程組的適用范圍,又提出了多種改進(jìn)形式的方程,這些方程的廣泛研究與應(yīng)用,對民生、經(jīng)濟(jì)發(fā)展等問題的研究具有較大的理論意義和應(yīng)用指導(dǎo)價(jià)值.在介紹了二維Boussinesq方程組的研究背景和現(xiàn)狀,以及本文所需的一些基本理論和常用不等式之后,本文主要研究了多個(gè)Boussinesq方程組模型的整體適定性,包含半粘性非線性Boussinesq方程組、帶有非齊次邊界的Boussinesq方程組以及分?jǐn)?shù)階Boussinesq方程組,并得到了一些有意義的結(jié)果.主要相關(guān)結(jié)論如下:(1)研究了無熱擴(kuò)散系數(shù)的Boussinesq方程組的初邊值問題.當(dāng)粘性系數(shù)為依賴于溫度的函數(shù)時(shí),該方程組的整體適定性仍是開放性問題.本章通過構(gòu)造方程組的近似解的方法,在討論相關(guān)收斂性之后,證得方程組準(zhǔn)強(qiáng)解的更高階正則性和唯一性.(2)利用對非齊次方程的傳統(tǒng)處理方法,研究了帶有非齊次邊界條件Boussinesq方程組解的適定性.通過重新計(jì)算方程組的擾動變量,將帶有擾動變量的方程組轉(zhuǎn)換成與其等價(jià)的可以用傳統(tǒng)方法計(jì)算的方程組,最終對方程組解的整體適定性進(jìn)行了討論.(3)研究了亞臨界條件下分?jǐn)?shù)階Boussinesq方程組的初邊值問題.充分利用α,β∈(23,1)的條件,討論了方程的持續(xù)正則性.同時(shí),我們還求得了其衰減估計(jì).結(jié)合現(xiàn)有的相關(guān)研究成果,為了完善分?jǐn)?shù)階二維Boussinesq方程組正則性的理論體系,還將其推廣至Sobolev空間中更進(jìn)一步地進(jìn)行了討論.(4)研究了次臨界條件下分?jǐn)?shù)階各向異性Boussinesq方程組的Cauchy問題.這一部分,以第四章的模型為基礎(chǔ),在周期域中次臨界情形下研究了只在水平方向耗散的分?jǐn)?shù)階Boussinesq方程組的Cauchy問題,證明了其強(qiáng)解的適定性問題,并討論了整體吸引子的存在性.最后,對主要工作再次進(jìn)行了總結(jié)并對未來的研究提出了展望.
[Abstract]:With the development of science, more and more hydrodynamic equations have emerged. In practical applications, the equations containing time variable t are called nonlinear evolution equations (group. Boussinesq equations have strong physical background and mathematical significance.It can describe the most prominent characteristics of rotation and stratification in atmospheric science and ocean motion.In recent years, in order to expand the scope of application of Boussinesq equations, many scholars have put forward a variety of improved equations, which are widely studied and applied to people's livelihood.The research of economic development and other issues has great theoretical significance and application guidance value.After introducing the research background and present situation of two-dimensional Boussinesq equations, and some basic theories and common inequalities needed in this paper, this paper mainly studies the global fitness of multiple Boussinesq equations, including semi-viscous nonlinear Boussinesq equations.The Boussinesq equations with nonhomogeneous boundaries and the fractional Boussinesq equations are obtained, and some meaningful results are obtained.The main conclusions are as follows: 1) the initial-boundary value problem of Boussinesq equations without thermal diffusion coefficient is studied.When the viscosity coefficient is a temperature-dependent function, the global fitness of the equations is still an open problem.In this chapter, the higher order regularity and uniqueness of the quasi strong solutions of the equations are obtained by constructing the approximate solutions of the equations, and after discussing the convergence of the correlation, we use the traditional methods to deal with the inhomogeneous equations.In this paper, the solution of Boussinesq equations with inhomogeneous boundary conditions is studied.By recalculating the perturbed variables of the equations, the equations with the perturbation variables are converted into the equivalent equations which can be calculated by the traditional method.Finally, the problem of initial boundary value of fractional Boussinesq equations under subcritical condition is discussed.The persistence regularity of the equation is discussed by using the conditions of 偽, 尾 鈭，

本文編號：1771032