本文關(guān)鍵詞：大初始擾動(dòng)下幾類可壓縮Navier-Stokes型方程組定解問(wèn)題的適定性及解的大時(shí)間漸進(jìn)行為　出處：《武漢大學(xué)》2017年博士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： 可壓縮Navier-Stokes方程組 內(nèi)流問(wèn)題 兩流體可壓縮Navier-Stokes-Poisson方程組 外流問(wèn)題 可壓縮Navier-Stokes-Korteweg方程組 大初始擾動(dòng) 整體適定性 解的大時(shí)間性態(tài) 粘性激波 邊界層解

【摘要】：關(guān)于以可壓縮Navier-Stokes方程為典型特例的帶耗散項(xiàng)的流體力學(xué)方程組定解問(wèn)題基本波(例如粘性激波、稀疏波、接觸間斷和邊界層解等)的非線性穩(wěn)定性的研究一直是近年來(lái)偏微分方程研究領(lǐng)域的一個(gè)熱點(diǎn)。關(guān)于這一問(wèn)題,在小初值擾動(dòng)情形下的相關(guān)結(jié)果已經(jīng)比較完善,但是對(duì)于大初始擾動(dòng)情形的情形,相應(yīng)的結(jié)論還不多見(jiàn)。本博士學(xué)位論文主要研究在大初始擾動(dòng)下幾類可壓縮Navier-Stokes型的方程組定解問(wèn)題的整體適定性以及其整體解大時(shí)間性態(tài)的精細(xì)刻畫(huà),所得到的結(jié)果包括在一類容許初始密度具有大的振幅(oscillations)的初始擾動(dòng)下一維等摘可壓縮Navier-Stokes方程內(nèi)流問(wèn)題弱粘性激波的非線性穩(wěn)定性、大初始擾動(dòng)下一維兩流體可壓縮Navier-Stokes-Poisson方程組外流問(wèn)題邊界層解的非線性穩(wěn)定性以及一維可壓縮Navier-Stokes-Korteweg方程組Cauchy問(wèn)題大初值整體光滑解的構(gòu)造等。本博士學(xué)位論文共分四章:第一章是緒論,在介紹國(guó)內(nèi)外同行在相關(guān)問(wèn)題中所取得的主要研究進(jìn)展的基礎(chǔ)上,我們給出了我們所擬研究的問(wèn)題以及所得到的結(jié)果。在第二章中,我們研究一維等熵可壓縮Navier-Stokes方程組的內(nèi)流問(wèn)題。對(duì)該問(wèn)題,Matsumura[120]給出了其整體解大時(shí)間漸進(jìn)行為的完整分類。至于這些分類的嚴(yán)格數(shù)學(xué)證明,在小初始擾動(dòng)的情形,Matsumura和Nishihara[127]得到了邊界層解以及由邊界層解和稀疏波所構(gòu)成的復(fù)合波的非線性穩(wěn)定性;施小丁[148]證明了超音速稀疏波的非線性穩(wěn)定性;至于粘性激波,黃飛敏、Matsumura和施小丁[65]證明了粘性激波以及由邊界層解和粘性激波所構(gòu)成的復(fù)合波的非線性穩(wěn)定性。而對(duì)大的初始擾動(dòng),文[29]得到了當(dāng)初始能量充分小但是密度函數(shù)具有大的振幅時(shí)邊界層解的非線性穩(wěn)定性并且得到了超音速稀疏波的整體非線性穩(wěn)定性。因此一個(gè)很自然的問(wèn)題是能否對(duì)一類大的初始擾動(dòng)得到粘性激波的非線性穩(wěn)定性?這是我們第二章所關(guān)心的主要問(wèn)題。在第二章中,通過(guò)利用能量方法和連續(xù)性技巧,我們對(duì)一類容許初始密度具有大的振幅的初始擾動(dòng)得到了其弱粘性激波的非線性漸近穩(wěn)定性(詳見(jiàn)定理2.1),整個(gè)分析的關(guān)鍵在于克服內(nèi)流邊界條件所導(dǎo)致的解的可能的增長(zhǎng)。第三章主要研究?jī)闪黧w一維可壓縮Navier-Stokes-Poisson方程組的外流問(wèn)題。對(duì)該問(wèn)題,文[26]研究了其邊界層解、稀疏波以及由邊界層解以及稀疏波所構(gòu)成的復(fù)合波的非線性穩(wěn)定性,文[186]進(jìn)一步得到了其整體解收斂到邊界層解的收斂率。值得指出的是,在文[26]中要求初始擾動(dòng)在某個(gè)Sobolev空間中的范數(shù)充分小,而文[186]則進(jìn)一步要求初始擾動(dòng)在某個(gè)加權(quán)的Sobolev空間中的范數(shù)充分小這一更強(qiáng)的小性要求。在第三章中,我們得到了兩流體一維可壓縮Navier-Stokes-Poisson方程組的外流問(wèn)題邊界層解在大初始擾動(dòng)條件下的非線性穩(wěn)定性,并在該非線性穩(wěn)定性結(jié)果的基礎(chǔ)上,進(jìn)一步得到了其外流問(wèn)題的整體解收斂到邊界層解的衰減估計(jì)。值得指出的是為了得到一維可壓Navier-Stokes-Poisson方程組的外流問(wèn)題的整體解收斂到邊界層解的衰減估計(jì),在非退化的情形,除了進(jìn)一步要求初始擾動(dòng)屬于某個(gè)加權(quán)的Sobolev空間外,我們對(duì)初始擾動(dòng)的要求與前面得到非線性穩(wěn)定性結(jié)果的要求一樣,但是對(duì)退化的情形,我們確實(shí)需要要求初始擾動(dòng)在某個(gè)加權(quán)的Sobolev空間中的范數(shù)充分小。與一維可壓縮Navier-Stokes方程的外流問(wèn)題相比較,問(wèn)題的關(guān)鍵在于如何控制由于電場(chǎng)項(xiàng)的出現(xiàn)而導(dǎo)致的一維可壓Navier-Stokes-Poisson方程組的外流問(wèn)題的解的可能的增長(zhǎng)。第四章主要研究一維非等摘可壓縮Navier-Stokes-Korteweg方程組的Cauchy問(wèn)題大初值整體光滑解的存在性。對(duì)于該模型的大初值整體適定性理論,就我們所知,只是對(duì)等熵情形有一些結(jié)果(見(jiàn)[2,6,9,38,51,155]及其所引文獻(xiàn)),至于非等熵的情形,還沒(méi)有見(jiàn)到相關(guān)的結(jié)果。在第四章中,我們得到了一維非等熵可壓縮Navier-Stokes-Korteweg方程組Cauchy問(wèn)題大初值整體解的存在性。與非等熵可壓縮Navier-Stokes方程一樣,關(guān)鍵在于如何得到密度函數(shù)和溫度函數(shù)的正的上下界估計(jì),但是Korteweg項(xiàng)的出現(xiàn)導(dǎo)致了一些分析上的困難。
[Abstract]:On the compressible Navier-Stokes equations as basic wave with typical examples of fluid mechanics equations of dissipative term solutions (such as viscous shock, rarefaction, contact discontinuity and boundary layer solution) on the stability of nonlinear partial differential equations in recent years has been a hot research field. On this issue, in the case of small initial perturbation results has been more perfect, but for the large initial disturbance situation, the corresponding conclusion is rare. This dissertation mainly studies several kinds of disturbance type compressible Navier-Stokes equations and the global well posedness of fine description of the global solutions to large time behavior in the initial, the results included in a class of admissible initial density with large amplitude (oscillations) under one-dimensional compressible Navier-Stokes equations in abstract flow problem of weak initial disturbance Nonlinear stability of viscous shock, large initial perturbation one-dimensional two fluid equations can be constructed to compress the Navier-Stokes-Poisson outflow boundary layer solutions of nonlinear stability and compressible Navier-Stokes-Korteweg equations Cauchy large initial global smooth solution. This dissertation consists of four chapters: the first chapter is the introduction, based on introducing the main research progress the domestic and foreign counterparts have related problem in the US, given our research questions and results. In the second chapter, we study the flow problem of one-dimensional isentropic compressible Navier-Stokes equations. For this problem, Matsumura[120] gives a complete classification of the large time asymptotic behavior of global solutions. As for strict mathematical proof of the classification, in the small initial perturbation, Matsumura and Nishihara[127] were obtained by boundary layer solution And composed of boundary layer solutions and sparse wave composite wave nonlinear stability; Shi Xiaoding [148] proved that the nonlinear stability of supersonic rarefaction wave; as for viscous shock, Huang Feimin, Matsumura and [65] proved that the application of Xiaoding complex nonlinear stability posed by viscous shock and boundary layer solution and the viscous shock on. The initial perturbation, paper [29] obtained when the initial energy is sufficiently small but density function with large amplitude nonlinear stability of boundary layer solutions and get the nonlinear stability of supersonic rarefaction wave. Therefore, a natural question is whether the nonlinear perturbation stability of viscous shock for a class of large initial? This is the main the second chapter concerns us. In the second chapter, by using the energy method and the continuity of skills, we have a large vibration of a class of admissible initial density The asymptotic stability of the weakly nonlinear viscous shock wave disturbance amplitude of the initial (see Theorem 2.1), the key analysis is to overcome the current boundary conditions of solution may increase. The third chapter mainly studies the problem of outflow of two fluid compressible Navier-Stokes-Poisson equations. For this problem, this paper studies the [26] the boundary layer solution, sparse wave and by boundary layer solution and constitute the composite wave wave nonlinear stability, the [186] further obtained the overall convergence to the boundary layer convergence rate. It is worth noting that, in the [26] in the initial disturbance in a norm in Sobolev space is small enough, small the requirements of [186] further requirements of the initial disturbance in a weighted norm in Sobolev space is sufficiently small that stronger. In the third chapter, we obtained two Navier-S compressible fluid The problem of outflow boundary layer solution in large initial perturbation nonlinear stability under the condition of tokes-Poisson equations, and based on the nonlinear stability results, further the outflow of the overall solution converges to the solution of boundary layer attenuation estimates. It is worth noting that in order to get the outflow problem for compressible Navier-Stokes-Poisson equations of the whole the solution converges to the boundary layer solution decays exponentially, in the non degenerate case, in addition to further requirements of the initial disturbance belong to a weighted Sobolev space, we have in front of the requirements and the initial disturbance to the nonlinear stability results of the requirements, but in the degenerate case, we do require initial perturbation in a weighted norm in Sobolev space is sufficiently small. The problem of outflow and one-dimensional compressible Navier-Stokes equations are compared, the key issue is how to control The problem of outflow of one dimensional due to the emergence of electric pressure Navier-Stokes-Poisson equation solution may increase. Existence of global smooth solution to the fourth chapter mainly studies the one-dimensional non Abstract compressible Navier-Stokes-Korteweg equations Cauchy large initial value. For the large initial value of the model overall well posedness theory, as far as we know there are some situations, just equal entropy results (see [2,6,9,38,51155] and references), as for the case of non isentropic case, also did not see the related results. In the fourth chapter, we obtain a one-dimensional isentropic compressible Navier-Stokes-Korteweg equations Cauchy large initial value. The existence of global solutions and nonisentropic can the Navier-Stokes equations, the key is how to get the upper and lower bounds of the density function and the function of temperature, but the Korteweg has caused some of the It's difficult.

【學(xué)位授予單位】：武漢大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175

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