大初始扰动下几类可压缩Navier-Stokes型方程组定解问题的适定性及解的大时间渐进行为

发布时间：2018-01-09 14:32

本文关键词：大初始扰动下几类可压缩Navier-Stokes型方程组定解问题的适定性及解的大时间渐进行为　出处：《武汉大学》2017年博士论文　论文类型：学位论文

【摘要】：关于以可压缩Navier-Stokes方程为典型特例的带耗散项的流体力学方程组定解问题基本波(例如粘性激波、稀疏波、接触间断和边界层解等)的非线性稳定性的研究一直是近年来偏微分方程研究领域的一个热点。关于这一问题,在小初值扰动情形下的相关结果已经比较完善,但是对于大初始扰动情形的情形,相应的结论还不多见。本博士学位论文主要研究在大初始扰动下几类可压缩Navier-Stokes型的方程组定解问题的整体适定性以及其整体解大时间性态的精细刻画,所得到的结果包括在一类容许初始密度具有大的振幅(oscillations)的初始扰动下一维等摘可压缩Navier-Stokes方程内流问题弱粘性激波的非线性稳定性、大初始扰动下一维两流体可压缩Navier-Stokes-Poisson方程组外流问题边界层解的非线性稳定性以及一维可压缩Navier-Stokes-Korteweg方程组Cauchy问题大初值整体光滑解的构造等。本博士学位论文共分四章:第一章是绪论,在介绍国内外同行在相关问题中所取得的主要研究进展的基础上,我们给出了我们所拟研究的问题以及所得到的结果。在第二章中,我们研究一维等熵可压缩Navier-Stokes方程组的内流问题。对该问题,Matsumura[120]给出了其整体解大时间渐进行为的完整分类。至于这些分类的严格数学证明,在小初始扰动的情形,Matsumura和Nishihara[127]得到了边界层解以及由边界层解和稀疏波所构成的复合波的非线性稳定性;施小丁[148]证明了超音速稀疏波的非线性稳定性;至于粘性激波,黄飞敏、Matsumura和施小丁[65]证明了粘性激波以及由边界层解和粘性激波所构成的复合波的非线性稳定性。而对大的初始扰动,文[29]得到了当初始能量充分小但是密度函数具有大的振幅时边界层解的非线性稳定性并且得到了超音速稀疏波的整体非线性稳定性。因此一个很自然的问题是能否对一类大的初始扰动得到粘性激波的非线性稳定性?这是我们第二章所关心的主要问题。在第二章中,通过利用能量方法和连续性技巧,我们对一类容许初始密度具有大的振幅的初始扰动得到了其弱粘性激波的非线性渐近稳定性(详见定理2.1),整个分析的关键在于克服内流边界条件所导致的解的可能的增长。第三章主要研究两流体一维可压缩Navier-Stokes-Poisson方程组的外流问题。对该问题,文[26]研究了其边界层解、稀疏波以及由边界层解以及稀疏波所构成的复合波的非线性稳定性,文[186]进一步得到了其整体解收敛到边界层解的收敛率。值得指出的是,在文[26]中要求初始扰动在某个Sobolev空间中的范数充分小,而文[186]则进一步要求初始扰动在某个加权的Sobolev空间中的范数充分小这一更强的小性要求。在第三章中,我们得到了两流体一维可压缩Navier-Stokes-Poisson方程组的外流问题边界层解在大初始扰动条件下的非线性稳定性,并在该非线性稳定性结果的基础上,进一步得到了其外流问题的整体解收敛到边界层解的衰减估计。值得指出的是为了得到一维可压Navier-Stokes-Poisson方程组的外流问题的整体解收敛到边界层解的衰减估计,在非退化的情形,除了进一步要求初始扰动属于某个加权的Sobolev空间外,我们对初始扰动的要求与前面得到非线性稳定性结果的要求一样,但是对退化的情形,我们确实需要要求初始扰动在某个加权的Sobolev空间中的范数充分小。与一维可压缩Navier-Stokes方程的外流问题相比较,问题的关键在于如何控制由于电场项的出现而导致的一维可压Navier-Stokes-Poisson方程组的外流问题的解的可能的增长。第四章主要研究一维非等摘可压缩Navier-Stokes-Korteweg方程组的Cauchy问题大初值整体光滑解的存在性。对于该模型的大初值整体适定性理论,就我们所知,只是对等熵情形有一些结果(见[2,6,9,38,51,155]及其所引文献),至于非等熵的情形,还没有见到相关的结果。在第四章中,我们得到了一维非等熵可压缩Navier-Stokes-Korteweg方程组Cauchy问题大初值整体解的存在性。与非等熵可压缩Navier-Stokes方程一样,关键在于如何得到密度函数和温度函数的正的上下界估计,但是Korteweg项的出现导致了一些分析上的困难。
[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.

【学位授予单位】：武汉大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175

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