非齐次边值条件下高维Navier-Stokes方程解的存在性

发布时间：2018-04-16 16:33

本文选题：Navier-Stokes方程 + 非齐次边值　；参考：《哈尔滨工业大学》2017年硕士论文

【摘要】：Navier-Stokes方程作为流体力学的基本方程之一,具有悠久的历史。它描述了粘性流体的运动,在流体力学的各个领域有普遍应用。平静的水流,湍急的小溪,飞机周围的气流,盘旋的飓风,以及爆破产生的冲击波等现象都可以通过Navier-Stokes方程来分析。本文主要考虑带有对称结构的高维多连通区域上,满足非齐次Direchlet边值条件的Navier-Stokes方程解的存在性。Navier-Stokes方程作为七个“千禧问题”之一,受到很多关注,也得到了一些成果。对Navier-Stokes方程的研究已经从各个角度全面展开。其中,不论是对高维空间中Navier-Stokes方程的研究,还是在多连通区域上对Navier-Stokes方程的研究,都取得了很多重要的结果。一方面,数学家们一度认为,对5维稳态Navier-Stokes方程的研究,有可能为最终解决3维发展的Navier-Stokes方程带来思路。另一方面,2015年Korobkov,Pileckas和Russo证明了在一般的2维多连通区域上,满足非齐次Direchlet边值条件的Navier-Stokes方程解的存在性。他们的结果获得了极大关注,发表在Annals of Math上。因此,在高维多连通区域上,讨论解的存在性,是一个很有意思的数学问题。如果边值函数在多连通区域的每一个连通分支上的流量都为零,那么该边值函数可以延拓为某函数的旋度。对于仅满足相容性条件的边值函数,上述延拓不能实现。本文将构造Virtual Drain函数,使其集中了边值函数在边界每一个连通分支上的全部流量,并且其支集可以限制在对称平面的某个小邻域内。原边值函数减去Virtual Drain函数后,便可以光滑地延拓为某函数的旋度。最后将问题转化为证明一个Leray不等式,从而可以证明方程的所有可能的解一致有界,再通过Leray-Schauder度理论,即可证明Navier-Stokes方程弱解的存在性。
[Abstract]:As one of the basic equations of fluid mechanics, Navier-Stokes equation has a long history.It describes the motion of viscous fluid and is widely used in various fields of fluid mechanics.Phenomena such as calm currents, fast-flowing streams, airflow around aircraft, circling hurricanes, and blast shock waves can all be analysed by Navier-Stokes equation.In this paper, as one of the seven millennium problems, the existence of solutions of the Navier-Stokes equation satisfying the nonhomogeneous Direchlet boundary value condition on the high dimensional multiconnected domain with symmetric structure is considered. As one of the seven "millennial problems", many attentions have been paid and some results have been obtained.The study of Navier-Stokes equation has been carried out from all angles.Among them, many important results have been obtained in both the study of Navier-Stokes equation in high-dimensional space and the study of Navier-Stokes equation in multi-connected domain.On the one hand, mathematicians once thought that the study of the 5-dimensional steady-state Navier-Stokes equation might bring some ideas for the solution of the Navier-Stokes equation.On the other hand, in 2015, Korobkovan Pileckas and Russo proved the existence of solutions to the Navier-Stokes equation satisfying the inhomogeneous Direchlet boundary value condition in a general 2-dimensional multiconnected domain.Their results received great attention and were published in the Annals of Math.Therefore, it is an interesting mathematical problem to discuss the existence of solutions in high dimensional multiconnected domains.If the flow of the boundary value function on each connected branch of the multi-connected region is zero, the boundary value function can be extended to the curl of a function.For the boundary value function which only satisfies the compatibility condition, the above extension can not be realized.In this paper, the Virtual Drain function is constructed so that it concentrates all the traffic of the boundary value function on each connected branch of the boundary, and its branch set can be restricted to a small neighborhood of the symmetric plane.When the original boundary value function subtracts the Virtual Drain function, it can be smoothly extended to the curl of a certain function.Finally, the problem is transformed into proving a Leray inequality, so that all possible solutions of the equation can be proved uniformly bounded, and the existence of weak solutions of the Navier-Stokes equation can be proved by the theory of Leray-Schauder degree.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.8

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