三维轴对称不可压MHD方程组解的性质研究

发布时间：2018-06-29 14:04

本文选题：轴对称流体 + 不可压流体　；参考：《北京工业大学》2016年博士论文

【摘要】：磁流体动力学(Magnetohydrodynamics,简称MHD)是研究等离子体和磁场相互作用的物理学分支,其基本方程是由流体力学中的Navier-Stokes方程和电动力学中的Maxwell方程组成.受到前辈Thomas Y.Hou、Lei Zhen、Li Congming、Chae Dongho和Lee Jihoon等关于三维不可压轴对称Navier-Stokes方程研究工作的启发,本文利用能量模估计、Marcinkiewicz乘子定理、Fourier变换、加权Calderon-Zygmund估计、截断函数法、嵌入定理、Serrin准则等方法和重要不等式如H¨older不等式、Calderon-Zygmund不等式、Sobolev内插不等式、Poincare不等式和Young不等式等,探讨三维不可压轴对称MHD方程组解的存在性,稳定性及正则性.第一章,绪论,主要介绍了磁流体动力学模型的基本概念,研究进展,并给出了本文的主要研究内容及研究结论.第二章,三维轴对称不可压MHD模型的推导,将直角坐标系下的三维不可压MHD方程组经过柱坐标变换,转化为柱坐标系下的轴对称不可压MHD方程组,并且推导出三类相互等价的MHD方程组.提出了三类特解以及这三类特解所对应的MHD方程组.第三章考虑特解uθ=Br=Bz=0.首先,通过使速度场u的径向分量ur满足加权的Serrin-Prodi型条件来获得更高的正则性,从而得到速度场和磁场的所有分量都是正则的.其次,由于三维不可压MHD方程组具有有限能量,并且有着光滑初值条件的解在有限时间的奇异性问题仍然是个公开的问题.本文通过研究一组大的各项异性初值问题,并且根据此类初值的Lp范数来获得其解的整体有界性.揭示了由于角磁场和角旋度场的相互作用所引起的动力学增长.最后,通过引入一个Banach空间,建立关于速度场和涡旋的Calderon-Zygmund不等式,使用标准的截断函数方法.得到了若urr满足Serrin条件,则解光滑.第四章考虑特解Br=Bz=0.若速度场u的径向分量ur及其负部ur-满足比Serrin-Prodi条件更加一般的加权Serrin-Prodi条件从而使其获得了更高的正则性,则弱解为正则的.其次,本章通过研究一族各向异性小初值问题得到其解的整体有界性,还得到了此类特解的大初值问题解的整体有界性.最后,本章通过引入光滑的截断函数,利用卷积类型的奇异积分算子的加权不等式,加权H¨older不等式,Young不等式,Gagliardo-Nirenberg不等式等方法.得到了若ur满足普通的Serrin条件时,则解光滑.第五章考虑特解Bθ=0.本章首先对不可压轴对称MHD方程的速度方程和磁场方程做旋度运算,得到了一组新的旋度与流密度函数的演化方程组.然后,引入R3空间标准的磨光算子,利用能量模估计的方法,截断函数法,Serrin准则,Sovolev内插不等式,嵌入定理和分部积分等方法,得到了若旋度的角分量及流密度函数的角分量满足一定条件,则解为光滑的.第六章考虑一般解的情况,引入一组新的二维模型,通过这族二维模型可以构造出一族三维模型的精确解,并且得到了此二维模型解的整体光滑性.
[Abstract]:Magnetohydrodynamics (MHD) is a branch of physics to study the interaction between plasma and magnetic fields. The basic equation is composed of the Navier-Stokes equation in the fluid mechanics and the Maxwell equation in the electrodynamics. The predecessors are Thomas Y.Hou, Lei Zhen, Li Congming, Chae Dongho and Maxwell This paper makes use of energy mode estimation, Marcinkiewicz multiplier theorem, Fourier transformation, weighted Calderon-Zygmund estimation, truncated function method, embedding theorem, Serrin criterion and other important inequalities such as H & older inequality, Calderon-Zygmund inequality, Sobolev interpolation inequality, Poincare, and Poincare, using energy mode estimation. The existence, stability and regularity of the solution of three dimensional non compressible axisymmetric MHD equations are discussed. Chapter 1, introduction, the basic concepts and research progress of magnetohydrodynamic model are introduced, and the main research contents and research conclusions of this paper are given. The second chapter, the second chapter, the push of the three-dimensional axisymmetric and incompressible MHD model In this way, the three-dimensional incompressible MHD equations in the rectangular coordinate system are transformed into axisymmetric incompressible MHD equations under cylindrical coordinates, and three classes of equivalent MHD equations are derived. Three kinds of special solutions and the MHD equations corresponding to these three kinds of special solutions are proposed. The third chapter considers the special solution u theta =Br=Bz=0. first, and passes through the special solution u theta =Br=Bz=0.. The radial component ur of the velocity field u satisfies the weighted Serrin-Prodi type condition to obtain higher regularity, so that all the components of the velocity field and the magnetic field are regular. Secondly, because the three dimensional incompressible MHD equations have finite energy, and the singularity of the solution with the smooth initial condition is still a common problem in the finite time. In this paper, we study a group of large number of anisotropic initial values and obtain the global boundedness of the solution according to the Lp norm of such initial values. The dynamic growth caused by the interaction of angular magnetic field and angular rotation field is revealed. Finally, by introducing a Banach space, the Calderon- of the velocity field and vortex is established. Zygmund inequality, using the standard truncation function method. It is obtained that if URR satisfies the Serrin condition, the solution is smooth. The fourth chapter considers that the radial component ur of the special solution Br=Bz=0. and its negative part ur- satisfy the weighted Serrin-Prodi condition more general than the Serrin-Prodi condition so that it obtains higher regularity, then the weak solution is regular. Secondly, this chapter obtains the global boundedness of the solution of the small initial value problem of a family and obtains the global boundedness of the solution of the large initial value problem of this kind of special solution. Finally, by introducing the smooth truncation function, the weighted inequality of the weighted H 'older inequality, Young inequality, G is weighted by the weighted inequality of the singular integral operator of the convolution type. Agliardo-Nirenberg inequality and other methods. If ur satisfies the common Serrin condition, the solution is smooth. In the fifth chapter, considering the special solution B theta =0. this chapter first makes a rotation operation on the velocity equation and the magnetic field equation of the non pressure symmetric MHD equation, and obtains a set of new rotational and flow density function evolution equations. Then, the R3 space standard is introduced. By using the method of energy mode estimation, the method of the energy mode estimation, the truncated function method, the Serrin criterion, the Sovolev interpolation inequality, the embedding theorem and the partial integral method, the solution is smooth if the angular component of the curl and the angular component of the flow density function are satisfied. The sixth chapter takes into account the general solution and introduces a new set of two dimensional modes. In this model, the exact solution of a family of three-dimensional models can be constructed through this two-dimension model, and the global smoothness of the solution of the two-dimensional model is obtained.
【学位授予单位】：北京工业大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175

【相似文献】