發(fā)布時(shí)間：2018-06-29 14:04

  本文選題：軸對(duì)稱(chēng)流體 + 不可壓流體�。� 參考：《北京工業(yè)大學(xué)》2016年博士論文

【摘要】：磁流體動(dòng)力學(xué)(Magnetohydrodynamics,簡(jiǎn)稱(chēng)MHD)是研究等離子體和磁場(chǎng)相互作用的物理學(xué)分支,其基本方程是由流體力學(xué)中的Navier-Stokes方程和電動(dòng)力學(xué)中的Maxwell方程組成.受到前輩Thomas Y.Hou、Lei Zhen、Li Congming、Chae Dongho和Lee Jihoon等關(guān)于三維不可壓軸對(duì)稱(chēng)Navier-Stokes方程研究工作的啟發(fā),本文利用能量模估計(jì)、Marcinkiewicz乘子定理、Fourier變換、加權(quán)Calderon-Zygmund估計(jì)、截?cái)嗪瘮?shù)法、嵌入定理、Serrin準(zhǔn)則等方法和重要不等式如H¨older不等式、Calderon-Zygmund不等式、Sobolev內(nèi)插不等式、Poincare不等式和Young不等式等,探討三維不可壓軸對(duì)稱(chēng)MHD方程組解的存在性,穩(wěn)定性及正則性.第一章,緒論,主要介紹了磁流體動(dòng)力學(xué)模型的基本概念,研究進(jìn)展,并給出了本文的主要研究?jī)?nèi)容及研究結(jié)論.第二章,三維軸對(duì)稱(chēng)不可壓MHD模型的推導(dǎo),將直角坐標(biāo)系下的三維不可壓MHD方程組經(jīng)過(guò)柱坐標(biāo)變換,轉(zhuǎn)化為柱坐標(biāo)系下的軸對(duì)稱(chēng)不可壓MHD方程組,并且推導(dǎo)出三類(lèi)相互等價(jià)的MHD方程組.提出了三類(lèi)特解以及這三類(lèi)特解所對(duì)應(yīng)的MHD方程組.第三章考慮特解uθ=Br=Bz=0.首先,通過(guò)使速度場(chǎng)u的徑向分量ur滿足加權(quán)的Serrin-Prodi型條件來(lái)獲得更高的正則性,從而得到速度場(chǎng)和磁場(chǎng)的所有分量都是正則的.其次,由于三維不可壓MHD方程組具有有限能量,并且有著光滑初值條件的解在有限時(shí)間的奇異性問(wèn)題仍然是個(gè)公開(kāi)的問(wèn)題.本文通過(guò)研究一組大的各項(xiàng)異性初值問(wèn)題,并且根據(jù)此類(lèi)初值的Lp范數(shù)來(lái)獲得其解的整體有界性.揭示了由于角磁場(chǎng)和角旋度場(chǎng)的相互作用所引起的動(dòng)力學(xué)增長(zhǎng).最后,通過(guò)引入一個(gè)Banach空間,建立關(guān)于速度場(chǎng)和渦旋的Calderon-Zygmund不等式,使用標(biāo)準(zhǔn)的截?cái)嗪瘮?shù)方法.得到了若urr滿足Serrin條件,則解光滑.第四章考慮特解Br=Bz=0.若速度場(chǎng)u的徑向分量ur及其負(fù)部ur-滿足比Serrin-Prodi條件更加一般的加權(quán)Serrin-Prodi條件從而使其獲得了更高的正則性,則弱解為正則的.其次,本章通過(guò)研究一族各向異性小初值問(wèn)題得到其解的整體有界性,還得到了此類(lèi)特解的大初值問(wèn)題解的整體有界性.最后,本章通過(guò)引入光滑的截?cái)嗪瘮?shù),利用卷積類(lèi)型的奇異積分算子的加權(quán)不等式,加權(quán)H¨older不等式,Young不等式,Gagliardo-Nirenberg不等式等方法.得到了若ur滿足普通的Serrin條件時(shí),則解光滑.第五章考慮特解Bθ=0.本章首先對(duì)不可壓軸對(duì)稱(chēng)MHD方程的速度方程和磁場(chǎng)方程做旋度運(yùn)算,得到了一組新的旋度與流密度函數(shù)的演化方程組.然后,引入R3空間標(biāo)準(zhǔn)的磨光算子,利用能量模估計(jì)的方法,截?cái)嗪瘮?shù)法,Serrin準(zhǔn)則,Sovolev內(nèi)插不等式,嵌入定理和分部積分等方法,得到了若旋度的角分量及流密度函數(shù)的角分量滿足一定條件,則解為光滑的.第六章考慮一般解的情況,引入一組新的二維模型,通過(guò)這族二維模型可以構(gòu)造出一族三維模型的精確解,并且得到了此二維模型解的整體光滑性.
[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.
【學(xué)位授予單位】：北京工業(yè)大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類(lèi)號(hào)】：O175

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