两类特殊方程组的唯一性问题

发布时间：2018-03-05 06:32

本文选题：Naiver-Stokes方程　切入点：连续唯一性　出处：《华中科技大学》2015年硕士论文　论文类型：学位论文

【摘要】：唯一性问题一直是偏微分方程研究中的重要问题.本文主要研究了两类方程组的唯一性问题.第一部分研究Navier-Stokes方程组的解在Rn×[0,L]上的连续唯一性问题；第二部分则对由Maxwell方程衍化而来的Schrodinger方程组在有界区域DN映射下的系数唯一性做了论证.对Navier-Stokes方程解的数学理论研究一直是数学界和物理界的热点问题.本文为简化方程,特别地引入Navier-Stokes方程解的旋度ρ,通过变换将原方程约化为关于旋度ρ的方程.再对简化后的方程选择适当的权函数,在Rn×[0,L]上对ρ进行带权函数的估计.在估计过程中,利用了反向唯一性方法克服了区域带来的估计困难,最终得到ρ的加权函数估计,进而得到ρ在Rn×[0,L]上的连续唯一性,再结合已知引理最终得到Navier-Stokes方程的解在Rn×[0,L]上的连续唯一性.接下来讨论Schrodinger方程组在有界区域DN映射下的系数唯一性问题,即为逆边值问题.逆边值问题来源于电阻抗成像技术的逆问题,将电阻抗成像技术的逆问题转化为对Schrodinger方程的逆边值问题的研究时,对边界的选取是此问题的重点.本文首先将二阶Schrodinger方程组转化为四阶方程,利用位势有界的四阶Schrodinger方程的Carleman估计和复几何光学解的构造,得到关于位势q的积分恒等式.然后选取合适的实解析振荡函数,应用微局部分析的理论,最终得到其系数的唯一性.
[Abstract]:The uniqueness problem has always been an important problem in the study of partial differential equations. In this paper, the uniqueness of two kinds of equations is studied. The first part is the continuous uniqueness of the solutions of Navier-Stokes equations on rn 脳 [0L]. In the second part, the uniqueness of the coefficients of the Schrodinger equations derived from the Maxwell equation is proved under the bounded domain DN mapping. The mathematical theory of the solution of the Navier-Stokes equation has always been a hot issue in the field of mathematics and physics. In particular, the curl 蟻 of the solution of the Navier-Stokes equation is introduced, and the original equation is reduced to the equation of the curl 蟻 by transformation. Then the proper weight function is chosen for the simplified equation, and the weighted function is estimated on the rn 脳 [0L]. In this paper, the inverse uniqueness method is used to overcome the difficulty of estimating the region, and finally the weighted function estimation of 蟻 is obtained, and then the continuous uniqueness of 蟻 on R n 脳 [0L] is obtained. Finally, combining the known Lemma, we obtain the continuous uniqueness of the solution of the Navier-Stokes equation on rn 脳 [0L]. Then we discuss the uniqueness of the coefficients of the Schrodinger equations under the bounded domain DN mapping. The inverse boundary value problem originates from the inverse problem of the electrical impedance imaging technology. The inverse problem of the electrical impedance imaging technique is transformed into the inverse boundary value problem of the Schrodinger equation. In this paper, the second order Schrodinger equations are first transformed into fourth-order equations, and the Carleman estimates of the potential bounded fourth-order Schrodinger equations and the construction of complex geometric optical solutions are used. The integral identity about potential Q is obtained, and then the proper real analytical oscillatory function is selected, and the uniqueness of its coefficient is obtained by applying the theory of differential local analysis.
【学位授予单位】：华中科技大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175.2

【相似文献】