多维Landau-Lifshitz方程的δ型黏性解、Blow up解和整体解
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摘要
关于多维Landau-Lifshitz方程，1986年周毓麟、郭柏灵就不具Gilbert项情形证明了它的整体弱解的存在性。1999年Chang Naiheng、Jalal Shatak和Uhlenbeck考虑了它的2-维柱对称情形的初值问题，在小初始能量条件下，他们证明了它存在一个整体光滑解。最近丁时进、郭柏灵又证明了它的弱解的部分正则性。它是否存在整体光滑解仍然是一个悬而未决的重要的公开问题。
为了探索多维(n≥2)Landau-Lifshitz方程组的整体光滑解的存在性，我们在这篇论文中研究下列四个课题：
第一、我们证明具多向效应场和Dirichlet边界条件的Landau-Lifshitz方程静态解的存在性，并建立Landau-Lifshitz方程解的稳定性。
第二、为了研究多维Landau-Lifshitz方程的解的存在性和极限行为，我们引入称之为δ-黏性解等的新概念，给出一些相关性质。作为应用，我们利用这些性质证明取值于三维单位球面的n维Landau-Lifshitz方程存在光滑解，我们还证明存在两个不相交的开子集使得这个光滑解在这两个集合之一内任一紧子集上趋于(0.1,0) 、在另一集合之内任一紧子集上趋于(0,-1,0)，这个光滑解在这两个集合的界面的一些点趋于(0,0,1)。
第三、给出具一定初边值条件的二维Landau-Lifshitz方程的一些精确的取值于单位球面的整体光滑解，同时给出一簇初值，这簇初值使得Landau-Lifshitz方程有一簇精确的取值于单位球面的整体光滑解，这簇整体光滑解构成一个连续统。因此本课题和上一课题意味着我们部分地回答了Landau-Lifshitz方程整体光滑解的存在性问题。
第四、给出具一定初边值条件的多维Landau-Lifshitz方程的一些精确的Blow up解。
For generalized Landau-Lifhsitz equations in multidimensions,Zhou Yulin and Guo Doling showed the global existence of weak solution in the case without Gilbert term in 1986. Nai-Heng Chang,J. Shatah,K. Uhlenbeck considered the initial value problem for the 2-dimensional cylindrical symmetric case In 1999,they proved that there exists one global smooth solution under the energy small initial condition. Recently,Ding Shijin and Guo Doling obtain the partial regularity of the weak solutions,whether it has global smooth solution is still an important open problem.
To investigate the global existence of the smooth solution for Landau-Lifshitz equation in multidimensions (n > 2),we study the following four topics in this thesis:
First,we prove the existence of the solutions of the static Landau-Lifshitz equation with multi-direct effective field and with Dirichlet boundary condition,and establish the stability of the solution of Landau-Lifshitz equation.
Second,we introduce some new concepts such as J-viscosity solutions etc and provide some relative properties in order to study the existence and limiting behavior of the solution of the multidimensional Landau-Lifshitz equations. By virtue of these results we prove that there exists a smooth solution of the multidimensional Landau-Lifshitz equation with values in unit sphere. We show also that there are two disjoint open subsets such that the solution tends to (0,1,0) and (0,-1.0) on their arbitrary inner compact sets respectively,and to (0,0,1) somewhere in the interface which separates the two open subsets.
Third,we present some exact global solutions with values in unit sphere for 2-dimensional Landau-Lifshitz equations with certain initial-boundary conditions,
Next,we give a tuft of initial data which generate a group of exact global smooth solutions with values in the unit sphere. These solutions,as we shall prove later,form a continuum. This topic combined with the second one imply that we partly reply the open problem about the existence of global smooth solution for multidimensional Landau-Lifshitz equations with initial-boundary conditions.
Fourth,We present some exact blow up solutions for n-dimensional Landau-Lifshitz equations with certain initial-boundary conditions.
为了探索多维(n≥2)Landau-Lifshitz方程组的整体光滑解的存在性，我们在这篇论文中研究下列四个课题：
第一、我们证明具多向效应场和Dirichlet边界条件的Landau-Lifshitz方程静态解的存在性，并建立Landau-Lifshitz方程解的稳定性。
第二、为了研究多维Landau-Lifshitz方程的解的存在性和极限行为，我们引入称之为δ-黏性解等的新概念，给出一些相关性质。作为应用，我们利用这些性质证明取值于三维单位球面的n维Landau-Lifshitz方程存在光滑解，我们还证明存在两个不相交的开子集使得这个光滑解在这两个集合之一内任一紧子集上趋于(0.1,0) 、在另一集合之内任一紧子集上趋于(0,-1,0)，这个光滑解在这两个集合的界面的一些点趋于(0,0,1)。
第三、给出具一定初边值条件的二维Landau-Lifshitz方程的一些精确的取值于单位球面的整体光滑解，同时给出一簇初值，这簇初值使得Landau-Lifshitz方程有一簇精确的取值于单位球面的整体光滑解，这簇整体光滑解构成一个连续统。因此本课题和上一课题意味着我们部分地回答了Landau-Lifshitz方程整体光滑解的存在性问题。
第四、给出具一定初边值条件的多维Landau-Lifshitz方程的一些精确的Blow up解。
For generalized Landau-Lifhsitz equations in multidimensions,Zhou Yulin and Guo Doling showed the global existence of weak solution in the case without Gilbert term in 1986. Nai-Heng Chang,J. Shatah,K. Uhlenbeck considered the initial value problem for the 2-dimensional cylindrical symmetric case In 1999,they proved that there exists one global smooth solution under the energy small initial condition. Recently,Ding Shijin and Guo Doling obtain the partial regularity of the weak solutions,whether it has global smooth solution is still an important open problem.
To investigate the global existence of the smooth solution for Landau-Lifshitz equation in multidimensions (n > 2),we study the following four topics in this thesis:
First,we prove the existence of the solutions of the static Landau-Lifshitz equation with multi-direct effective field and with Dirichlet boundary condition,and establish the stability of the solution of Landau-Lifshitz equation.
Second,we introduce some new concepts such as J-viscosity solutions etc and provide some relative properties in order to study the existence and limiting behavior of the solution of the multidimensional Landau-Lifshitz equations. By virtue of these results we prove that there exists a smooth solution of the multidimensional Landau-Lifshitz equation with values in unit sphere. We show also that there are two disjoint open subsets such that the solution tends to (0,1,0) and (0,-1.0) on their arbitrary inner compact sets respectively,and to (0,0,1) somewhere in the interface which separates the two open subsets.
Third,we present some exact global solutions with values in unit sphere for 2-dimensional Landau-Lifshitz equations with certain initial-boundary conditions,
Next,we give a tuft of initial data which generate a group of exact global smooth solutions with values in the unit sphere. These solutions,as we shall prove later,form a continuum. This topic combined with the second one imply that we partly reply the open problem about the existence of global smooth solution for multidimensional Landau-Lifshitz equations with initial-boundary conditions.
Fourth,We present some exact blow up solutions for n-dimensional Landau-Lifshitz equations with certain initial-boundary conditions.
引文
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