三维薛定谔方程组的线性Profile分解

Linear Profile decomposition of three-dimensional Schr dinger equations

为了研究线性薛定谔方程组解Strichartz估计的紧性缺失问题,针对H•^(1)(R^(3))×H•^(1)(R^(3))中的三维线性薛定谔方程组的有界解向量序列,使用解序列的Profile分解方法,构造为解向量子列的(1/√h_(n))U(t-t_(n))/h_(n)^(2),(x-x_(n))/h_(n)类型的Profile分解和.其中,U是线性薛定谔方程组的解向量,在Strichartz范数估计下具有一个很小的余项.首先确定伸缩变换参数序列族,利用傅里叶变换和迭代的思想确定Profile分解族.其次验证了Profile分解和的收敛性,说明了Strichartz范数下余项的收敛性.最后证明了当线性薛定谔方程组的解序列有界时,都可以分解为解向量子列和的形式.

In order to study the defect of compactness problem of the Strichartz estimates for the solution of the linear Schr dinger equations,for the bounded solution vector sequences of the three-dimensional linear Schr dinger equations in H•^(1)(R^(3))×H•^(1)(R^(3)),the profile decomposition method of the solution sequences was used to construct a(1/√h_(n))U(t-t_(n))/h_(n)^(2),(x-x_(n))/h_(n)-type Profile decomposition sum of the solution vector subsequences.Among them,U is the solution vector of the linear Schr dinger equations,with a small remainder under the estimate of the Strichartz norm.Firstly,the sequence family of stretching transformation parameters was determined,and the Profile decomposition family was determined using the ideas of Fourier transform and iteration.Secondly,the convergence of the Profile decomposition sum was verified,the convergence of the remainder under the Strichartz norm was demonstrated.Finally,it was proved that when the solution sequences of the linear Schr dinger equations are bounded,it can be decomposed into a form of the sum of the solution vector subsequences.