Abstract: This paper is concerned with the stability of a parallel flow of the compressible Navier-Stokes equation in a cylindrical domain. The spectrum of the linearized operator is analyzed for the purpose of the study of the nonlinear stability. It is shown that if the Reynolds and Mach numbers are sufficiently small, then the linearized semigroup is decomposed into two parts; one behaves like a solution of a one dimensional heat equation as time goes to infinity and the other one decays exponentially. Some estimates related to the spectral projections are established, which will also be useful for the study of the nonlinear problem.
Abstract: We consider the variable selection problem in multivariate linear models where the predictors are given as functions and the responses are scalars, with the help of sparse regularization. Observations corresponding to the predictors are supposed to be measured repeatedly at discrete time points, and then they are treated as smooth functional data. Parameters included in the functional multivariate linear model are estimated by the penalized least squared method with the l1/l2 type penalty. We construct a blockwise coordinate descent algorithm for deriving the estimates of the functional multivariate linear model. A tuning parameter which control the degree of the regularization is decided by information criteria. In order to investigate the effectiveness of the proposed method we apply it to the analysis of simulated data and real data.