The Green?s Function for the Initial-Boundary Value Problem of One-Dimensional Navier-Stokes Equation

Abstract

We study an initial-boundary value problem for the one-dimensional Navier-Stokes Equation. The point-wise structure of the fundamental solution for the initial value problem is first established. The estimate within finite Mach number area is based on the long wave-short wave decomposition. The short wave part describes the propagation of the singularity while the long wave part is shown to decay exponentially. A weighted energy estimate method is applied outside the finite Mach number area. With the Green's identity, we are able to relate the Green's function for the half space problem to the full space problem. The crucial step is to calculate the Dirichlet-Neumann map that constructs the Neumann boundary data from the known Dirichlet boundary data. Here we apply and modify the method in \cite{LiuYu4}. The full structure of the boundary data is thus determined. Thus the Green's function for the initial-boundary value problem is obtained. At last, we write the representation of the solution to the nonlinear problem which is a perturbation of a constant state by Duhamel's principle. We introduce a Picard's iteration for the representation and make an ansatz assumption according to the initial data given. We then verify our ansatz to obtain the asymptotic behavior of our solution.