Some linear parabolic system in Besov spaces
Volume 81 / 2008
Banach Center Publications 81 (2008), 567-612
MSC: Primary 35K45, 35K50; Secondary 35K40.
DOI: 10.4064/bc81-0-36
Abstract
We study the solvability in anisotropic Besov spaces $B_{p,q}^{{\sigma\over2},\sigma}(\Omega^T)$, $\sigma\in\mathbb R_+$, $p,q\in(1,\infty)$ of an initial-boundary value problem for the linear parabolic system which arises in the study of the compressible Navier-Stokes system with boundary slip conditions.
The proof of existence of a unique solution in $B_{p,q}^{{\sigma\over2}+1,\sigma+2}(\Omega^T)$ is divided into three steps:
$1^\circ$ First the existence of solutions to the problem with vanishing initial conditions is proved by applying the Paley-Littlewood decomposition and some ideas of Triebel. All considerations in this step are performed on the Fourier transform of the solution.$2^\circ$ Applying the regularizer technique the existence is proved in a~bounded domain.
$3^\circ$ The problem with nonvanishing initial data is solved by an appropriate extension of initial data.
