Global existence in critical spaces for flows of compressible viscous and heat-conductive gases
type: Item
instance of: scholarly article
title: Global existence in critical spaces for flows of compressible viscous and heat-conductive gases
zbMATH Open document ID: 1018.76037
author: Raphaël Danchin
published in: Archive for Rational Mechanics and Analysis
publication date: 2003-09-11T00:00:00Z
review text: The author studies the following system modelling viscous heat-conductive gases: \[ \partial_t \rho + {div} (\rho u) = 0, \] \[ \partial_t (\rho u) + {div} (\rho u \otimes u) - {div} \tau + \nabla P = \rho f, \] \[ \partial_t (\rho (\frac{|u|^2}{2} + e)) + {div} (u(\rho(\frac{|u|^2}{2} + e)+ P)) = {div} (\tau \cdot u - q) + \rho f\cdot u, \] \[ \tau = \lambda {div} u Id + 2 \mu Du, \quad q = -k \nabla {\mathcal T}. \] The pressure law is \[ P(\rho,{\mathcal T}) = \pi_0(\rho) + {\mathcal T} \pi_1 (\rho), \] with \(\partial_\rho P(\overline \rho,\overline {\mathcal T}) > 0\) and~\( P(\overline \rho,\overline {\mathcal T}) \partial_\tau P(\overline \rho,\overline {\mathcal T}) >0 \) while \(\overline \rho,\overline {\mathcal T}\) is a constant stable equilibrium (with zero velocity). The aim is to prove global existence, for small enough data and force, in scaling invariant spaces of Besov type. Those spaces are of course critical for the equations. In order to do so, the author introduces hybrid Besov spaces whose regularity depends on whether the frequency is high or low. Littlewood-Paley theory is at the core of the analysis. An iterative method in time-average versions of the above spaces is followed, where again low and high frequencies are treated differently (in order to use both decay and smoothing effects of the equations).
reviewed by: Isabelle Gallagher
Mathematics Subject Classification ID: 76N10
Mathematics Subject Classification ID: 35Q35
zbMATH DE Number: 1697676
zbMATH Keywords: critical spaces
zbMATH Keywords: viscous heat-conductive gases
zbMATH Keywords: global existence
zbMATH Keywords: scaling invariant spaces
zbMATH Keywords: hybrid Besov spaces
zbMATH Keywords: regularity
zbMATH Keywords: Littlewood-Paley theory
zbMATH Keywords: iterative method
MaRDI profile type: Publication
full work available at URL: s002050100155
OpenAlex ID: W2383986130
DOI: 10.1007/S002050100155
description: scientific article; zbMATH DE number 1697676