Global existence in critical spaces for flows of compressible viscous and heat-conductive gases
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).
scientific article; zbMATH DE number 1697676