Existence of global strong solutions in critical spaces for barotropic viscous fluids

Boris Haspot (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

This paper is dedicated to the study of viscous compressible barotropic fluids in dimension $N\geq2$. We address the question of the global existence of strong solutions for initial data close to a constant state having critical Besov regularity. First, this article shows the recent results of \cite{CD} and \cite{CMZ} with a new proof. Our result relies on a new a priori estimate for the velocity that we derive via the intermediary of \textit{the effective velocity}, which allows us to cancel out the coupling between the density and the velocity as in \cite{H2}. Second we improve the results of \cite{CD} and \cite{CMZ} by adding as in \cite{CD} some regularity on the initial data in low frequencies. In this case we obtain global strong solutions for a class of large initial data which rely the results of D. Hoff in \cite{5H4,Hoffnouv1, Hoffnouv2} and those of \cite{CD,CMZ}. We conclude by generalizing these results for general viscosity coefficients.

Domaines: Equations aux dérivées partielles [math.AP]

Boris Haspot. Existence of global strong solutions in critical spaces for barotropic viscous fluids. Archive for Rational Mechanics and Analysis, 2011, 202 (Issue 2), pp.427-460. ⟨hal-00778831⟩