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

This paper is dedicated to the study of viscous compressible barotropic fluids in dimension N ≥ 2. 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 [7] and [11] with a new proof. Our result relies on a new a priori estimate for the velocity that we derive via the intermediary of the effective velocity, which allows us to cancel out the coupling between the density and the velocity as in [21]. Second we improve the results of [7] and [11] by adding as in [7] 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 [25, 28, 29] and those of [7, 11]. We conclude by generalizing these results for general viscosity coefficients.