A Global Existence Result for the Compressible Navier–Stokes Equations in the Critical Lp Framework

The present paper is dedicated to the global well-posedness issue for the barotropic compressible Navier–Stokes system in the whole space $${\mathbb{R}^d}$$ with d ≧ 2. We aim at extending the work by Danchin (Inventiones Mathematicae 141(3):579–614, 2000) to a critical framework which is not related to the energy space. For small perturbations of a stable equilibrium state in the sense of suitable Lp-type Besov norms, we establish the global existence. As a consequence, like for incompressible flows, one may exhibit a class of large highly oscillating initial velocity fields for which global existence and uniqueness holds true. In passing, we obtain new estimates for the linearized and the paralinearized systems which may be of interest for future works on compressible flows.

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