Critical regularity issues for the compressible Navier–Stokes system in bounded domains
Raphaël Danchin, Patrick Tolksdorf
Abstract
Abstract We are concerned with the barotropic compressible Navier–Stokes system in a bounded domain of $$\mathbb {R}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> (with $$d\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> ). In a critical regularity setting , we establish local well-posedness for large data with no vacuum and global well-posedness for small perturbations of a stable constant equilibrium state. Our results rely on new maximal regularity estimates—of independent interest—for the semigroup of the Lamé operator, and of the linearized compressible Navier–Stokes equations.