Fourier Analysis Methods for the Compressible Navier-Stokes Equations

In the last three decades, Fourier analysis methods have known a growing importance in the study of linear and nonlinear PDEs. In particular, techniques based on Littlewood-Paley decomposition and paradifferential calculus have proved to be very efficient for investigating evolutionary fluid mechanics equations in the whole space or in the torus. The present survey overviews results based on Fourier analysis and paradifferential calculus, for the compressible Navier-Stokes equations. Focus is on the initial value problem in the case where the fluid domain is ℝ d $$ \mathbb {R}^{d} $$ (or the torus 𝕋 d $$ \mathbb {T}^{d} $$ ) with d ≥ 2 and on some asymptotic properties of global small solutions.