Global existence in critical spaces for compressible Navier-Stokes equations. (English) Zbl 0958.35100

Danchin, R.

Invent. Math. 141, No. 3, 579-614 (2000).

The author investigates global solutions for isentropic compressible fluids in \(\mathbb{R}^n, n\geq 2\) using critical spaces, i.e. spaces in which the associated norm is invariant under the transformation \((\rho, u)\mapsto (\rho(l\cdot), lu(l\cdot))\) (up to a constant independent of \(l\)). In the settings of homogeneous Besov spaces with minimal regularity \(n/2\) for the velocity \(u\) and \(n/2-1\) for the density \(\rho\), with initial data \((u_0,\rho_0)\) close to a stable equilibrium \((0, \bar\rho)\), and under the additional assumption \(\rho_0-\bar\rho\in B^{n/2-1}\) he shows existence, regularity, and uniqueness (the latter for \(n\geq 3\), for \(n=2\) one needs higher regularity).

Reviewer: Uwe Kähler (Aveiro)

MSC:

35Q30	Navier-Stokes equations
76N10	Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q35	PDEs in connection with fluid mechanics

Keywords:

global strong solutions; isentropic compressible fluids; Besov spaces