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

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