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Local theory in critical spaces for compressible viscous and heat-conductive gases. (English) Zbl 1007.35071
Using homogeneous Littlewood-Paley decomposition, harmonic analysis and a priori estimates in Besov spaces, the author examines local existence and uniqueness of solutions for a general model of Newtonian viscous heat-conductive gas under low regularity assumptions on initial data (the velocity and temperature may be discontinuous). The solution is sought in the whole \(n\)-dimensional space, but the author claims that the results can be easily extended to the periodic case. Local well-posedness is shown to hold in Besov-type spaces which are critical with respect to a scaling of the equations, provided that the initial density is close enough to a positive constant. When initial data are more regular, the local well-posedness takes place for any initial density uniformly bounded from zero. This former result lies on new estimates for linear heat equation with non-constant diffusion coefficient. Finally, the author proves some Bernstein-like inequalities for solutions in the whole space.

MSC:
35Q30 Navier-Stokes equations
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
80A20 Heat and mass transfer, heat flow (MSC2010)
35B35 Stability in context of PDEs
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