Well-posedness and exponential decay for the Navier–Stokes equations of viscous compressible heat-conductive fluids with vacuum
Abstract
This paper is concerned with the Cauchy problem of Navier–Stokes equations for compressible viscous heat-conductive fluids with far-field vacuum at infinity in ℝ3. For less regular data and weaker compatibility condition than those proposed by Cho–Kim [Existence results for viscous polytropic fluids with vacuum, J. Differ. Equ. 228 (2006) 377–411], we first prove the existence of local-in-time solutions belonging to a larger class of functions in which the uniqueness can be shown to hold. The local solution is in fact a classical one away from the initial time, provided the initial density is more regular. We also establish the global well-posedness of classical solutions with large oscillations and vacuum in the case when the initial total energy is suitably small. The exponential decay estimates of the global solutions are obtained.
Communicated by Zhouping Xin
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