For the three-dimensional full compressible Navier–Stokes system describing the motion of a viscous, compressible, heat-conductive, and Newtonian polytropic fluid, we establish the global existence and uniqueness of classical solutions with smooth initial data which are of small energy but possibly large oscillations where the initial density is allowed to vanish. Moreover, for the initial data, which may be discontinuous and contain vacuum states, we also obtain the global existence of weak solutions. These results generalize previous ones on classical and weak solutions for initial density being strictly away from a vacuum, and are the first for global classical and weak solutions which may have large oscillations and can contain vacuum states.
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X.-D. Huang is partially supported by the National Center for Mathematics and Interdisciplinary Sciences, CAS, and by President Fund of Academy of Mathematics Systems Science, CAS, No. 2014-cjrwlzx-hxd and NNSFC Grant Nos. 11471321, 11371064, 11671412, and 11688101; J. Li is partially supported by the National Center for Mathematics and Interdisciplinary Sciences, CAS, and NNSFC Grant Nos. 11371348, 11688101, and 11525106.
Communicated by P.-L. Lions
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Huang, X., Li, J. Global Classical and Weak Solutions to the Three-Dimensional Full Compressible Navier–Stokes System with Vacuum and Large Oscillations. Arch Rational Mech Anal 227, 995–1059 (2018). https://doi.org/10.1007/s00205-017-1188-y