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On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities

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Abstract

We study the Navier-Stokes equations for compressible barotropic fluids in a bounded or unbounded domain Ω of R 3. We first prove the local existence of solutions (ρ,u) in C([0,T*]; (ρ ∞ +H 3(Ω)) × under the assumption that the data satisfies a natural compatibility condition. Then deriving the smoothing effect of the velocity u in t>0, we conclude that (ρ,u) is a classical solution in (0,T **)×Ω for some T ** ∈ (0,T *]. For these results, the initial density needs not be bounded below away from zero and may vanish in an open subset (vacuum) of Ω.

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Authors and Affiliations

  1. Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan

    Yonggeun Cho

  2. Department of Mathematics, Sogang University, Seoul, 121-742, Korea

    Hyunseok Kim

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  1. Yonggeun Cho
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  2. Hyunseok Kim
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Correspondence to Yonggeun Cho.

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Cho, Y., Kim, H. On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities. manuscripta math. 120, 91–129 (2006). https://doi.org/10.1007/s00229-006-0637-y

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  • DOI: https://doi.org/10.1007/s00229-006-0637-y

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