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Pointwise Space-Time Behavior of a Compressible Navier-Stokes-Korteweg System in Dimension Three

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Abstract

The Cauchy problem of compressible Navier-Stokes-Korteweg system in ℝ3 is considered here. Due to capillarity effect of material, we obtain the pointwise estimates of the solution in an H4-framework, which is different from the previous results for the compressible Navier-Stokes system in an H6-framework [24, 25]. Our result mainly relies on two different descriptions of the singularity in the short wave of Green’s function for dealing initial propagation and nonlinear coupling respectively. Our pointwise results demonstrate the generalized Huygens’ principle as the compressible Navier-Stokes system. As a corollary, we have an Lp estimate of the solution with p > 1, which is a generalization for p ≥ 2 in [33].

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Authors and Affiliations

  1. Department of Mathematics, Donghua University, Shanghai, 201620, China

    Xiaopan Jiang  (姜晓盼) & Zhigang Wu  (吴志刚)

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  1. Xiaopan Jiang  (姜晓盼)
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  2. Zhigang Wu  (吴志刚)
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Correspondence to Zhigang Wu  (吴志刚).

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Supported by Natural Science Foundation of China (11971100) and Natural Science Foundation of Shanghai (22ZR1402300).

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Jiang, X., Wu, Z. Pointwise Space-Time Behavior of a Compressible Navier-Stokes-Korteweg System in Dimension Three. Acta Math Sci 42, 2113–2130 (2022). https://doi.org/10.1007/s10473-022-0522-0

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  • DOI: https://doi.org/10.1007/s10473-022-0522-0

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