POINTWISE SPACE-TIME BEHAVIOR OF A COMPRESSIBLE NAVIER-STOKES-KORTEWEG SYSTEM IN DIMENSION THREE

doi:10.1007/s10473-022-0522-0

Abstract

Abstract: The Cauchy problem of compressible Navier-Stokes-Korteweg system in $\mathbb{R}^3$ 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 L^p estimate of the solution with p > 1, which is a generalization for p ≥ 2 in [33].

Key words: Navier-Stokes-Korteweg system, Green’s function, Large time behavior

CLC Number:

35B40

Xiaopan JIANG, Zhigang WU. POINTWISE SPACE-TIME BEHAVIOR OF A COMPRESSIBLE NAVIER-STOKES-KORTEWEG SYSTEM IN DIMENSION THREE[J].Acta mathematica scientia,Series B, 2022, 42(5): 2113-2130.

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References

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