Abstract
This paper proves a Serrin’s type blow-up criterion for the 3D density-dependent Navier-Stokes-Korteweg equations with vacuum. It is shown that if the density ϱ and velocity field u satisfy \(\Vert\nabla\varrho\Vert_{L^{\infty}(0,T;W^{1,q})}+\Vert u\Vert_{L^{s}(0,T;L_{\omega}^{r})}<\infty\) for some q > 3 and any (r, s) satisfying 2/s + 3/r ⩽ 1, 3 < r ⩽ ∞, then the strong solutions to the density-dependent Navier-Stokes-Korteweg equations can exist globally over [0, T]. Here L rω denotes the weak Lr space.
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This work was supported by the start-up grant from Zhengzhou University.
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Li, H. A Blow-Up Criterion for the Strong Solutions to the Nonhomogeneous Navier-Stokes-Korteweg Equations in Dimension Three. Appl Math 66, 43–55 (2021). https://doi.org/10.21136/AM.2020.0228-19
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DOI: https://doi.org/10.21136/AM.2020.0228-19