Geodesic
A blow-up criterion for the strong solutions to the nonhomogeneous Navier-Stokes-Korteweg equations in dimension three
Applications of Mathematics, Tome 66 (2021) no. 1, pp. 43-55.

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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 $\rho $ and velocity field $u$ satisfy $\|\nabla \rho \|_{L^{\infty }(0,T; W^{1,q})} + \| u\|_{L^s(0,T; L^r_{\omega })} \infty $ for some $q>3$ and any $(r,s)$ satisfying $2/s+3/r \le 1$, $3 $ then the strong solutions to the density-dependent Navier-Stokes-Korteweg equations can exist globally over $[0,T]$. Here $L^r_{\omega }$ denotes the weak $L^r$ space.
DOI : 10.21136/AM.2020.0228-19
Classification : 35D35, 35Q35, 76D45
Keywords: Navier-Stokes-Korteweg equations; capillary fluid; blow-up criterion; vacuum; strong solutions 
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Li, Huanyuan. A blow-up criterion for the strong solutions to the nonhomogeneous Navier-Stokes-Korteweg equations in dimension three. Applications of Mathematics, Tome 66 (2021) no. 1, pp. 43-55. doi : 10.21136/AM.2020.0228-19. http://geodesic.mathdoc.fr/articles/10.21136/AM.2020.0228-19/

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