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Keywords:
Navier-Stokes-Korteweg equations; capillary fluid; blow-up criterion; vacuum; strong solutions
Summary:
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 $2/s+3/r \le 1$, $3 <r \le \infty ,$ 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.
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