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A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations. (English) Zbl 1141.35432
Summary: Let $(\rho, u)$ be a strong or smooth solution of the nonhomogeneous incompressible Navier-Stokes equations in $(0, T^*) \times \Omega$, where $T^*$ is a finite positive time and $\Omega$ is a bounded domain in $\Bbb R^3$ with smooth boundary or the whole space $\Bbb R^3$. We show that if $(\rho, u)$ blows up at $T^* $, then $ \int_0^{T^*}|u (t)|_{L_w^r (\Omega)}^s \, dt = \infty$ for any $(r,s)$ with $\frac 2s +\frac 3r =1$ and $ 3 < r \le \infty$. As immediate applications, we obtain a regularity theorem and a global existence theorem for strong solutions.

35Q30Stokes and Navier-Stokes equations
35B40Asymptotic behavior of solutions of PDE
76D03Existence, uniqueness, and regularity theory
76D05Navier-Stokes equations (fluid dynamics)
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