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Zbl 1141.35432
Kim, Hyunseok
A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations.
(English)
[J] SIAM J. Math. Anal. 37, No. 5, 1417-1434 (2006). ISSN 0036-1410; ISSN 1095-7154/e

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.
MSC 2000:
*35Q30 Stokes and Navier-Stokes equations
35B40 Asymptotic behavior of solutions of PDE
76D03 Existence, uniqueness, and regularity theory
76D05 Navier-Stokes equations (fluid dynamics)

Keywords: blow-up criterion; nonhomogeneous incompressible Navier-Stokes equations

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