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 \(\mathbb R^3\) with smooth boundary or the whole space \(\mathbb R^3\). 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 \( 3 < r \leq \infty\). As immediate applications, we obtain a regularity theorem and a global existence theorem for strong solutions.


35Q30 Navier-Stokes equations
35B40 Asymptotic behavior of solutions to PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
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