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Serrin-type criterion for the three-dimensional viscous compressible flows. (English) Zbl 1241.35161

The authors consider a global 3-dimensional compressible Navier-Stokes equation and give a blow-up criterion as follows. They consider the following system: \[ \begin{cases} \partial_t \rho +\mathrm{div}(\rho u)=0,\\ \partial_t (\rho u)+\mathrm{div}(\rho u \times u)-\mu\Delta u -(\mu +\lambda )\nabla (\mathrm{div}u)+\nabla P(\rho )=0. \end{cases} \] Here \(\rho (x,t)\) is the density, \(u(x,t)\) is the fluid velocity, and the pressure \(P\) is given by \(P(\rho )=a\rho^{\gamma}\). In addition, the constants \(\mu\) and \(\lambda\) are the shear viscosity and the bulk viscosity, respectively, which the authors admit to be different. It is assumed that \(\mu >0,\;\lambda + \frac{2\mu }{3}\geq 1\) by a physical reason. They consider the initial condition \((\rho ,u)|_{t=0}=(\rho=0 ,u_0 )\), together with a boundary condition \(u(x,t)\rightarrow 0,\;\rho (x,t)\rightarrow\tilde{\rho}\) when \(|x|\rightarrow \infty\), for some constant \(\tilde{\rho}\geq 0\). They prove that if \(\frac{2}{s} +\frac{3}{r}\leq 1,\;3<r\leq \infty\), and a solution belonging to a kind of Sobolev spaces has a finite life-span \(T^*\), then both \(||\mathrm{div}\;u||_{L^2 (0,T;L^{\infty})} +||\rho^{1/2}u||_{L^s (0,T;L^r )}\) and \(||\rho ||_{L^{\infty} (0,T;L^{\infty})} +||\rho^{1/2}u||_{L^s (0,T;L^r )}\) blow-up at \(T=T^*\). This is a generalization of a similar result due to J. Serrin [Arch. Ration. Mech. Anal. 9, 187–195 (1962; Zbl 0106.18302)] for an incompressible Navier-Stokes equation. Furthermore, in the case that either the shear viscosity coefficient is suitably large or there is no vacuum, they prove that Serrin’s condition on the velocity can be removed in this criterion.

MSC:

35Q35 PDEs in connection with fluid mechanics
35B65 Smoothness and regularity of solutions to PDEs
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35B44 Blow-up in context of PDEs

Citations:

Zbl 0106.18302
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