A blow-up criterion of strong solutions to the 2D compressible Navier-Stokes equations (Q547326)

The authors prove a blowup criterion in terms of an upper bound for the density for strong solutions to two-dimensional compressible Navier-Stokes equations in a bounded domain with homogeneous Dirichlet boundary conditions for the velocity: \[ \begin{aligned} &\partial_t \rho+\mathrm{div}(\rho u)=0,\\ &\partial_t (\rho u)+\mathrm{div}(\rho u\otimes u)+\nabla P(\rho)=\mu\Delta u+(\mu+\lambda)\nabla \mathrm{div} u,\\ &(\rho,u)|_{t=0}=(\rho_0,u_0),\\ &u|_{\partial\Omega}=0, \end{aligned} \] where the initial vacuum is allowed. The blowup criterion is established by introducing a new quantity \(w=u-v\), where \(v\) is the solution to the Lamé system \[ \begin{aligned} &\mu\Delta v+(\mu+\lambda)\nabla \mathrm{div} v=\nabla p(\rho),\\ &v|_{\partial\Omega}=0, \end{aligned} \] and by adopting a logarithmic estimate for this Lamé system.