Existence of solutions for the compressible Navier-Stokes equations
The author considers the initial value problem for the compressible, isentropic Navier-Stokes equations, i.e. \[ \rho_t+\text{div}(\rho u)=0,\;u_t-\rho^{-1}\mu\Delta u-\rho^{-1}(\mu+\mu')\nabla\text{ div }u+(u\cdot\nabla)u+\rho^{-1}\nabla p(\rho)=0\text{ in }\mathbb{R}^n\times(0,T), \] \[ \rho(.,0)=\overline\rho+\widetilde\rho_0,\quad u(.,0)=u_0\quad\text{in }\mathbb{R}^n, \] where \(n\geq 2\), \(\mu>0\), \(\mu'+{2\over 3}\mu\geq 0\), \(p'(\rho)>0\) and \(\overline\rho>0\) is a constant. He proves local existence of a strong solution and -- under the condition that \(\rho\geq K>0\) -- determines which norm blows up if the strong solution does not exist globally in time.
scientific article; zbMATH DE number 940893