In this paper,we prove some results concerning blow-up of viscous compressible reactive (selfgravitating) flows when the initial density is compactly supported and the other initial value satisfy proper conditions.It extends the work of Xin and Cho to the case of viscous compressible reactive self-gravitating flows equations.We control the lower bound of second moment by total energy and obtain the precise relationship between the size of the support of initial density and the existence time.
We investigate the nonlinear instability of a smooth steady density profile solution to the threedimensional nonhomogeneous incompressible Navier-Stokes equations in the presence of a uniform gravitational field,including a Rayleigh-Taylor steady-state solution with heavier density with increasing height(referred to the Rayleigh-Taylor instability).We first analyze the equations obtained from linearization around the steady density profile solution.Then we construct solutions to the linearized problem that grow in time in the Sobolev space H k,thus leading to a global instability result for the linearized problem.With the help of the constructed unstable solutions and an existence theorem of classical solutions to the original nonlinear equations,we can then demonstrate the instability of the nonlinear problem in some sense.Our analysis shows that the third component of the velocity already induces the instability,which is different from the previous known results.
