Blow-up criterion for the 3D viscous polytropic fluids with degenerate viscosities

Abstract

In this paper, the Cauchy problem of the 3D compressible Navier-Stokes equations with degenerate viscosities and far field vacuum is considered. We prove that the L ∞ norm of the deformation tensor D(u) (u: the velocity of fluids) and the L 6 norm of ∇ log ρ (ρ: the mass density) control the possible blow-up of regular solutions. This conclusion means that if a solution with far field vacuum to the Cauchy problem of the compressible Navier-Stokes equations with degenerate viscosities is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of D(u) or ∇ log ρ as the critical time approaches; equivalently, if both D(u) and ∇ log ρ remain bounded, a regular solution persists.

Non-global solution for visco-elastic dynamical system with nonlinear source term in control problem

“…However, we cannot expect the same for the nonlinear dynamical systems see [23], [22] and [21] as examples. Hence a lot of interest has been paid to the finite-time blow-up phenomena [26], [11], [4], not only for the parabolic type model [26], [21], [22], but also for the hyperbolic type model, and also the Schrödinger model [25]. Furthermore, combing the results from both sides of global existence and finite-time blow-up, we are also interested in the so-called sharp conditions [21], [23], [22].…”