32 research outputs found
Blowup Criterion for the Compressible Flows with Vacuum States
We prove that the maximum norm of the deformation tensor of velocity
gradients controls the possible breakdown of smooth(strong) solutions for the
3-dimensional compressible Navier-Stokes equations, which will happen, for
example, if the initial density is compactly supported \cite{X1}. More
precisely, if a solution of the compressible Navier-Stokes equations is
initially regular and loses its regularity at some later time, then the loss of
regularity implies the growth without bound of the deformation tensor as the
critical time approaches. Our result is the same as Ponce's criterion for
3-dimensional incompressible Euler equations (\cite{po}). Moreover, our method
can be generalized to the full Compressible Navier-Stokes system which improve
the previous results. In addition, initial vacuum states are allowed in our
cases.Comment: 17 page
Stability of Transonic Shock Solutions for One-Dimensional Euler-Poisson Equations
In this paper, both structural and dynamical stabilities of steady transonic
shock solutions for one-dimensional Euler-Poission system are investigated.
First, a steady transonic shock solution with supersonic backgroumd charge is
shown to be structurally stable with respect to small perturbations of the
background charge, provided that the electric field is positive at the shock
location. Second, any steady transonic shock solution with the supersonic
background charge is proved to be dynamically and exponentially stable with
respect to small perturbation of the initial data, provided the electric field
is not too negative at the shock location. The proof of the first stability
result relies on a monotonicity argument for the shock position and the
downstream density, and a stability analysis for subsonic and supersonic
solutions. The dynamical stability of the steady transonic shock for the
Euler-Poisson equations can be transformed to the global well-posedness of a
free boundary problem for a quasilinear second order equation with nonlinear
boundary conditions. The analysis for the associated linearized problem plays
an essential role
A Blow-Up Criterion for Classical Solutions to the Compressible Navier-Stokes Equations
In this paper, we obtain a blow up criterion for classical solutions to the
3-D compressible Naiver-Stokes equations just in terms of the gradient of the
velocity, similar to the Beal-Kato-Majda criterion for the ideal incompressible
flow. In addition, initial vacuum is allowed in our case.Comment: 25 page