Exponential Stability of Spherically Symmetric Solutions to Nonlinear Non-autonomous Compressible Navier-Stokes Equations

2015 ◽  
pp. 175-197
Author(s):  
Yuming Qin ◽  
Xin Liu ◽  
Taige Wang
Keyword(s):  
Exponential Stability ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Spherically Symmetric ◽  
Navier Stokes Equations ◽  
Spherically Symmetric Solutions

VACUUM STATE FOR SPHERICALLY SYMMETRIC SOLUTIONS OF THE COMPRESSIBLE NAVIER–STOKES EQUATIONS

2006 ◽  
Vol 03 (03) ◽  
pp. 403-442 ◽  
Author(s):  
ZHOUPING XIN ◽  
HONGJUN YUAN
Keyword(s):  
Weak Solutions ◽  
Stokes System ◽  
Stokes Equations ◽  
Vacuum State ◽  
Navier Stokes ◽  
Spherically Symmetric ◽  
Sufficient Condition ◽  
Navier Stokes Equations ◽  
Vacuum States ◽  
Spherically Symmetric Solutions

We study the properties of vacuum states in weak solutions to the compressible Navier–Stokes system with spherical symmetry. It is shown that vacuum states cannot develop later on in time in a region far away from the center of symmetry, provided there is no vacuum state initially and two initially non-interacting vacuum regions never meet each other in the future. Furthermore, a sufficient condition on the regularity of the velocity excluding the formation of vacuum states is given.

A note on the solution of the Navier-Stokes equations for a spherically symmetric expansion into a very low pressure

1973 ◽  
Vol 59 (2) ◽  
pp. 391-396 ◽  
Author(s):  
N. C. Freeman ◽  
S. Kumar
Keyword(s):  
Shock Wave ◽  
Reynolds Number ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Low Pressure ◽  
Navier Stokes ◽  
Spherically Symmetric ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Type Flow

It is shown that, for a spherically symmetric expansion of a gas into a low pressure, the shock wave with area change region discussed earlier (Freeman & Kumar 1972) can be further divided into two parts. For the Navier–Stokes equation, these are a region in which the asymptotic zero-pressure behaviour predicted by Ladyzhenskii is achieved followed further downstream by a transition to subsonic-type flow. The distance of this final region downstream is of order (pressure)−2/3 × (Reynolds number)−1/3.

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